Handbook of Quantum Logic and Quantum Structures

HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM STRUCTURES

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HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM STRUCTURES

Edited by

KURT ENGESSER Universitat Konstanz, Konstanz, Germany

DOV M. GABBAY King's College London, Strand, London, UK

DANIEL LEHMANN The Hebrew University of Jerusalem, Jerusalem, Israel

ELSEVIER Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier Radanveg 29, PO Box 2 11, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2007 Copyright 02007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected] you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions,and selecting Obtainingpermission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made

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FOREWORD

More than a century ago Hilbert posed his unsolved (and now famous), 23 problems of mathematics. When browsing through the Internet recently I found that Hilbert termed his Sixth Problem non-mathematical. How could Hilbert call a problem of mathematics non-mathematical? And what does this problem say? In 1900, Hilbert, inspired by Euclid's axiomatic system of geometry, formulated his Sixth Problem as follows: To find a few physical axioms that, similar to the axioms of geometry, can describe a theory for a class of physical events that i s as large as possible. The twenties and thirties of the last century were truly exciting times. On the one hand there emerged the new physics which we call quantum physics today. On the other hand, in 1933, N. A. Kolmogorov presented a new axiomatic system which provided a solid basis for modern probability theory. These milestones marked the entrance into a new epoch in that quantum mechanics and modern probability theory opened new gates, not just for science, but for human thinking in general. Heisenberg's Uncertainty Principle showed, however, that the micro world is governed by a new kind of probability laws which differ from the Kolmogorovian ones. This was a great challenge to mathematicians as well as to physicists and logicians. One of the responses to this situation was the, now famous, 1936 paper by Garret Birkhoff and John von Neumann entitled "The logic of quantum mechanics", in which they suggested a new logical model which was based on the Hilbert space formalism of quantum mechanics and which we, today, call a quantum logic. G. Mackey asked the question whether every state on the lattice of projections of a Hilbert space could be described by a density operator; and his young student A. Gleason gave a positive answer to this question. Although this was not part of Gleason's special field of interest, his theorem, now known as Gleason's theorem, had a profound impact and is rightfully considered one of the most important results about quantum logics and structures. Gleason's proof was non-trivial. When John Bell became familiar with it, he said he would leave this field of research unless there would be a simpler proof of Gleason's theorem. Fortunately, Bell did find a relatively simple proof of the partial result that there exists no two-valued measure on a three-dimensional Hilbert space. An elementary proof of Gleason's theorem was presented by R. Cooke, M. Keane and W. Moran in 1985. In the eighties and nineties it was the American school that greatly enriched the theory of quantum structures. For me personally Varadarajan's paper and subsequently his book were the primary sources of inspiration for my work together with

vi

Foreword

Gleason's theorem. The theory of quantum logics and quantum structures inspired many mathematicians, physicists, logicians, experts on information theory as well as philosophers of science. I am proud that in my small country, Czechoslovakia and now Slovakia, research on quantum structures is a thriving field of scientific activity. The achievements characteristic of the eighties and nineties are the fuzzy a p proaches which provided a new way of looking at quantum structures. A whole hierarchy of quantum structures emerged, and many surprising connections with other branches of mathematics and other sciences were discovered. Today we can relate phenomena first observed in quantum mechanics to other branches of science such as complex computer systems and investigations on the functioning of the human brain, etc. In the early nineties, a new organisation called International Quantum Structures Association (IQSA) was founded. IQSA gathers experts on quantum logic and quantum structures from all over the world under its umbrella. It organises regular biannual meetings: Castiglioncello 1992, Prague 1994, Berlin 1996, Liptovsky Mikulas 1998, Cesenatico 2001, Vienna 2002, Denver 2004, Malta 2006. In spring 2005, Dov Gabbay, Kurt Engesser, Daniel Lehmann and Jane Spurr had an excellent idea -to ask experts on quantum logic and quantum structures to write long chapters for the Handbook of Quantum Logic and Quantum Structures. It was a gigantic task to collect and coordinate these contributions by leading experts from all over the world. We are grateful to all four for preparing this monumental opus and to Elsevier for publishing it. When browsing through this Handbook, in my mind I am wandering back to Hilbert's Sixth Problem. I am happy that this problem is in fact not a genuinely mathematical one which, once it is solved, brings things t o a close. Rather it has led to a new development of scientific thought which deeply enriched mathematics, the understanding of the foundations of quantum mechanics, logic and the philosophy of science. The present Handbook is a testimony to this fact. Those who bear witness to it are Dov, Kurt, Daniel, Jane and the numerous authors. Thanks to everybody who helped bring it into existence. Anatolij DvureEenskij, President of IQSA July 2006

vii

EDITORIAL PREFACE

There is a wide spread slogan saying that Quantum Mechanics is the most successful physical theory ever. And, in fact, there is hardly a physicist who does not agree with this. However, there is also the reverse. Not only is Quantum Mechanics unprecedently successful but it also raises fundamental problems which are equally unprecedented not only in the history of physics but in the history of science in general. The most profound problems that Quantum Mechanics raises are conceptual in nature. What is the proper interpretation of Quantum Mechanics? This is a question touching on most fundamental issues, and it is, at this stage, safe to say that there is no answer to this question yet on which physicists and philosophers of science could agree. It is, moreover, no exaggeration to say that the problem of the conceptual understanding of Quantum Mechanics constitutes one of the great intellectual puzzles of our time. The topic of the present Handbook is, though related to this gigantic issue, more modest in nature. It can, briefly, be described as follows. Quantum Mechanics owes is tremendous success to a mathematical formalism. It is the mathematical and logical investigation of the various aspects of this formalism that constitutes the topic of the present Handbook. This formalism, the core of which is the mathematical structure of a Hilbert space, received its final elegant shape in John von Neumann's classic 1932 book "Mathematische Grundlagen der Quantenmechanik" . In 1936, John von Neumann published, jointly with the Harvard mathematician Garret Birkhoff, a paper entitled "The logic of quantum mechanics". In the Introduction the authors say: "The object of the present paper is to discover what logical structure we may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic". The idea of the paper, which was as ingenious as it was revolutionary, was that the Hiibert space formalism of Quantum Mechanics displayed a logical structure that could prove useful to the understanding of Quantum Mechanics. Birkhoff and von Neumann were the first to put forward the idea that there is a link between logic and (the formalism of) Quantum Mechanics, and their now famous paper marked the birth of a field of research which has become known as Quantum Logic. The Birkhoff-von Neumann paper triggered, after some time of dormancy, a rapid development of quantum logical research. Various schools of thought emerged. Let us, in this preface, highlight just a few of the milestones in this development.

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Editorial Preface

In hi famous essay "Is logic empirical?" Putnam put forward the view that the role of logic in Quantum Mechanics is similar to that of geometry in the theory of relativity. On this view, logic is as empirical as geometry. Putnam's revolutionary thesis triggered a discussion which was highly fruitful not only for Quantum Logic but for our views on the nature of logic in general. Another school of thought was initiated by Piron's "Axiomatique Quantique". This school, which has become known as the Geneva school, aimed at reconstructing the formalism of Quantum Mechanics from first principles. Piron's student Diederik Aerts continued this in Brussels. The achievements of the GenevaBrussels school are reflected in various chapters written by Aerts and former students of his. In Italy it was Enrico Beltrametti and Maria Luisa Dalla Chiara, just t o mention two names, who founded another highly influential school which is well represented in this Handbook. Highly sophisticated efforts resulted in linking the logic of Quantum Mechanics to mainstream logic. Just to give a flavour of this, let us mention that Nishimura studied Gentzen type systems in the context of Quantum Logic. Abramsky and Coecke in Oxford and Sernadas in Lisbon as well as others established the connection with Categorial Logic and Linear Logic, and the connection with NonMonotonic Logic was made by the editors. Prior to this, another direction of research had focused on the lattice structures relevant to Quantum Logic. Essentially, this field of research was brought to fruition in the USA by the pioneering work of Foulis and Greechie on orthomodular lattices. Moreover, we have to mention the Czech-Slovak school which was highly influential in establishing the vast field of research dealing with the various abstract Quantum Structures, which constitute the topic of a whole volume of this Handbook. Let us in this context just mention the names of Anatolij DvureEenskij and Sylvia Pulmannovi in Bratislava and Pave1 P t i k in Prague. The editors are happy and grateful to have succeeded in bringing together the most eminent scholars in the field of Quantum Logic and Quantum Structures for the sake of the present Handbook. We cordially thank all the authors for their contributions and their cooperation during the preparation of this work. Most of the authors are members of the Internatonal Quantum Structures Association (IQSA). We would like to express our deep gratitude to IQSA and in particular to its President, Professor Anatolij DvureEenskij, for cooperating closely with us during the preparation of this Handbook. The present Handbook is an impressive document of the intellectual achievements which have been reached in the study of the logical and mathematical structures arising from Quantum Mechanics. We hope that it will turn out to be a milestone on the road that will ultimately lead to the solution of one of the great intellectual puzzles of our time, namely the understanding of Quantum Mechanics.

Editorial Preface

ix

Finally, we would like to thank our Handbook and Publications Administrator Mrs Jane Spurr for her excellent and dedicated work on yet another Handbook of Logic. Thanks too, to our publishers Elsevier and especially to Arjen Sevenster and Andy Deelen, without whose efforts the book would not get onto shelves. The Editors: Kurt Engesser, Dov Gabbay and Daniel Lehmann Konstanz, London, and Jerusalem September 2006

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CONTRIBUTORS Enrico Beltrametti Dipartimento di Fisica, Universita' di Genova, via Dodecaneso 33, 16164 Genova, Italy. beltrameQge.infn.it David Buhagiar Department of Mathematics, Faculty of Science,University of Malta, MsidaMSD.06, Malta. david.buhagiarQum.edu.mt Emmanuel Chetcut i Department of Mathematics, Junior College, University of Malta, Msida MSD.05, Malta. emanuel.chetcutiQum.edu.mt Georges Chevalier UniversitC Claude Bernard, bltiment de mathkmatiques, 43 boulevard du 11novembre 1918, 69622 Villeurbanne cedex, France. chevalie(0math.univ-lyonl.fr Ferdinand Chovanec Department of Natural Sciences, Academy of Armed Forces, P.O. Box 45, SK03101 Liptovsky Mikulas, Slovakia chovanecQaoslm.sk Anatolij DvureEenskij Mathematical Institute, Slovak Academy of Sciences, Stefiinikova 49, SK-81473 Bratislava, Slovakia. [email protected] Kurt Engesser University of Konstanz, 78457 Konstanz, Germany. [email protected] David J . Foulis 1 Sutton Court, Amherst, MA 01002, USA. foulis(0math.urnass.edu

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Contributors

Dov M. Gabbay Department of Computer Science, King's College London, Strand, London WC2R 2LS, UK. dov.gabbayQkcl.ac.uk Richard J. Greechie Department of Mathematics and Statistics, Louisiana Tech University, Railroad Avenue, Ruston, LA 71272, USA. greechieQcoes.latech.edu

Stan Gudder Department of Mathematics, University of Denver, 2360 S. Gaylord St, Denver, CO 80208, USA. [email protected] Jan Hamhalter Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University, Technickd 2, 166 27 Prague 6, Czech Republic. [email protected]

John Harding Department of Mathematical Sciences, New Mexico State University, P O Box 30001, Las Cruces, NM 88003-8001, USA. jhardingQnmsu.edu FrantiSek K6pka Department of Engineering Fundamentals, Faculty of Electrical Engineering, University of Zilina, Workplace Liptovsky Mikulas, SK-031101 Liptovsky Mikulas, Slovakia kopkaQlm.utc.sk Daniel Lehmann School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. 1ehmannQcs.huji.ac.il RenC Mayet 13 Rue Saint Antoine, 69003 Lyon, France. rene.mayetQwanadoo.fr Norman D. Megill Boston Information Group, 19 Locke Lane, Lexington, MA 02420, USA. nmQalum.mit.edu

Contributors

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Mirko Navara Center for Machine Perception, Departtment of Cybernetics, Faculty of Electrical Engineering, Czech Technical University, Technicki 2, 166 27 Praha 6, Czech Republic. navara(0math.feld.cvut.cz Mladen PaviEi6 Faculty of Civil Engineering, University of Zagreb, Kariceva 26, POB 217, HR10001 Zagreb, Croatia. mpavicicQgrad.hr Pave1 P t i k Czech Technical University, Faculty of Electrical Engineering, Department of Mathematics, Technicki 2, 16627-Prague 6, Czech Republik. ptakQnath.fe1d.cvut.c~ Sylvia Pulmannovi Mathematical Institute, Slovak Academy of Sciences, Stefiianikova 49, SK-81473 Bratislava, Slovakia. pulmannhat savba.sk

.

Jaroslaw Pykacz Institute of Mathematics, University of Gdarisk, Wita Stwosca 57, 80-952 Gdarisk, Poland pykaczQnath.univ.gda.pl Isar Stubbe Departement Wiskunde en Informatica, University of Antwerp, Middelheimlaan 1, B-2020 Antwerp, Belgium. isar.stubbeQua.ac.be Bart Van Steirteghem Departamento de Matemitica, Instituto Superior TCcnico, 1049-001 Lisboa, Portugal. bvansQnath.ist.ut1.pt Mingsheng Ying State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P R China. yingmshQnai1.tsinghua.edu.cn

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CONTENTS Foreword Anatolij DvureEenskij

v

Editorial Preface Kurt Engesser, Dov Gabbay and Daniel Lehmann

vii

List of Authors

xi

New Quantum Structures Anatolij DvureEenskij

1

Quantum Structures and Fuzzy Set Theory Jaroslaw Pykacz

55

Algebraic and Measure-theoretic Properties of Classes of Subspaces of an Inner Product Space David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

75

Quantum Probability Stan Gudder Quantum Logics as Underlying Structures of Generalized Probability Theory Pave1 PtAk and Sylvia Pulmannov6 Quantum Logic and Partially Ordered Abelian Groups David J. Foulis and Richard J. Greechie Quantum Structures and Operator Algebras Jan Hamhalter Constructions of Quantum Structures Mirko Navara D-Posets Ferdinand Chovanec and FrantiSek K6pka

xvi

Contents

Wigner's Theorem and its Generalizations Georges Chevalier Propositional Systems, Hilbert Lattices and Generalized Hilbert 477 Spaces Isar Stubbe and Bart Van Steirteghem Equations and Hilbert Lattices Renk Mayet The Source of the Orthomodular Law John Harding Starting from the Convex Set of States Enrico Beltrametti Quantum Logic and Automata Theory Mingsheng Ying Quantum Logic and Quantum Computation Mladen PaviEi6 and Norman D. Megill Index

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HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

1

NEW QUANTUM STRUCTURES Anatolij DvureEenskij

1 INTRODUCTION

About a century ago D. Hilbert presented his famous program for 20th century mathematics at the Second International Mathematical Congress in Paris, 1900. There, he formulated his famous 23 open mathematical problems. This program was to be extremely influential in the development of 20th century mathematics and constituted a source of deep and fundamental ideas. Many of the important problems that occupied mathematicians and specialists in other fields in the 20th century can be traced back to Hilbert's problems, and many of the great and fundamental results we have at our disposal today were in fact stimulated by this program. One of Hilbert's most interesting ideas, which lies between mathematics and physics and is thus in the center of our considerations, is his Sixth Problem: To find a few physical axioms that, similar t o the axzoms of geometry, can describe a theory for a class of physical events that i s as large as possible. The year 2000 was acclaimed by UNESCO as the World Year of Mathematics. By that time almost all of Hilbert's problems had been solved. Therefore, the Clay Institute of Mathematics, Cambridge, Massachusetts, USA, was looking for new open problems in mathematics, and on May 24, 2000, Profs M. Atiyah and J. Tate presented, again in Paris, a new challenge for the 21st century: 7 open problems to be considered the greatest in mathematics. The solution of each of these problems will be awarded 1 000 000 USD. In the new list there are also problems in connection with quantum mechanics, Yang-Mills theory and the hypothesis on mass differences. So, again, we see that physics and in particular quantum mechanics provide a motivation for deep mathematical problems as it was with Hilbert's problems. It is safe to say that Hilbert's Sixth Problem will continue motivating research in quantum structures even at the beginning of the third millennium. We try to present some ideas inspired by Hilbert's Sixth Problem and describe some new directions followed today in research on quantum structures. Classical physics in the nineteenth and twentieth century was, from the logical point of view, governed by the paradigm introduced by G. Boole [Boole, 18541. In fact, the basic structure underlying the logic of classical physics was that of a Boolean algebra.

2

Anatolij Dvureenskij

In the thirties, the situation in both physics and mathematics was very inv, teresting. A.N. Kolmogorov published his fundamental work [ ~ o l m o ~ o r o19331 in which he, for the first time, axiomatized modern probability theory using the framework of Boolean algebras. However, although the Kolmogorovian model turned out to be very important for mathematics, it did not correctly describe the situation that arose in those days in connection with what at that time was called the new physics and what we nowadays call quantum mechanics. This is due to the Heisenberg uncertainty principle [Heisenberg, 19301, which asserts that the position x and the momentum p of an elementary particle cannot be measured simultaneously with arbitrarily prescribed accuracy. If Amp and A,x denote the inaccuracies of the measurement of the momentum p and position x respectively in a state m, then

where h = h/27r and h is Planck's constant. A similar relation holds for the energy-life time

It is interesting to note that we encounter relations similar to the uncertainty relations also in other scientific areas, for example in the social sciences, in psychology, in research on the human brain, in the study of automata. Psychologists know that, when we examine a patient through questions, the patient's psychological stress and the accuracy of his answers are, essentially, connected via (1.1). It was J. von Neumann [von Neumann, 19321 who made the first fundamental step towards a rigorous mathematical framework of the new mechanics as well as an interpretational logic for it. The core of this framework is the concept of a complex separable Hilbert space. In 1936, G. Birkhoff and J. von Neumann published their paper [~irkhoffand von Neumann, 19361 where they showed that the set of assertions about a certain quantum mechanical system has a structure different from that of a Boolean algebra, and they suggested that this structure be a projective geometry. Central to the description a physical system in quantum mechanics is the concept of measurement the description of which in turn relies on the concept of a state. A state is mathemtically represented as a probability measure on the physical system. The system L(H) of all closed subspaces of a real, complex, or quaternionic ~ , in his investigaHilbert space is of particular interest. G. Mackey [ ~ a c k e19631 tion on the mathematical foundations of quantum mechanics posed the following problem: Describe the set of all states on the quantum logic L(H) for a separable real or complex Hilbert space. A.M. Gleason [Gleeason, 19571 published the answer to Mackey7sproblem in 1957, and this result led to decisive progress in research on Hilbert space quantum mechanics. In addition, Gleason's theorem also influenced

New Quantum Structures

3

purely mathematical research and the theorem was extended to many cases and 19931. structures, for an overview on Gleason's theorem see [~vure~enskij, A turbulent development started in the sixties. There appeared many important results which can be found in numerous monographs, e.g. [Beltrametti and Cassinelli, 1981; Birkhoff, 1967, DvureEenskij, 1993, DvureEenskij and Pulmannovb, 2000; Gudder, 1979; Kalmbach, 1983; Maeda, 1980, Piron, 1976; Ptbk and PulmannovL, 1991, Varadarajan, 19681. New structures appeared such as orthomodular posets and orthoalgebras. Today all these structures, which were inspired by attempts at clarifying the mathematical foundations of quantum mechanics, are called quantum structures. In the early nineties an international association called International Quantum Structure Association (IQSA) was founded. It organizes Biennial Congresses where mathematicians, physicists, logicians, experts on quantum information, and philosophers interested in quantum structures can meet. It was in the nineties when the Slovak and Italian schools contributed an important new concept, namely that of a D-poset (difference poset), or equivalently, a weak orthoalgebra. Later the American school came up with the concept of an effect algebra. All these equivalent partial algebraic structures combine algebraic ideas with fuzzy set ideas. It turned out that this framework is suited for treating the space of effect operators of a Hilbert space as well as the important structures called MV-algebras. The concept of an MV-algebra was introduced by C.C. Chang [Chang, 19581 in 1958. These structures generalize two-valued logic, i.e. the logic having the truth values 0 and 1only, to an infinite-valued logic taking values from the whole , showed that there real interval [O,l]. Later in 1986, D. Mundici [ ~ u n d i c i19861 is a natural equivalence between MV-algebras and Abelian unital lattice-ordered groups with a strong unit u in the sense that any MV-algebra is isomorphic to an interval [O,u] of some Abelian unital lattice-ordered group with strong unit u. Today, MV-algebras constitute an important class of substructures of other quantum structures. For example, the compatible part (i.e. "classical one") of a lattice-ordered effect algebra forms an MV-subalgebra. For a nice survey of MV-algebras see [Cignoli et al., 20001. Recently, G. Georgescu and A. Iorgulescu [ ~ e o r ~ e s cand u Iogulescu, 20011 introduced the concept of a pseudo MV-algebra, which is a non-commutative generalization of the concept of an MV-algebra (now also called GMV-algebra). It is interesting that certain psychological processes may be viewed as illustrating such a non-commutative generalization of twevalued reasoning. The following experiment was performed in clinical medicine in connection with the transplantation of human organs. Two groups of people were asked the same two questions: (1) Do you agree to dedicate your organs for medical transplantation after your death? (2) Do you agree to accept organs of a donor, if neccesary? When the order of questions was changed in the second group, the number of positive answers to the first question was higher in the second group than in the first group. Today, there even exists a programming language [Baudot, 20001 based on a non-commutative logic.

Anatolij Dvurdenskij

4

Clearly, quantum mechanics provides an example. Quantum mechanical measurements are in general non-commutative; the result of an experiment may d e pend on the order of the measurements. Consider, for example, a beam of particles which are prepared in a certain state, and which are sent through a sequence of three polarizing filters Fl,Fz,F3. It is well-known that the order of the filters makes in general a difference. For example, let the filter be polarizing in planes perpendicular to the particle beam, such that FI polarizes vertically, Fz horizontally and F3 at a 45' angle. If we place the filters in the order Fl, Fz,F3,then no particles are detected, but in the order Fl,F3,F2,particles are detected; the difference is due to quantum interference. 20021 presented a represenFor pseudo MV-algebras, the author [~vure~enskij, tation via intervals [0,u], where u is a strong unit in an t-group not necessarily Abelian, thus generalizing a famous result of Mundici [Mundici, 19861 for MValgebras. In 2001, A. DvureEenskij and T. Vetterlein introduced the concept of a pseudo effect algebra and proved an analogue of the interval representation for these structures on the assumption of a special version of the Riesz decomposition property. In the present paper we concentrate on these non-commutative quantum struct ures. From the measure-theoretical point of view, the quantum structures mentioned above are important in the sense that on these structures a measure theory as well as a probability theory can be built up. This will be done below. We will see that, in the endeavour of establishing connections between different areas of mathematics we were led to beautiful mathematical constructions serving a useful unifying function. They serve as unifying tools in the study of diverse structures, such as algebraic structures, lattice ordered and partially ordered groups, probability theory, many-valued logic etc. and can be of use in the broader context of quantum mechanics, quantum computation and non-Kolmogorovian probability theory.

2 QUANTUM LOGICS, EFFECT ALGEBRAS AND D-POSETS

2.1

Orthomodular lattices and orthomodular posets

After the publication of the famous paper by Birkhoff and von Neumann [Birkhoff and von Neumann, 19361 there appeared a whole hierarchy of quantum structures. At the beginning there were orthomodular posets. We recall that an orthomodular poset (OMP, for short) is a poset L with two distinguished elements 0 5 1 and a unary operation ', called an orthocomplementation such that, for all a, b E L, we have (i) a" = a; (ii) b' 5 a' whenever a 5 b; (iii) a V a' = 1;

New Quantum Structures

(iv) a V b E L whenever a 5 b'; (v) b = a V (b A a') whenever a 5 b (orthomodular law). We write a I b whenever a 5 b', and we say that a and b are orthogonal or mutually exclusive. If an OMP is a lattice, we call it an orthomodular lattice (OML for short); if it is a a-complete lattice, we called it a quantum logic; if ai exists in L for any sequence of mutually orthogonal elements {G)of L, L is said to be an orthea-complete logic. See [~eltramettiand Cassinelli, 1981; Birkhoff, 1967; DvureEenskij, 1993; DvureEenskij and Pulmannovii, 2000; Gudder, 1979; Kalmbach, 1983; Maeda, 1980; Piron, 1976; Ptiik and Pulmannovii, 1991; Varadarajan, 19681. The most important example of a quantum logic is the quantum logic L ( H ) of a real, complex or quaternionic Hilbert space H , where L(H) consists of all closed subspaces M of H , and M I denotes its orthogonal complement; the partial order is set-theoretical inclusion. This example is of basic importance for so-called Hilbert space quantum mechanics. An equivalent structure is the system P ( H ) of all projectors on H. We say that two elements a and b of an OMP L are compatible if there exist three mutually orthogonal elements a l l bl, c in L such that a = a1Vc and b = bl Vc. We recall that an OMP L is a Boolean algebra iff all elements of L are compatible. This property shows that a system of mutually compatible elements generates a Boolean subalgebra of L. Compatibility is a very important property because the maximal sets of mutually compatible elements, called blocks, form Boolean subalgebras, and every OMP can be covered by blocks. So blocks constitute a locally classical part of an OMP, where the classical Kolmogorovian theory holds. States are, mathematically, represented as probability measures. Recall that a state on an OMP L is a mapping m : L + [0, 11such that m(1) = 1,and m(aVb) = m(a) m(b) whenever a Ib. If m is a-additive on L, i.e., m(Vi aj) = m(ai) for any sequence of mutually orthogonal elements {aj) for which Vi ai is defined in L, we call it a a-additive state. Similarly we define a completely additive state. It is well-known that every Boolean algebra admits a state, in fact plenty of states. For an OML this neesdn't be true. We have examples of stateless OMLs, [~reechie, 19711. An important construction of OMLs is due to Greechie [Greechie, 19681 using the technique pasting blocks. An overview on the latest development , on this topic see [ ~ a v a r a19761. D.J. Foulis and C.H. Randall [Foulii and Randall, 19721 introduced orthoalgebras which are more general than OMPs. We recall that an orthoalgebra is a set L with two distinguished elements O , 1 , and with a partial binary operation + : L x L t Lsuch that f o r a l l a , b , c ~ L w e h a v e

VEl

xi

+

+ + +

+

(OAi) if a b E L, then b + a E L and a b = b + a (commutativity); (OAii) i f b + c E L a n d a + ( b + c ) E L, t h e n a + b ~ L a n d( a + b ) + c ~L, and a (b c) = (a b) + c (associativity); (OAii) for any a E L there is a unique b E L such that a b is defined, and a + b = 1 (orthocomplement ation);

+

+

Anatolij DvureEenskij

6

(OAiv) if a + a is defined, then a = 0 (consistency). If the assumptions of (OAii) are satisfied, we write a + b + c for the element (a b) c = a (b c) in L. We introduce a partial order 5 on L via a 5 b iff a + c = b for some c E L, and we write a Ib 8 a + b exists in L. A state is a mapping m : L + [O,1] such that m(1) = 1 and m(a + b) = m(a) + m(b). For example, if we put a b := a V b whenever a Ib in an OMP, then we see that any OMP is an orthoalgebra.

+ +

+ +

+

2.2 Eflect algebras and D-posets In the early nineties, two former students of mine, F. KGpka and F. Chovanec [KGpka and Chovanec, 19941 introduced a new structure called a diflerence poset, D-poset for short. In these structures difference of comparable elements is a primary notion. A D-poset or a poset with difference is a system (P; I , @ ,0 , l ) consisting of a partially ordered set P with a partially defined binary operation 8 satisfying the following conditions for all a, b, c E P. (Dl) b 8 a is defined if and only if a 5 b; (D2) If a 5 b, then b 8 a 5 band b e ( b 8 a ) = a ; (D3) I f a < b < c , t h e n c 8 b 5 ~ 8 a a n d ( c 8 a ) ~ ( c 8 b ) = b 8 a .

+

+,

If P is a D-poset, then we can define a partial operation, such that a b = c iff c 8 b = a, and (P,+,0 , l ) is an effect algebra. Effect algebras as introduced by D. Foulis and M.K. Bennett [~oulisand Bennett, 19941, see also [Giuntini and Greuling, 19891, with addition as a primary notion, are structures equivalent to D-posets. We recall that an effect algebra is a non-empty set E with two distinguished elements 0, 1, and with a partial binary operation + : E x E + E such that, for all a, b, c E E, we have

+

+

+

+

(EAi) if a b E E, then b a E E and a b = b a (commutativity); (EAii) i f b + c € E and a + ( b + c ) E E , t h e n a + b ~E and ( a + b ) + c ~ E and , a (b c) = (a b) c (associativity); (EAiii) for any a E E there is a unique b E E such that a b is defined, and a b = 1 (orthocomplementation);

+ +

+ +

+

+

+

(EAiv) if 1 a is defined, then a = 0 (zereone law). Let a and b be two elements of an effect algebra E. We say that (i) a is orthogonal to b and write a I b iff a + b is defined in E; (ii) a is less than or equal to b and write a 5 b 8 there exists an element c E E such that a Ic and a c = b (in this case we also write b 2 a and c = b 8 a); (iii) b is the orthocomplement of a iff b is a (unique) element of E such that b I a and a + b = 1 and it is written as a'. It is clear that if E is an effect algebra, then (E, 5,8 , 0 , 1 ) is a D-poset. We recall that every orthoalgebra is an effect algebra, and an effect algebra is an orthoalgebra iff a I a implies a = 0.

+

New Quantum Structures

7

A basic example of effect algebras is the system E(H) of all Hermitian operators on a Hilbert space H between the zero operator and the identity. E(H) contains the system of all projectors, and E(H) is not a lattice. We recall that E(H) is a basic tool in secalled Hilbert space quantum mechanics. All algebraic structures defined above are, nowadays, called quantum structures. Another prototypical example of effect algebras is the following. Let (G, u) be an Abelian unital po-group with a strong unit u, and if

+,

+,

is endowed with the restriction of the group addition then (I'(G, u); 0, u) is an effect algebra. We say that an effect algebra E satisfies (i) the Riesz interpolation property, (RIP) for short, if, for all XI, xz, yl, yz in E, xi 5 yj for all i,j implies there exists an element z E E such that xi 5 z 5 yj for all i, j ; (ii) the Riesz decomposition property, (RDP) for short, if x 5 yl yz implies that there exist two elements X I , xz E E with XI 5 yl and xz 5 yz such that x = XI xz. We recall that (1) if E is a lattice, then E trivially satisfies (RIP); the converse is not true as we see below. (2) E has (RDP) iff, x l +xz = yl y2 implies that there 2 E such that XI = ~ 1 1 ~ 1 25, 2 = C Z ~ CZZ, exist four elements ell, cl2, C Z ~~, 2 E y1 = ell czl, and yz = clz czz, [ ~ v u r e ~ e n s kand i j Pulmannovi, 2000, Lemma 1.7.51. (3) (RDP) implies (RIP), but the converse is not true (e.g. if E = L(H), the system of all closed subspaces of a Hilbert space H , then E is a complete lattice but without (RDP)). On the other hand, every finite poset with (RIP) is a lattice. A partially ordered Abelian group (G; +, 0) is said to satisfy the Riesz decomposition property provided, given x, yl, y2 in GS such that x 5 yl yz, there exist X I , x2 in G+ such that x = x l xz and x j 5 yj for each j = 1,2. This condition is by [Goodearl, 1986, Prop 2.11 equivalent to the following two equivalent conditions:

+

+

+

+ +

+

+

+

+

(a) Given XI, 2 2 , yl, y2 in G such that xi 5 yj for all i,j , there exists z in G such that xi 5 z 5 yj for all i,j . (b) Given xl,x2, yl, y2 in G+such thatx1+x2 = yl+yz, thereexist zll,z12, z21, z22 in G+ such that xi = zil zi2 for each i and yj = z~ z2j for each j .

+

+

According to [Goo], a group G with the %esz decomposition property is said to be an interpo2ation group. It is clear that if (G,u) is a unital interpolation group, then E = I'(G,u) has (RDP) . We recall that by a universal group for an effect algebra E we mean a pair (G, 7) consisting of an additive Abelian group G and a G-valued measure 7 : E -+ G (i.e., ~ ( ab) = ~ ( a ) ~ ( b whenever ) a b is defined in E ) such that the following

+

+

+

lAn element u E G+ is said to be a strong unit for a po-group G , if given an element g E G , there is an integer n 2 1 such that -nu 5 g 5 nu.

8

Anatolij DvureEenskij

conditions hold: (i) 7(E) generates G. (ii) If H is an additive Abelian group and 4 : E -+ H is an H-valued measure, then there is a group homomorphism qS : G + H such that 4 = qS o 7. According to [~oulisand Bennett, 19941, every effect algebra possesses a universal group. Ravindran [Ravindran, 19961 ([Dvure~enskijand PulmannovB, 2000, Theorem 1.17.171) proved the following important result. THEOREM 1. Let E be an effect algebra with the Riesz decomposition property. Then there exists a unital interpolation group (G, u) with a strong unit u such that r ( G , u) i s isomorphic to El and there i s a G-valued injective measure 7 such that (G, 7) i s a universal group for E. One can even prove categorical equivalence. Let REA be the category of effect algebras with the Riesz decomposition property, i.e. whose objects are effect algebras satisfying the %esz decomposition property whose morphisms are the homomorphisms of effect algebras, and let UCGZ be the category of unital Abelian Ggroups with interpolation with the morphisms taken to be the Lhomomorphisms preserving the distinguished strong unit. The mapping r : ULGZ -+ REA defined by (G, u) cr r(G, u), and r(h) = hlr(G, u ) , is a functor. THEOREM 2. The functor r defines a categorical equivalence between the category ULGZ of unital Abelian t-groups with interpolation and the category REA. Moreover, if h i s a morphism of unital l-groups with interpolation, then h is injective if and only if r(h) i s injective, and h i s surjective if and only if r ( h ) is surjective. As a corollary we have that every effect algebra with (RDP) has at least one state (the definition of a state is the same as that for orthoalgebras), moreover, every state on r(G, u) is the restriction of a unique.state on (G, u). We recall that a state on (G, u) is any mapping s : G --+ W such that (i)s(g h) = s(g) s(h), g, h E G, (ii) s(g) 2 0 whenever g 2 0, and (iii) s(u) = 1. We recall that a poset (E; 5 ) is an antilattice if only comparable elements of E have an infimum or a supremum. It is clear that any linearly ordered poset is an antilattice and every finite effect algebra with (RIP) is a lattice. The Ravindran result can also be reformulated in the following form:

+

+

THEOREM 3. Every effect algebra E with ( R D P ) i s a subdirect product of antilattice effect algebras with (RDP), and all existing meets and joins in E are preserved in the subdirect product.

2.3 Perfect efect algebras In this subsection, we define a class of perfect effect algebras and we show that they are categorically equivalent to Abelian po-groups. Let G be a directed Abelian po-group and define the lexicographical product

New Quantum Structures

9

where Z is the group of all integers. Then the element (1,O) is a strong unit in the po-group G(Z) and E(G) := F(G(Z), (1, 011, (2.1) is an effect algebra. Every element a E E(G) is of the form either a = (1, -g) or a = (0, g), where g E G+. In addition, if G is a directed interpolation group, then G(Z) is an interpolation group, [~oodearl,1986, Cor 2.121, and E(G) satisfies the %esz interpolation property. Let a be any element of an effect algebra E and n an integer (n 2 0). We define recursively Oa:=O, l a = a , ( n + l ) a = n a + a , n L 1 ,

+

supposing that n a and n a a are defined in E . An ideal of an effect algebra E is a non-empty subset I of E such that (i) x t: E, y E I, x < y imply x E I, and (ii) ifx,y E I and x + y is defined in E, then x + y E I. An ideal I is said to be a Riesz ideal if, for x E I, a, b E E and x < a b, there exist al, bl E I such that x < a1 bl and a1 5 a and bl 5 b. For example, if E has (RDP), then any ideal of E is Riesz. Let A be a subset of an effect algebra E satisfying (RDP). Then the ideal Io(A) of E generated by A is the set

+

+

We denote by Z(E) the set of all ideals of E; then {0), E E Z(E). We recall that if E is linearly ordered, then Z(E) is linearly ordered with respect to setIzbe two ideals of E and assume a € Il \ I2 theoretical inclusion. Indeed, let 11, and b E I2\ Il. We can assume that, e.g., a b which gives a E 12, a contradiction. An element a is said to be infinitesimal if na is defined in E for any integer n 2 1, and denote by Infinit(E) the set of all infinitesimals of E. Then (i) 0 E Infinit(E), (ii) if b E E, a E Infinit(E) and b 5 a, then b E Infinit(E), and (iii) 1 $ Infinit(E). A proper ideal I of an effect algebra E is said to be mmzmal if is not a proper subset of some proper ideal of E (or equivalently, I is a value of 1). By Zorn's lemma E possesses at least one maximal ideal, and let M ( E ) be the set of all maximal ideals of E . We define the radical of E, Rad(E), by

<

It can be proved that if a A b is defined in an effect algebra E, then (a - (a A b)) A (b - (a A b)) = 0. PROPOSITION 4. Let E be an effect algebra satisfying (RDP). Then Infinit (E)

Rad(E).

Anatolij DvureEenskij

10

Proof. Suppose that a 4 R a d ( E ) . T h e n there exists a maximal ideal I o f E such that a 4 I . Hence, t h e ideal I o ( I , a ) generated b y I and a must coincide with E. Therefore, there exist elements z E I , a:, .. . ,a: E E with a: 5 a (i = 1 , . . . , k ) such that 1 = z + a: + . . . + a:. I f a E Infinit(E), t h e n k a 5 a' and one would have z + a: + . . . + a: = a + a', so that z = a (a' - (a: . . . a:)) 2 a, whence a E I , a contradiction. Hence, a 4 Infinit(E) and Infinit(E) C R a d ( E ) . rn

+

+ +

It is useful t o recall that it is possible t o have inequality i n (2.2). Inspired b y this, we say that an effect algebra E has t h e Rad-property if Infinit ( E ) = R a d ( E ) . T H E O R E M 5. Assume that E is an effect algebra with ( R D P ) of one of the following kinds: ( 1 ) E i s a lattice. ( 2 ) E i s a simple effect algebra. (3) E = E ( G ) := I'(G(Z), (1,0 ) ) , where G ( Z ) i s the lexicographical product of Z with a directed Abelian po-group G with interpolation. ( 4 ) For any x E El x 5 1 - x or 1 - x 5 x. Then E has the Rad-property. I n addition, in (3), Infinit(E) i s the unique maximal ideal of E. PROPOSITION 6. Let E be an effect algebra with ( R D P ) satisfying the Radproperty. Then R a d ( E ) i s a commutative semigroup with respect to such that

+

( i ) 0 i s a neutral element, (ii) the cancellation law holds in R a d ( E ) , (iii) if a b = 0 for a , b E R a d ( E ) , then a = b = 0 . ( i v ) R a d ( E ) n R a d l ( E ) = 012 ( v ) if a, b E R a d ' ( ~ ) then , a + b i s not defined in El ( v i ) if a E R a d ( E ) and b E ad'(^), then a 5 b.

+

+

+

Proof. Assume a, b E R a d ( E ). T h e n a = a1 c and b = bl c, where a l l bl, c E E and a1 bl c E E . T h e n a l l bl, c E R a d ( E ) . B y t h e Rad-property, we have a1 bl c E Infinit(E) = R a d ( E ) , therefore, (a1 bl c) (a1 bl c) E E which gives a b is defined in E, consequently, a b E R a d ( E ) . T h e conditions (i)-(iii) are now evident. (iv) Assume a E R a d ( E ) n R a d t ( E ) . T h e n a E R a d l ( E ) , so that a' E R a d ( E ) which gives 1 = a + a' E R a d ( E ) , a contradiction. ( v ) Assume a b E E. T h e n a 5 b' E R a d ( E ) , which implies a E R a d ( E ) , a contradiction. ( v i ) I f b E R a d 1 ( E ) ,t h e n b' E R a d ( E ) , which implies a+ b' is defined in R a d ( E ) . Hence a 5 (b')' = b.

+ + + + +

+ + + +

+

2~A f

E, then A' := {a' : a E A).

+ +

New Quantum Structures

11

We say that an effect algebra E with (RDP) and the Rad-property is perfect if, for any element a E E either a E Rad(E) or a' E Rad(E). For example, E(G), where E(G) is defined by (2.1) for a directed po-group G with interpolation is a perfect algebra. In the sequel we show that every perfect effect algebra is of this nature. Let PEA be the category of all perfect effect algebras where the objects are perfect effect algebras and the morphisms are the homomorphisms of effect algebras. Let GZ be the category of Abelian po-groups, where the objects are directed Abelian po-groups with interpolation and the morphisms are the homomorphisms of po-groups, i.e. positive homomorphisms of po-groups. We define a mapping E : GZ -+ PEA by

and if h is a morphism of Abelian po-groups, we set

where g E G+ . Then E is a functor and, moreover, it is a faithful and full functor. PROPOSITION 7. Let E be a perfect effect algebra. T h e n there i s a unique (up t o isomorphism) directed Abelian po-group G with inte~polationsuch that E E(G).

Proof. Let E be a perfect effect algebra. By Proposition 6, Rad(E) is a cancellative semigroup satisfying the conditions of Birkhoff [Fuchs, 1963, Thm 11.41 which guarantee that Rad(E) is a positive cone of a unique (up to isomorphism) po-group G. Since E has (RDP), G is a directed Abelian po-group with interpolation. Without loss of generality, we can assume that any element g E G can be written in the form g = gl - gz with gl,gz E Rad(E). The mapping y : Rad(E) -t G given by y(a) := a, a E Rad(E),

+

is injective and preserves in Rad(E). Define E(G) by (2.1). By Theorem 5, E(G) is a perfect effect algebra, and define a mapping f : E -+ E(G) by f (a) = (0, a) and f (a') = (1, -a) if a E Rad(E). Then f is an isomorphism of the effect algebras E and E(G). rn We now present the main result on categorical equivalence of the category of perfect effect algebras and the category of directed Abelian po-groups with interpolation. 2006al). E i s a categorical equivalence of the cateTHEOREM 8 ([~vure~enskij, gory GZ of directed Abelian po-groups with interpolation and the category PEA of perfect effect algebras. THEOREM 9. Let E = r ( G , u), where (G, u) i s a n Abelian unital po-group with (RDP). T h e n E is a n antilattice if and only if G is a n antilattice.

Anatolij DvureEenskij

12

As a corollary, we have the following result. 2006al). Every perfect effect algebra E is a subdiTHEOREM 10 ([~vure~enskij, rect product of antilattice perfect effect algebras.

2.4

MV-algebras and compatibility in eflect algebras

MV-algebras entered mathematics by Chang [Chang, 19581 in 1958. In the sequel we show that a subsytem of an effect algebra which corresponds t o a Boolean subsystem of an OMP, i.e. a compatible block, is just an MV-subalgebra. We recall that an MV-algebra is an algebra M := (M; @, @,* ,O,l) of type (2,2,1,0,0) such that, for all a, b,c E M, we have (MI) (M2) (M3) (M4) (M5) (M6) (M7) (M8)

a@b=b@a; (a @ b) @ c = a @ (b @ c); a@O=a; a@l=l; (a*)* = a; a @ a* = 1; O* = 1; (a* @ b)* @ b = (a @ b*)* @ a.

+

If we define a partial operation + on M in such a way that a b is defined in E iff a 5 b*, then a b := a @ b, then (M; 0 , l ) is an effect algebra with the Riesz decomposition property. MV-algebras appeared in connection with effect algebras in many natural ways: , showed that starting from an AF C*-algebra we can Mundici [ ~ u n d i c i 19861 obtain an countable MV-algebra and, conversely, any countable MV-algebra can be derived in such a way. Bennett and Foulis [Bennett and Foulis, 19951 introduced @-symmetric effect algebras which are exactly the MV-algebras, and the Boolean D-posets of Chovanec and KGpka [Chovanec and KGpka, 19921 are MV-algebras too. Suppose that (G, u) is an Abelian unital l-group, and on r ( G , u) we define

+

+,

Then (I'(G, u),.@,@,* ,0, u) is an MV-algebra. , gives that r ( G , u) is a protoThe famous theorem of Mundici [ ~ u n d i c i19861 typical example of MV-algebras. Let MY be the category of MV-algebras, whose objects are MV-algebras and morphisms are homomorphisms of MV-algebras, and let ULB be the category of Abelian unital l-groups whose objects are Abelian unital l-groups and morphisms are l-homomorphism of unital l-groups preserving distinguished strong units. THEOREM 11 ([Mundici, 19861). Every MV-algebra M is isomorphic to r ( G , u) for some Abelian unital l-group (G, u) .

New Quantum Structures

13

Moreover, r defines a categorical equivalence between the category of MV-algebras,

M V , and the category UCB. MV-algebras play a similar role with respect to effect algebras as Boolean algebras do with respect to orthomodular posets - they describe maximal sets of v~, mutually (strongly) compatible elements. Moreover, RieEanovB [ ~ i e ~ a n o20001 recently proved an important result to the effect that every lattice ordered effect algebra can be covered by MV-subalgebras. Two elements a and b of an effect algebra E are said to be (i) compatible and write a ct b if there exist three elements a l , bl, c E E such that a = al+c, b = bl +c and a1 + bl + c E E, and (ii) strongly compaible and we write a A b if there are three elements a l , bl, c E E such that a = a1 c, b = bl c, a1 A bl = 0 and a1 bl + c E E. We recall that (i) if a t% b, then a tt b, (ii) a tt b (a A b) implies b * a C a); (iii) 0 A a 1; (iv) if a 5 b, then a b, (b = (b - a) a, (b a=O+a).

+

+

+

-

+

We show that if a A b, then the corresponding elements a l , bl, c are uniquely determined in E with (RIP). If a and b are only compatible, there is possible to find more triples of a l , bl, c satisfying a = a1 c, b = bl c and a1 bl c E E.

+

+

+ +

PROPOSITION 12. Let an effectalgebra E satisfy (RIP). If a = a1 +c, b = bl +c withal Abl = O andal + b l + c E E, t h e n a A b = c , a V b = a l +bl +c.

<

<

< <

Proof. We have c a, b. If d a, b, there exists E E such that c, d do a, b. Hence 4 - c 5 a - c = al, - c 5 b - c = bl, so that 4 - c 5 a1 A bl = 0. Put u = a1 bl c. Then u 2 a, b. Assume e 2 a, b. There exists eo E E such that u , c eo a, b. Then u - eo u - a = bl and u - eo u - b = a1 so that u - eo = 0 and u = eo e. rn

+ +

> >

<

<

Let E = (0, a, b, c, d, I ) , where the addition

a b c d l '

a b c d l O 1

d c l x x a c

c l x x d x l x x x x x l x x x x x x x b c d l d a b O

<

+ is defined in the table.

Anatolij DvureEenskij

14

Then E is an effect algebra which is not a lattice and without (RIP), but all elements of E are strongly compatible and e.g. c d and cVd E E but cAd 6E aswellasa&b,aAb~EbutaVb$E. THEOREM 13. Let an effect algebra E satisfy (RIP). The following statements are equivalent. (i) a ~ b a n d a A b ~ E . (ii) a o b a n d a v b E. ~

C

(iii) a b. (v) a V b , a A b E E a n d ( a V b ) - b = a - ( a ~ b ) . (iv) a V b , a A b ~ E and ( a V b ) - a = b-(aAb). PROPOSITION 14. In an antilattice effect algebra E with (RIP) a only if a A b E E.

b if and

We say that an effect-algebra E satisfies the difference-meet property, (DMP) for short, if x 5 y, x A z E E and y Az E E imply (y -x) A z E E. For example every latticeordered effect algebra satisfies the difference-meet property. On the other hand, Example 15 gives an example of an antilattice effect algebra with (RIP) and (RDP) where the difference-meet property fails. In addition, it can be shown that there exist two elements a, b E E such that a A b, a tt b' but a #+b', and b & b'. EXAMPLE 15. Let G be the additive group W2 with the positive cone of all (x, y) such that either x = y = 0 or x > 0 and y > 0. Then u = (1,l) is a strong unit for G. The effect algebra E = r(G, u) is an antilattice having (RIP) and (RDP) but E is not a lattice. THEOREM 16 ([DvureEenskij, 2002al). Every block of an effect algebra E with (RIP) and (DMP) is an effect subalgebra of E which is an MV-algebra. Moreover, any such effect algebra is the set-theoretical union of its blocb. THEOREM 17 ([Dvure~enskij,2002al). Every effect algebra E satisfying (RIP) and (DMP) is a set-theoretical union of MV-algebras. THEOREM 18 ([~vure~enskij, 2002al). Every effect algebra E satisfying (RIP) is the set-theoretical union of blocks which are distributive sublattices of E.

2.5 Visualixation o j eflect algebras with (RDP) In the present subsection, we show that effect algebras with (RDP) and MValgebras can be represented and visualized by a set of automorphisms of an antilattice or a linearly ordered set. Let (R, 5 ) be a non-empty antilattice, and let A(R) be the set of all automorphisms a : R + R preserving the partial order 5. Then A(R) can be converted into a pegroup such that the groupaddition is the composition of automorphisms,

15

New Quantum Structures

the order on A(R) is defined via a 5 P iff (w)a 5 (w)P for all w E R, and the neutral element is the identity function on R. Holland [Holland, 19631 proved the basic result that every &group can be embedded into the &group A(R) for some linearly ordered set 51, and Glass [Glass, 1972, Thm 541 generalized this result to directed po-groups satisfying (RIP) showing that every such a pegroup can be embedded into the po-group A(R) for some antilattice $2. We show that a similar result can also be proved for effect algebras. Namely, we prove that every pseudo effect algebra E satisfying (RDP) can be represented as a set of automorphisms of an antilattice R. Such a representation allows us to present effect algebras in a visualized form which can be useful for a more precise investigation of effect algebras. THEOREM 19 ([Dvure~enskij,2002~1). Every effect algebra E with (RDP) can be represented as a n effect algebra of automorphisms of A(R) for some antilattice set R such that all finite infima and suprema existing in E are preserved. If M is an MV-algebra, then M is an effect algebra with (RDP) and then its visualization has the following form. COROLLARY 20. Every MV-algebra M can be represented as a n MV-algebra of automorphisms of A(R) for some linearly ordered set R. The case of the visualization of pseudo effect algebras will be studied below, Subsection 4.6.

2.6 Loomis-Sikorski theorem for eflect algebras The Loomis-Sikorski theorem plays an important role in the study of Boolean a-algebras, see [Sikorski, 19641. Its generalization to a-complete MV-algebras (i.e. MV-algebras which are a a-complete lattice) was proved independently by 20001 and by Mundici [ ~ u n d i c i19991. , It was shown the author [~vure~enskij, that a-complete MV-algebras are always a-homomorphic images of tribes, i.e., of a-complete MV-algebras of fuzzy sets. In [~uhagiaret aL, 20061, we generalized the Loomis-Sikorski theorem to monotone a-complete effect algebras proving that the latter are a-homomorphic images of effect-tribes which are monotone a-complete effect algebras of [0, 11-valued functions with (RDP). Of crucial help was a delicate result of Choquet [Alf, p. 491 which says that the set of extremal states is always a Baire space (not necessarily compact as in the case of MV-algebras). An effect algebra E is monotone a-complete provided that for every ascending (descending) sequence XI 5 xz 5 . . (xl 2 x2 2 . ) in E which is bounded above (below) in E has a supremum (infimum) in E.

.

.

THEOREM 21. Let (G, u ) be a n interpolation group with strong unit. (1) (G, u) is monotone a-complete if and only if E = r ( G , u) is monotone acomplete.

16

Anatolij DvureEenskij

(2) If E = r(G, u) is monotone a-complete, then G is Archimedean. LEMMA 22. Let E be a monotone a-complete effect algebra with (RDP). Let a, b E E. The following two statements are equivalent. (i) s(a) = 0 for each s E Exts(E) implies a = 0. (ii) s(a) = s(b) for each s E Exts(E) implies a = b. Given a compact convex subset K of a topological vector space, we denote by Aff(K) the collection of all affine continuous functions on K . Of course, Aff(K) is an Archimedean partially ordered group and 1 is its strong unit, which is a subgroup of C(K), the system of all continuous real-valued functions on K . Unlike C(K), the space Aff(K) need not to be a lattice. The following lemma ( [ ~ u n c eet al., 1999, Lem 71, [Wright, 1973, Cor 31) will play a crucial role in our investigation. LEMMA 23. Let {a,} be a monotone descending sequence of nonnegative finctions in Aff(K), and let a(x) = limn a,(x) for any x E K . Then A, a, = 0 in Aff(K) if and only if {x E d K : a(%) > 0) is a meager subset in dK, where d K is the set of extreme points of K. The following representation theorem for monotone a-complete interpolation 16.151. groups with strong unit is from [~oodearl,19861

or

THEOREM 24. Let (G, u) be a nonzero monotone a-complete intelpolation group with strong unit and set A = {f E Aff(S(G, u)) : f (s) E s(G) for all discrete s E Exts(G, u)}. Then the natural mapping 4 : G --+ Aff(S(G,u)) provides an isomorphism of (G, u) onto (A, 1) (as ordered groups with strong unit). REMARK 25. In Theorem 24 we have that A is monotone u-complete and, moreover, if f € Aff(S(G, u)), f 5 4(gi), gj € G, for any integer i 2 l, then there is a g E G such that f 5 4(g) 5 4(gi) for any i, see [Goodearl, 1986, Lem 16.171. Therefore, suprema and infima existing in A are the same as those taken in Aff (S(G, 21)). An effect-tribe on a set a # 0 is any system 7 C [O,l]" such that (i) 1 E 7, (ii) i f f E 7, then 1- f E 7, (iii) if f , g E 7 , f 5 1-g, then f + g € 7, and (iv) for any sequences {f,} of elements of 7 such that f, /' f (pointwisely), then f E 7. It is evident that any effect-tribe is a monotone a-complete effect algebra. We recall that given a family B of [O, 11-valued functions on R there is a minimal effect-tribe, '&(B), on 0 generated by B. The following result is a Loomis-Sikorski theorem for monotone a-complete effect algebras with (RDP). Its proof uses the previous theorems.

17

New Quantum Structures

THEOREM 26 ( [ ~ u h a ~ iet a ral., 20061). Let E be a monotone a-complete effect algebra with (RDP). Then there are a convex space 51, an effect-tribe 7 of [O,l]valued functions on 51 with (RDP) and a a-homomorphism h from 7 onto E. An analogous result for a-complete MV-algebras needs the following notion. We recall that a tribe on 51 # 0 is a collection 7 of fuzzy sets from [O,l]" such that (i) 1 E 7, (ii) i f f E 7,then 1- f E 7,and (iii) if {f,}, is sequence from 7,then min{Cz==lf,, 1) E 7 . The Loomis-Sikorski theorem for a-complete MV-algebras is as follows. 2000; Mundici, 19991). Every a-complete MVTHEOREM 27 ([~vure~enskij, algebra M is a a-homomorphic image of a tribe of fuzzy sets.

2.7 Product egect algebras We now introduce product effect algebras. Using the representation theorem for effect algebras with the Riesz decomposition property we show that they are categorically equivalent to unital perings with interpolation. A product on an effect algebra E = (E; 0,1) is any total binary operation . on E such that, for all a, b, c E E, the following holds: If a b is defined in E, thena.c+b-candc.a+c.bexistin E and

+,

+

and we say that E with a product . is a product eflect algebra, and we write E = (E; +, .,0 , l ) . An element u of a product effect algebra E is said to be a unity, ifa.u=u.a=aforany a E E . A product . on E is (i) associative if (a . b) - c = a . (b . c), a, b, c E E ; (ii) commutative if a . b = b - a, a, b E E . It is worth saying that i f . is a product on E, then (iii) a . 0 = 0 = 0 . a; (iv) i f a s b , thenfor any C E E a, . c s b . c a n d c . a I c . b .

-

+

+

Property (iii) follows easily from the following: a 0 = a . (0 0) = a . 0 a - 0 , and the cancellation property gives a . 0 = 0. Similarly, 0 . a = 0. We recall that every effect algebra E possesses at least one commutative and associative product, namely the zero product, i.e. a . b = 0 for all a, b E E . We recall that a po-ring is a ring (R; .,0) such that (i) (R; 0) is an additive 0 and a . a 0 for any c, a 0. If u is a strong Abelian po-group, and c a unit for R, i.e., for any a E R there is an integer n 1 such that a I nu, and if u - u 5 u, then the effect algebra

>

+,

>

>

>

+,

18

Anatolij DvureEenskij

is a product effect algebra with the product . which is the restriction of the ring product . to E x E; the product . is commutative or associative on r ( R , u) whenever so is . on R. For example, if (W; +, .,0) is the ring of the real numbers, then the standard interval [O,1] := r(W, 1) is a product effect algebra. We now present the two main results of this subsection the proof of which uses the basic representation theorem 2.

+,

THEOREM 28 ([DvureEenskij, 2002dl). Let (E; .,0 , l ) be a product effect algebra with the Riesz decomposition property. T h e n there exists a unique (up t o isomorphism) unital po-ring (R, u) satisfying the Riesz decomposition property with the product - and with u u 5 u such that E r ( R , u) and q5(a . b) = q5(a) . q5(b), where q5 i s a n isomorphism of E onto r ( R , u ) preserving the product .. I f . is commutative or associative, so i s . o n R. Denote by PRUDE^ the category of product effect algebras, i.e. its objects are product effect algebras and its morphisms are homomorphisms of effect algebras also preserving .. We denote by R the category of associative unital interpolation po-rings (R, u) with a distinguished strong unit u such that u . u u, and its morphisms are homomorphisms of po-groups which preserve . and the distindefined guished strong units. We denote by rRthe map from R into by ~ R ( ( R+, ; . , 0 , 5 , u ) ) := (r(R,u); -,O,u), and r ~ ( f := ) f lr(R, u ) . THEOREM 29 ([~vure~enskij, 2002dl). T h e functor defines a categorical equivalence of the category R of unital associative interpolation po-rings with a strong unit u such that u . u 5 u and the category PRODEA of product effect algebras.

<

+,

+,

PROPOSITION 30. Let P be a proper ideal of a product effect algebra (E; ., 0 , l ) with the (RDP). If al ~p a2 and bl N p b2, then a1 .bl ~p a2.b2. I n addition, E/P is a product effect algebra with the induced product . defined wia

[ a ] p - [ b ] p : = [ a . b ] ~(,a , b E E ) , and the canonical mapping f : E + EIP given by f ( a ) = [ a l p , a E E , i s a surjective homomorphism preserving all existing joins, meets and the product in E. THEOREM 31 ([~vure~enskij, 2002dl). Every product effect algebra (E; +, .,0 , l ) with the (RDP) hawing 1 as a product unity i s a subdirect product of antilattice product effect algebras with the (RDP). Moreover, we can find a n injective homomorphism into the subdirect product that preserves all existing meets, joins and the product in E. We now present some effect algebras that admit products. PROPOSITION 32. If e is unity of a product effect algebra E , then e = 1. Moreover, l i s u n i t y f o r . i f a n d o n l y i i f o r a n y a E E, a . 1 2 a a n d 1 . a L a . PROPOSITION 33. A finite effect algebra E which i s a lattice and has (RDP) admits a product . such that a . 1 = a = 1 . a for a n y a E E if and only if E

New Quantum Structures

19

i s a Boolean algebra, i.e., a V a' = 1 for any a E E . If this i s the case, then a . b = a A b E E. We recall that G+ of a po-group G is Archimedean if, for some a, b E G+ with n a 5 b for any n 2 1, a = 0. PROPOSITION 34. Let E be a perfect effect algebra and define a binary operation as follows. For a , b E Rad(E), let

-

Then . i s a commutative product o n E such that a 1= a = 1. a for any a E E . Let E = I'(Z x,l G, (1,O)) and let G+ be Archimedean. Then o n E there is a unique product . such that a - 1 = a = 1. a for any a E E; this product i s defined by (2.3). 3 PSEUDO MV-ALGEBRAS Recently, G. Georgescu and A. Iorgulescu [Georgescu and Iogulescu, 20011 introduced pseudo MV-algebras (sometimes also called GMV-algebras) which are a non-commutative generalizations of MV-algebras introduced by C.C. Chang [Chang, 19581 and which may be viewed as constituting an extension of twovalued logic. Such a non-commutative generalization of two-valued reasoning can be found e.g. in psychological processes. J. Rachiinek [~achiiek,20021 introduced non-commutative MV-algebras, which in fact are equivalent to pseudo MValgebras. Today we can observe an interest in noncommutative reasoning. On the other hand, the author's representation theorem [Dvure~enskij,20021 allows for a bridge between different areas of mathematics: the generalized Mundici functor, I?, allows us to study unital t-groups via pseudo MV-algebras, in particular finer varieties, or more precisely, their analogues. For some open problems on pseudo i j Holland, to appear]. MV-algebras, see [ ~ v u r e ~ e n s kand

3.1 Elements of pseudo MV-algebras In this subsection we present the basic notions of the theory of pseudo MValgebras. DEFINITION 35. A pseudo MV-algebra is an algebra ( M ; @ , - ," ,O,1) of type (2,1,1, 0,O) such that the following axioms hold for all x, y, z E M with an additional binary operation @ defined via

(Al) x @ ( y @ z) = (x @ y ) @ z ;

Anatolij DvureEenskij

(A2) (A3) (A4) (A5) (A6) (A7) (A8)

x@O=O@x=x; x@l=l@x=l; 1" = 0; 1- = 0; (x- €3 y-)" = (x" @ yN)- ; x @ ( x " O y ) = y @ ( y N O ~ ) = ( ~ O ~ - ) @ ~ = ( ~ ~ ~ - ) @ ~ ~ X O ( X - @ ~=) ( x @ y N ) O y ; (x-)" = x.

If we define x 5 y iff x- @ y = 1, then 5 is a partial order such that M is a distributive lattice with x V y = x @ (x" y) and x A y = x @ (x- @ y). For basic properties of pseudo MV-algebras see [Georgescu and Iogulescu, 20011. A pseudo MV-algebra M is an MV-algebra iff x @ y = y @ x for all x, y E M. For example, if u is a strong unit of a (not necessarily Abelian) 1-group G,

and

then (I'(G, u); @,- ," ,0, u) is a pseudo MV-algebra [Georgescu and Iogulescu, 20011. Let (M; @,- ," ,0,1) be a pseudo MV-algebra. Define a partial binary operation on M via: x y is defined iff x 5 y-, and in this case

+

+

+ +

It is clear that x y is defined iff y 5 xN. In addition a 5 b iff there exists c E M such that a c = b iff there exists d E M such that d a = b. Moreover, for a 5 b, we define two partial subtractions, \ and / by a / b := a"Obandb\[email protected]+a/b=b=b\a+a. We recall that the pseudo MV-algebra (I'(G, u); @,- ," ,O, u) is an archetype of pseudo MV-algebras by the following basic representation theorem for pseudo MV-algebras which was proved in [Dvure~enskij,1993, Thm 3.9, Thm 6.41 and which generalizes a famous result of Mundici [Mundici, 19861 for MV-algebras, see Theorem 11:

+

THEOREM 36 ([~vure~enskij, 20021). For any pseudo MV-algebra M, there exists a unique (up to isomorphism) unital 1-group G with a strong unit u such that M I'(G,u). The functor I' defines a categorical equivalence of the category of pseudo MV-algebras with the category of unital 1-groups. Moreover, zf h is a given

New Quantum Structures

21

isomorphism of M with r ( G , u), ( G ,h ) is the universal group for M : that is, for any partially ordered group ( K ; <,0) and any order and preserving mapping g : M 4 K there is a homomorphism of partially ordered groups g' : G -+ K such that g = g'o h.

+

+,

W e present two independent proofs of this basic statement: ( 1 ) using the above derived partial addition + and the Bosbach representation o f semiclans, and ( 2 ) using the word technique for pseudo effect algebras with (RDP2),see subsection 4.2. be derived partial addition. The following properties PROPOSITION 37. Let hold in any pseudo MV-algebra M .

+

(i) x + O = x = O + x f o r a n y x E M . (ii) x + x W = 1 = x - + x f o r a n y x ~ M . (iii) If x- y = 1, then y = x . (iv) If x y = 1, then y = x" and x = y-. ( v ) Let x y and ( x y) z be defined i n M . Then y defined i n M , and ( x + y ) z = x + ( y + z).

+ + +

+ +

+z

and x

+ ( y + z ) are

+

Similarly, if y+z and x+(y+z) are defined in M , so arex+y and (x+y)+z. (vi) y x + b = y, then b = x " O y. (vii) If x + yl = x y2, then yl = y2. (viii) If xl y = 2 2 y, then X I = xz. (ix) I f y = a + x , t h e n a = y @ x - . (x) I f x A y = 0 , thenx+y and y+x are defined i n M , a n d x + y = x V y = y+x.

+

+ +

PROPOSITION 38. (i) If a+ bl and a+b2 exist in M , then a+ (bl V bz) is defined in M , and a

+ (bl V b2) = ( a + bl) V ( a + b2).

+ a and b2 + a exist in M , then (bl V b2)+ a is defined in M , and (bl V bz) + a = (bl + a ) V (b2+ a). (iii) If a + bl and a + b2 exist i n M , then a + (bl A b2) is defined i n M , and a + (bl A b2) = ( a + bl) A ( a + b2). (iv) If bl + a and b2 + a exist i n M , then (bl A bz) + a is defined i n M , and

(ii) If bl

(bi Abz) + a = (bi + a ) A (bz + a). PROPOSITION 39. For all x , y of a pseudo MV-algebra M we have:

Anatolij DvureEenskij

22

(a) x O ( x A y ) - = x O y - . (b) X O ( X A Y ) - + Y = X O Y - + ~ = X V Y . ( c ) ( x A y)"O y = x" O y, X V y = x + ( x A y)"O y. (d) ( X V Y ) O ~ = - XO(ZAY)-=xo~-. (e) y" O ( x V y) = y" O x = ( x A y)" O x. PROPOSITION 40. In any pseudo MV-algebra M we have the following cancellation laws. ( i ) If a

+ bl I a + b

2,

then bl 5 b2.

+ b 5 a2 + b, then a1 5 az. (iii) If a + b is defined i n M , and if a1 5 a and bl 5 b, then a1 + bl is defined in (ii) If a1

M , andal+bl I a + b . In our subsequent reasoning the notion o f a semiclan introduced b y Bosbach will be crucial. W e say that ( C ;A, +) is a semiclan i f it is a A-semilattice and a partial groupoid with respect t o such that the following axioms are satisfied:

+

( C l ) I f a 5 b, then there exist x , y E C such that b = a

+ x and b = y + a.

(C3) I f a + x , a + y E C , t h e n ( a + x ) ~ ( a + y )= a + ( x ~ y ) a, n d i f x + a , y+aE C , then ( x a ) A ( y a ) = ( x A y) a.

+ + + (C4) a + b, ( a + b) + c E C i f fb + c, a + (b + c) E C , and i n this case we have ( a + b ) + c = a+ (b+c).

I t is clear that if G is an e-group, then (G+;A, +) is a semiclan, and b y Bosbach [Bosbach, 1981, p. 3211, for any semiclan ( C ;A, +) there exists an e-group G with t h e positive cone G+ such that C can be embedded into some semiclan ( G S ;A, +) preserving +, A, and all existing V in C . PROPOSITION 41. Let ( M ;@,- ," ,0,1) be a pseudo MV-algebra and let + be defined as above. Then ( M ;A, +) is a semiclan, and, moreover, there exist an f-group Go with the positive cone G$ and an injective mapping f : M -+ G$ such A, and V . that f preserves

+,

Proof. Using Propositions 3 7 4 0 , we see that ( M ;A, +) is a semiclan. Using the crucial result o f Bosbach [Bosbach, 1981, p. 3211 mentioned above, we have the ¤ assertion i n question.

New Quantum Structures

Denote by Xo := f (M) and

Then Xo is a lattice which is a sublattice of G.: What we claim below is that X is also a sublattice of G;: for x, y E X , the expression x A y E X means that x A y taken in the l-group Go is an element of X . It can be proved that (i) X is an l-semigroup of G$, and is left and right cancellative, (ii) 0 = f (0) is the neutral element, (iii) if x y = 0 for x, y E X then x = 0 = y, and (iv) x X = X x for x E X . Applying now the Birkhoff theorem, [~uchs,1963, Thm 11.41, we see that X is a positive cone of an l-group G. By the construction of X , u := f (1) is a strong unit for G, and r ( G , u ) = f (M) which gives a sketch of the proof of the following theorem, that is the first part of Theorem 36. For an alternative proof using the word technique, see Subsection 4.2.

+

+

+

THEOREM 42. Let (M; @,- ," ,0,1) be a pseudo MV-algebra. Then there exists an I-group G with strong unit u such that M and r(G, u) are isomorphic pseudo MV-algebras. Using this result, we can prove the basic representation theorem for pseudo MV-algebras which generalizes the famous theorem of Mundici, Theorem 11. As a consequence of the representation theorem, we have the following import ant results. We say that a pseudo MV-algebra M is Archimedean if the existence of nu := a1 . . . a,, where a1 = . . = a, = a for any integer n 2 1, entails that a = 0. It is known that if M is a a-complete lattice, then M is Archimedean

+ +

PROPOSITION 43. Let M = I'(G,u) be a pseudo MV-algebra. Archimedean if and only if G is an Archimedean l-group.

Then M is

THEOREM 44. Any Archimedean pseudo MV-algebra is commutative, i.e. an MV-algebra. In particular, every finite pseudo MV-algebra is commutative. We denote by Z(M), P(M), and N ( M ) the sets of all ideals, prime ideals and normal ideals of a pseudo MV-algebra M respectively. Similarly, let C(G), P(G) and L(G) be the sets of all convex l-subgroups, prime subgroups (i.e., prime convex l-subgroups) and l-ideals of an l-group G. We say that an ideal of a pseudo MV-algebra M is a subset I of M such that (i) 0 E I, (ii) if x, y E I, then x @ y E I and y @ x €1, and (iii) if x E I, y E M, and y 5 x, then Y E I . An ideal I of M is said to be (i) normal if x @ I = I @ x for any x E M,3 (ii) prime if x A y E I entails x E I or y E I , (iii) mmimal if I is a proper ideal of M and is not included in any proper ideal of M different from I. 3

~ define e x $ I := {x $ y : y E I); similarly we define I $ x.

Anatolij DvureEenskij

24

Let ( G ,u) b e a unital l-group and let M = r ( G ,u). Using the basic representation o f pseudo MV-algebras b y unital l-groups, Theorem 36, we get the following result. THEOREM 45 ( [ ~ v u r e ~ e n s k20011). ij, Let ( G ,u) be a unital l-group and let M = r ( G , u). For any ideal I of M , we assign 4 ( I ) := {x E G : 1x1 A u E I ) . ( 1 ) 4 ( I ) is a convex l-subgroup of G generated by I . The mapping 4 : I H 4 ( I ) from Z ( M ) to C ( G ) defines a one-to-one mapping preserving set-theoretical inclusion. The inverse mapping $ is given by

(3) The restriction of 4 to P ( M ) or N ( M ) gives a bijection between P ( M ) and C ( G ) , and N ( M ) and C(G),respectively, which preserves set-theoretical inclusion.

3.2 States on pseudo MV-algebras In this subsection, we introduce states and state-morphisms on pseudo MV-algebras, and we exhibit their close connections with normal and normal maximal ideals o f pseudo MV-algebras, A state o n a pseudo MV-algebra M is a mapping m : M + [O, 11 such that ( i ) m ( 1 ) = 1, and (ii) m ( a b) = m ( a ) m ( b ) whenever a b is defined in M . I f M is a commutative pseudo MV-algebra (i.e. an MV-algebra), then M possesses at least one state. For pseudo MV-algebras this is not the case. T h e basic properties o f states are the following. PROPOSITION 46. Let m be a state o n a pseudo MV-algebra M . Then, for all a , b E M , we have

+

+

+

(i) m ( 0 ) = 0. (ii) If a 5 b, then m ( a ) 5 m ( b ) and m ( b @ a-) = m ( b ) - m ( a ) = m ( a W @ b). (iii) m ( a - ) = 1 - m ( a ) = m ( a N ) . (iv) m ( a = ) = m ( a ) = m ( a c ) . ( v ) m ( a V b) m ( a A b) = m ( a ) m(b). (vi) m ( a @b) + m ( b @ a ) = m ( a ) + m ( b ) . (vii) The set Ker(m) := { a E M : m ( a ) = 0 )

+

is a normal ideal of M .

+

New Quantum Structures

25

(viii) a / K e r ( m ) = b/Ker(m) i f and only if m ( a ) = m ( a A b) = m ( b ) . (ix) There i s a unique state 6i o n M / K e r ( m ) such that %([a])= m ( a ) , [a] E M / K e r ( m ) , where [a]is the coset in M / K e r ( m ) determined by a E M . ( x ) m ( a @b) = m ( b @a). A mapping m from a pseudo MV-algebra M into the standard MV-algebra [O, 11 such that, for all a, b E M , m ( a @ b) = m ( a ) @w m ( b ) , (ii) m ( a - ) = m ( a W )= 1 - m ( a ) , and (iii)m ( 1 ) = 1 is said t o be a state-morphism. THEOREM 47. Let m be a state on a pseudo MV-algebra M . Then m is a statemorphism i f and only if Ker(m) is a maximal ideal of M . Let ml and m2 be two measure-morphisms on a pseudo MV-algebra M such that K e r ( m l ) = Ker(m2). Then ml = ma. Conversely, let I be a normal and maximal ideal of a pseudo MV-algebra M . Then there is a unique state-morphism m on M such that Ker(m) = I . PROPOSITION 48. Let m be a state on a pseudo MV-algebra M . Then the following statements are equivalent:

(i) m is an extremal state on M . (ii) m is a state-morphism o n M . (iii) m ( x A y) = m i n { m ( x ) ,m ( y ) ) , x , y E M . Denote b y S ( M ) the set o f all states o n a pseudo MV-algebra M . T h e n S ( M ) is a convex set, and let E x t s ( M ) b e the set o f all its extremal states. A net { m a ) o f states o n M converges weakly t o a state m ifF m,(a) + m ( a ) for every a E M . I f S ( M ) # 0,then S ( M ) is a compact convex Hausdorff space with respect t o the weak topology, and the space o f all state-morphisms on M is a non-empty compact Hausdorff space. In view o f the Krein-Mil'man theorem, [Goodearl, 1986], any state is a weak limit o f convex linear combinations o f the set o f extremal points o f S(M). W e recall that b y categorical equivalence there exists a one-bone correspondence between the states o n r(G,u) and the states on (G,u). As we have seen, every MV-algebra admits a state. THEOREM 49. A pseudo MV-algebra M admits a state i f and only zf M admits a maximal ideal which is normal. In particular, every linearly ordered pseudo MV-algebra M possesses a unique state, and every normal-valued pseudo MV-algebra M admits a state. T h e following example is an oral communication from Prof. A. Glass. W e recall that a pseudo MV-algebra is said t o b e symmetric i f x- = x" for any x E M . E X A M P L E 50. There is a symmetric stateless pseudo MV-algebra M . Proof. Let A u t ( R ) denote the group o f automorphisms g : W + W, and set

26

Anatolij DvureEenskij

+

Then G is an &group with strong unit u : t cr t 1 (t E R) that is central. Indeed, if g E G, and Og = s > 0, let n E Z+ be an integer such that n > s. Then for all t E B,let m E Z with m < t < m 1. Then t ~ - ( ~ + '<) 0 and Oum < t. So < Og = s < n = Oun wpch gives tg = tu-("+')gu(m+l) < oUn+m+' < tun+' while u is central. Thus tg < tun+' for all t E R so g < un+'. Similarly, if Og 5 0, then u-(~'+') < g (since g-l < un'+l for some n' E Z+), which proves u is a strong unit for G. Let to E R and Gt, = {g E G : tog = to) be the stabilizer of to. Then u 4 Gto and if H is any subgroup of G strictly containing Gto,there is h E H with toh # to. By replacing h by h-l, we may assume toh > to. Now if toh > to 1, then h L u, and if toh < to 1, there is g E Gto c H with to 112 < tohg < to 1 and tohg > (to 1)h-l. This follows from the fact that Gt, = Gt,+l and h 4 Gt,, and there is an open unit interval I R with to E I and I C supp(h) = {t E R : th # h). Since h E G, we have ( I 1) := {t 1 : t E I) supp(h). So tohgh > to 1. Hence t ~ ( h g h )>~to 2 so u < (hgh)2 E H. Thus if H is a convex subgroup, then u E H. Consequently, Gt, is a maximal convex 4-subgroup of G not containing u. If f E G with tof = to 114, then fP1Gtof = Gto+l/4 # Gto SO Gto is not normal in G. On the other hand, (G, R) is a transitive and primitive periodic e-group [Glass, 1981, Ex.1.9.2, p. 931. By [Glass, 1999, Thm 6C], G has only trivial normal convex 4-subgroup, i.e., no maximal convex ideal of G is normal. Consequently, M = r(G, u) is a symmetric stateless pseudo MV-algebra. Since every stabilizer Gt is a convex maximal &subgroup of G, Gt n M is a maximal ideal of M , and Ker(M) = (0).

+

+

+

+

+

+

+

+

+

+

+

3.3 Perfect pseudo MV-algebras Pseudo MV-algebras are intervals in unital Lgroups and are categorically equivalent to unital l-groups. We now describe a subclass of pseudo MV-algebras, the subclass of perfect pseudo MV-algebras, which is categorically equivalent to the class of all l-groups. Such pseudo MV-algebras can be "split" into two two "slices". If G is an arbitrary (noncommutative) e-group, then

gives a (noncommutative) symmetric pseudo MV-algebra. Moreover, (i) M = {(O,g) : g E G+) U {(I, -g) : g E G+), (ii) {(O,g) : g E G+) rl{(l, -9) : g E G+) = 0, (iii) {(O,g) : g E G+)* = {(I, -g) : g E G f ) and {(I, -g) : g E Gf)* = {(O,g) : g E G + l , (iv) Infinit(M) = ((0, g) : g E G+) = Rad(M), (v) if g, h E G+, (1, -g) (1, -h) is not defined in M,

+

New Quantum Structures

27

(vi) (0, g) 5 (1,-h) for all g, h EG+ (vii) M admits a unique state, namely s(0, g) = 0 and s(1, -g) = 1for all g E G+. PROPOSITION 51. Let M be a symmetric pseudo MV-algebra with Rad(M) = Infinit(M). Then Rad(M) is an associative semigroup with respect to such that

+

(i) (ii) (iii) (iv) (v) (vi)

0 is the neutral element, the cancelation law holds in Rad(M), if a + b = 0 for a , b E Rad(M), then a = b = 0, Rad(M) n Rad(M)* = 0, i f a, b E Rad(M)*, then a b is not defined in M, i f a E Rad(M) and bE Rad(M)*, then a < b.

+

PROPOSITION 52. Let M be a symmetric pseudo MV-algebra. The following statements are equivalent: (i) (ii) (iii) (iv)

Rad(M) U Rad(M)* = M. Infinit(M) is a n ideal of M and Infinit(M) U Infinit(M)* = M. M admits a state and Infinit(M) U Infinit(M)* = M. There i s a proper ideal A of M, such that A U A* = M, A n A* = 0, and a 5 b whenever a E A and b E A*. In such a case, A is a unique maximal ideal of M. (v) Infinit(M) U Infinit(M)* = M and a 5 b whenever a E Infinit(M) and b E Infinit (M) * .

If one of (i)-(v) is satisfied, then Rad(M) = Infinit(M), Infinit(M) is a unique maximal ideal of M , and M admits a unique state, s, s ( a ) = 0 zf a E Infinit(M) and s ( a ) = 1 i f a E Infinit(M)*. I n addition, (vi)-(viii) are mutually equivalent, and each of (i)-(v) implies (vi). (vi) Rad, (M) # M and Rad,(M) U Rad, (M)* = M. (vii) M admits only a two-valued state. (viii) If I is a maxzmal and normal ideal of M, then M / I = r ( Z , 1). We say a (non-degenerate)symmetric pseudo MV-algebra M is perfect if Rad(M)U Rad(M)* = M. Let PBMV be the category of perfect pseudo MV-algebras, where the objects are perfect pseudo MV-algebras and the morphisms are homomorphisms of pseudo MV-algebras. Let B be the category of 1-groups, where the objects are &groups and the morphisms are homomorphisms of &groups. We define a mapping I : B t PBMV by

Anatolij DvureEenskij

28

and if h is a morphism of [-groups, we set

where g E G+. Then & is a functor. THEOREM 53. & is a categorical equivalence of the category 6 of [-groups and the category PBMV of perfect pseudo MV-algebras. In addition, suppose that h : E(G) + &(H) is a homomorphism of pseudo MV-algebras, then there is a unique homomorphism f : G -+ H of unital [-groups such that h = &( f), and (i) if h is surjective, so is f ; (ii) if h is an isomorphism, so is f . In the sequel we show that the variety V(PBMV) of pseudo MV-algebras generated by PGMV has a single perfect pseudo MV-algebra generator. THEOREM 54. Let G be a doubly transitive 1-group. Then V(PBMV) = V(&(G)). In particular, an identity holds in every perfect pseudo MV-algebra if and only if it holds in &(G). An example of a doubly transitive permutation [-group is the system of all automorphisrns, Aut(R), of the real line R, or the next example: Let u E Aut(R) be the translation tu = t 1, t E R, and

+

Then (BAut(R), u) is a doubly transitive unital [-permutation group, and according to l lass, 1999, Lem 10.3.11, the variety of pseudo MV-algebras generated by I'(BAut (B), u) is the variety of all pseudo MV-algebras. Finally, we note that the variety V(PQMV) is defined by the equation:

3.4 n-perfect pseudo MV-algebras Inspired by perfect pseudo MV-algebras, we extend this notion to the notion of a n-perfect pseudo MV-algebra. These structures are roughly speaking pseudo MV-algebras that can be "split" into n 1 "slices". Let n 2 1 be a fixed integer. A non-degenerate pseudo MV-algebra M is said to be n-perfect if there are nonempty subsets Mo, MI,. . . ,Mn of M such that

+

(a) Mi n Mj = 0 and Mi 5 Mj for all i < j , that is, if x E Mi and y E Mj, then x 5 Yl (b) M = MoU M I U . . . U M,,

New Quantum Structures

(c) M i = Mn-i = M y for any i = 0,1,. . . ,n, ( d ) i f x E Mi and y E M j , then x @ y E Miej. W e shall write M = (Mo,M I , . . . ,M,) for the perfect pseudo MV-algebra M . Let G be an l-group, and set

Then &,(G) is a symmetric n-perfect pseudo MV-algebra such that &,(G) := (GO,G I , .. . ,G,), where

submitted(a)]. The following results were proved in [Dvure~enskij, THEOREM 55. Let M = (Mo,M I , . . . ,M,) be an n-perfect pseudo MV-algebra.

+

+

( i ) Let a E Mi, b E M j . If i j < n, then a + b is defined in M ; i f a b is defined i n M , then i j 5 n. (ii) Mi M j = Mi+j whenever i + j < n. (iii) If a E Mi and b E M j , and i j > n, then a b is not defined in M . (iv) Given a E M I , there is a' E M I such that a' 5 a and nu' is defined in M . ( v ) Mo is a normal and mmimal ideal of M such that Mo + Mo = Mo, Mo = Rad(M) = Infinit(M). (vi) Mo is the unique maximal ideal of M , and M E M . (vii) M admits a unique state, namely s(Mi) = iln for each i = 0,1,. . . ,n. Then Mi = s-'({iln)) for any i = 0,1,. . . ,n. (viii) Let M = (Mh,M i , . . . ,MA) be another representation of M satisfying (a)( d ) , then Mi = Mi for each i = 0,1,. .. ,n.

+

+

+

+

By M we denote the system o f pseudo pseudo MV-algebras M such that every maximal ideal o f M is normal. W e note that the converse statement t o Theorem 55, that is, i f M is a (symmetric) pseudo MV-algebra, M E M , and M admits only one extremal state and this state is (n+ 1)-valued, then M is n-perfect, can be proved. Let us define PBMV,, the system of n-perfect pseudo MV-algebras (PBMV: symmetric n-perfect pseudo MV-algebras), V(PBMV,), the variety generated by all n-perfect pseudo MV-algebras. I t is clear that PBMV = P G M V ~ . Let Vpn and be the variety o f pseudo MV-algebras and the variety o f symmetric pseudo MV-algebras, respectively, satisfying the following two equations

v;~

+

( ( n 1) o x " ) ~= 2 O xn+l,

Anatolij DvureEenskij

30

for every integer p, 1 < p

< n, such that p is not

a divisor of n.

THEOREM 56. V(PQMV,) = Vpn = 7(Vpn), and V(PGMV:)

=

v:~.

An n-perfect pseudo MV-algebra M = (Mo, MI,. . . ,M,) = r(G, u) is said to be strong if there is a E Ml such that (i) a belongs to the commutative center of G, and (ii) nu = 1;such an element a is said to be a strong cyclic element of order n. Moreover, M enjoys unique extraction of roots of 1. Indeed, if there is b E M such that nb = 1, then a b = b a in the corresponding group G, which gives n(a-b)=na-nb=O that i s a = b . For example, En(G) is strong; the element a = (1,O) is that in question. On the other hand, every symmetric 1-perfect pseudo MV-algebra is strong, we take a = 1.

+

+

THEOREM 57. An n-perfect pseudo MV-algebra M is isomorphic to some E,(G) if and only if M is strong. In such a case, G is unique up to isomorphism of l-groups. Let SPQMV, denote the category of strong n-perfect pseudo MV-algebras, where objects are pairs (M, a) with a strong n-perfect pseudo MV-algebra M and a fixed strong cyclic element a E M of order n, and morphisms are pseudo MVhomomorphisms of pseudo MV-algebras preserving fixed strong cyclic elements. SPQMV, is a functor such that (E,(G), (1,O)) is an The mapping En : Q object of the category SPQMP,, and if h is a morphism of l-groups, then En (h) (4 9)

=

(0, h(g)) (i, h(g)) (n, -h(g))

if i = 0, g E G+, if 0 < i < n, g E G, if i = n, g E G+,

is a morphism of strong n-perfect pseudo MV-algebras. THEOREM 58. En is a categorical equivalence of the category Q of l-groups and the category SPQMV, of strong n-perfect pseudo MV-algebras. In addition, suppose that h : (En(G), (1,O)) 4 (E,(H), (1,O)) is a morphism of strong n-perfect pseudo MV-algebras, then there is a unique homomorphism f : G 4 H of l-groups such that h = E,(f), and (i) if h is surjective, so is f ; (ii) if h is an isomorphism, so is f THEOREM 59. Let G be a doubly transitive &-group. Then for the variety V(SPBMVn) generated by SPQMV,, we have V(SPQMV,) = V(En(G)). In particular, an identity holds in every strong n-perfect pseudo MV-algebra if and only if it holds in E,(G).

3.5 Pseudo Lukasiewicx BCK-algebras Lukasiewicz BCK-algebras are studied in [Dvure~enskijand PulmannovL, 20001. They are parts of the positive cones of Ggroups. If they admit a greatest element

New Quantum Structures

31

1, they are equivalent to MV-algebras. In the present subsection, we give a noncommutative generalization of the concept of a Lukasiewicz BCK-algebra called a pseudo Lukasiewicz BCK-algebra. A structure (E; 5, O , O ,O), where (E; <) is a poset with the smallest element 0, and 0 , Q are binary operations, is called a pseudo-LBCK algebra if, for all a, b, c E El the following holds.

(PLB1) (PLB2) (PLB3) (PLB4)

<

a b if and only if a @ b = 0 if and only if a Q b = 0. a @ ( a Q b) = bO (bQ a) and a Q ( a O b) = b Q (b@ a). (a@ b) Q c = (a O c) @ b. If c I a , b , then from a @ c = b@ c o r a @ c = b O c i t follows a = b.

If O = , we obtain Lukasiewicz BCK-algebras. The prototypical example for pseudo-LBCK algebras is the following. Let G be a lattice-ordered group, and let Go be a convex subset of G+ containing 0. Define for a, b E G+

Then (Go; 2, @ , O ,O), where 5 is the group order and 0 the group zero, is a pseudeLBCK algebra.

LEMMA 60. Let (E; 5, @ , Q ,0) be a pseudo-LBCK algebra. Then for a, b, c E E the following holds. (i) a O O = a O O = a a n d a O a = a O a = O . (ii) E is a lower semi-lattice; we have a A b = a @ (a O b) = a Q (a @ b). Furthermore, a O b = a @ (aAb) a n d a Q b = a Q (aAb). (iii) I f a I b , t h e n a @ c I b @ c a n d a O c I b O c . SupposecIa,b. If t h e n a O c I bO c o r a Q c I bO c, it followsa5 b. (iv) If a 5 b, then c @ b I c O a a n d c Q b I c Q a. (v) (vi) (vii) (viii)

In particular, a @ b, a b I a. a @ b s c i f and onlyzfaO c I b . (a@ c ) O (bO c) = ( a 0 b) A (a@ c) and ( a O c ) O (bQ c) = ( a @ b) A ( a 0 c). (cO a ) O (cO b) 5 bO a and (cQ a)@ ( c O b) b O a. (aA b)O c = (a@ c) A (bO c) and (aA b) Q c = ( a m c) A (bO c).

<

In [Bosbach, 19821, Bosbach introduced cone algebras as follows. A cone algebra is a structure (E; @ , Q ), where @ and Q are binary operations such that, for all a, b, c E E , 4 ~ use e here a notation slightly different from that in [Bosbach, 19821;our operations translate into Bosbach's by: a * b = b Q a and a :b = aO b for any elements a, b of a cone algebra.

Anatolij DvureEenskij

32

(cl) ( c O a ) O ( b O a ) = ( c O b ) O ( a O b ) , ( c O a ) O ( b O a ) = ( c O b ) O (aOb); (c2) b O ( a O a) = b O ( a m a) = b ; (c3) ( a 0 b) O c = (a O c) O b; ( ~ 4 )a O ( a O b) = bO (bO a). Moreover, for a, b E E, we let a 5 b iff a @ b = 0. and Vetterlein, 20021). THEOREM 61 ([~vure~enskij , 0) be a pseudo-LBCK algebra. Then E can be isomorphically Let (E; 5 , , embedded into the positive cone of an f-group G, such that infima are preserved. Assuming E G, the order 5 of E is the restriction of the group order to E , 0 is the group zero, and 0 , are given by (3.1).

4 PSEUDO EFFECT ALGEBRAS In this section, we generalize the notion of an effect algebra by dropping the assumption of commutativity: a b = b a. This notion was introduced by the author and Vetterlein [Dvure~enskijand Vetterlein, 2001; DvureEenskij and Vetterlein, 2001aI.

+

+

Pseudo eflect algebras and their basic properties

1

+

A partial algebra (E;+, 0, I), where is a partial binary operation and 0 and 1 are constants, is called a pseudo effect algebra ([DvureEenskij and Vetterlein, 2001; DvureEenskij and Vetterlein, 2001a]) if, for all a, b, c E E, the following holds

+

+ + +

+

+ +

(i) a b and (a b) c exist if and only if b c and a (b c) exist, and in this case ( a + b) c = a + (b+c). (ii) for any a E E, there is exactly one d E E and exactly one e E E such that a+d=e+a=l; (iii) if a b exists, there are elements d, e E E such that a b = d a = b e; (iv) if 1 a or a 1 exists, then a = 0.

+ +

+

+

+

+

If we define a 5 b iff there exists an element c E E such that a + c = b, then 5 is a partial ordering on E such that 0 5 a 5 1 for any a E E. If E is a lattice under 5, we say that E is a lattice pseudo effect algebra. If + is commutative, i.e.

+

+

+

if a b = b a, for all a, b E E such that b a E E, E is our well-known effect algebra. Let E be a pseudo effect algebra. Let / , \ be two partial binary operations on E such that, for a, b E E, a / b is defined iff b \ a is defined ifF a 5 b, and such that in this case we have

New Quantum Structures

Then a = ( b \ a ) / b = b \ (alb). If a 5 b 5 c, then

Let E = (E; +,O, 1) be a pseudo effect algebra. We define a- := 1 \ a and a" := a / 1 for any a E E. For example, if (G, u) is a unital (not necessary Abelian) po-group with a strong unit u (sometimes it is sufficient to assume just u > O), and

+

then (r(G, u); +,0, u) is a pseudo effect algebra if we restrict the group addition to r ( G , u). A pseudo effect algebra (E; +,0 , l ) is said to be an interval pseudo eflect algebra if there exists a unital po-group (G, u) such that (E; 0 , l ) is isomorphic to (r(G, u); +,0, u). For example, E(H) = F(B(H), I) is an interval effect algebra, where B(H) is the system of all Hermitian operators of a Hilbert space H, which is a po-group under addition and the usual order of operators A 5 B iff (Ax, x) 5 (Bx, x) for any x E H, and I is the identity operator. The conditions under which a pseudo effect algebra is an interval were studied in [ ~ v u r e ~ e n s kand i j Vetterlein, 2001; DvureEenskij and Vetterlein, 2001a]. For this purpose we need the following generalizations of the Riesz decomposition property:

+,

(a) For a, b E E, we write a corn b to mean that for all a1 5 a and bl and bl commute.

5 b, a1

(b) We say that E satisfies the Riesz interpolation property, (RIP) for short, if for any al, a2, bl, b2 E E such that al, a2 5 bl, b2 there is a c E E such that a1,az 5 c 5 b1,bz. (c) We say that E satisfies the weak Riesz decomposition property, (RDPo) for short, if for any a, b l , b2 E E such that a 5 bl b2 there are dl, d2 E E such that dl 5 bl, dz 5 bz and a = dl dz.

+

+

(d) We say that E satisfies the Riesz decomposition property, (RDP) for short, if for any al,a2,bl,b2 E Esuch thatal+a2 = bl+bz therearedl,d~,dg,dqE E such that dl d2 = a l , dg d4 = a2, dl d3 = bl, d2 d4 = b2.

+

+

+

+

34

Anatolij DvureEenskij

(e) We say that E satisfies the commutational Riesz decomposition property, (RDP1) for short, if for any a l , a2, bl, b2 E E such that a1 +az = bl b2 there are dl, d2,d3, d4 E E such that (i) dl d2 = a l , d3 d4 = a2, dl d3 = bl, d2 d4 = b2, and (ii) d2 corn d3.

+

+

+ +

+

(f) We say that E satisfies the strong Riesz decomposition property, (RDP2) for short, if for any al, a2, bl, b2 E E such that a1 a2 = bl b2 there are dl,dz,d3,d4 E E such that (i) dl d2 = a l , d3 dq = a2, dl d3 = bl, d2 d4 = b2, and (ii) d2 A d3 = 0.

+

+

+ +

+

+

Note that Riesz-type decomposition properties are, roughly speaking, another kind of distributivity properties; they are closely connected to the refinements of decompositions of unity. We have the implications (i) (RDP2) + (RDP1) + (RDP) + (RDPo) + (RIP), reverse ones can fail. The converse of any of these implications does not hold. (ii) E satisfies (RDP2) if and only if E is latticeordered and fulfils (RDPo). (iii) Let E be commutative. Then we have the implications (RDP2) +- (RDP1) (RDP) *(RDPo) +- (RIP). Any implication not shown here does not hold. If the positive cone G+ of a po-group G satisfies the analogous conditions, we say that G satisfies the corresponding type of the Riesz decomposition property. The following important representation theorem for pseudo effect algebras says that every pseudo effect algebra with (RDPl) is an interval.

*

LEMMA 62. Let (E; +,0 , l ) be a pseudo effect algebra satisfying (RDPo). (i) Let a, b, c E E and a + b exist. Then from a a n n c and b a n n c it follows a + b ann c. (ii) If a A b = 0, then a + b, b + a and a V b exist and are all equal. A further property of Riesz type involves any finite number of elements rather than a maximum of four. To make the collections of formulas that will appear in the sequel more easily readable, we use the following abbreviation. Let dij, ai, bj E E, 1 5 i 5 m, l s j s n . By dll . . . dl, -t a1

we mean d i l + j = 1,...,n.

.

.

a

+

din = ai for i = 1,...,m and d l j + . . . + & . 3 = bj for

New Quantum Structures

LEMMA 63. Let ( E ; 0 , l ) be a pseudo efect algebra satisfying (RDP~). Let

+,

where m, n 2 1. Then there are elements dll,. . . ,dm, E E such that

and such that for 1 5 i < m, 12 j


we have

4.2 Representation of pseudo eflect algebras In order to prove an analogue of the Mundici representation theorem for pseudo effect algebras, we will use the so-called word technique introduced by Baer [ ~ a e r , 19491 and Wyler [Wyler, 19661. This technique was also successfully applied to effect algebras satisfying the Riesz Interpolation Property [Ravindran, 19961. In the first step, we embed a given pseudo effect algebra into a semigroup. The semigroup will then, in a second step, be extended to a po-group. The word technique means the following. Let (E; 0 , l ) be a pseudo effect algebra.

+,

(i) A sequence A = (al, . . ., a,) of finite, but non-zero, length with entries from E is called a word in E. We denote by W(E) the set of all words; that is def

W(E) = {(al, .. .,an): al, ...,a, E E , n 2 1). We define an addition in W(E) as concatenation; that is

-

(ii) We call two words A and B of E directly similar, in symbols A B, if one of them has the form (al, ...,a,), n 2 2, and the other one has the form (a1, ...,ap ap+l, ...,an), 1i P < n. We call two words A and B similar, in symbols A 1: B , if there are words Ao, ..., Ak, k 2Osuch that A = A o -A1 - . . . - A k = B. In such acase we say that A and B are connected by a chain of length k.

+

We set for a l , ...,a, E E, n 2 1,

Anatolij DvureEenskij

and we put def

C(E) = {[al, ...,a,]: a l , ...,a, E E , n 2 1). It is can be proved that (i) Similarity in W(E) is an equivalence relation compatible with

+.

+ being the induced relation, (C(E); +) is a semigroup with the neutral element [0]. (ii) For a l , ..., a,, b E E, n >_ 1, (al,. . . , a,) (b) if and only if a1 . .. a, exists and equals b.

--

+ +

The crucial lemma needed for the representation theorem is the following. LEMMA 64. Let (E; +, 0 , l ) be a pseudo effect algebra satisfying (RDP1). Let where m, n 2 1. Then there are elements dll,. . . ,dm, E E such that

and such that, for 1 5 i

< m,

1 5 j < n we have

LEMMA 65. Let (E; +,0,1) be a pseudo effect algebra satisfying (RDP1). Then C(E) is a semigroup such that the following holds. (i) [O] is a neutral element. (ii) Let a, b E C(E). From a + b = [0] it follows a = b = [O]. (iii) Let a, b, c E C(E). From a b = a c it follows b = c;

+

+

+

from b + a = c a it follows b = c. (iv) For any pair a, b E C(E) there is a c E C(E) such that a and there is a b E C(E) such that a

+ b = b +b.

+ b = c + a,

Combining the above properties and the word technique, we obtain the basic representation theorem for pseudo effect algebras with (RDP1). THEOREM 66 ([Dvure~enskijand Vetterlein, 2001; DvureEenskij and Vetterlein, 2001al). Let (E;+,O, 1) be a pseudo effect algebra fulJilling (RDP1). Then L E :

New Quantum Structures

37

E + B(E), a I+ ([a],[O]) determines a n isomorphism between ( E ;+,0,1) and (r(B(E), (['I, ['I)); + 1 (['I, ['I), ([I], ['I)), where ([I],['I) is a strong unit B(E). In particular, E i s a n interval pseudo effect algebra. It can be shown that if (E; +,0,1) is a pseudo effect algebra fulfilling (RDPl), then (i) B(E) satisfies (RDPl), (ii) the embedding L E : E + g(E), a I+ ( [ a ][O]) , preserves infima and suprema existing in E , (iii) E is an effect algebra iff B(E) is commutative, (iv) E is linearly ordered iff B(E) is linearly ordered. In the sequel we show that r defines again a categorical equivalence, which generalizes Theorem 2. Let RPEA denote the category whose objects are the pseudo effect algebras fulfilling (RDP1) and whose morphisms are the homomorphisms of these structures. Let RPUB denote the category whose objects are the unital pegroups fulfilling (RDP1) and whose morphisms are the homomorphisms of these structures. Then the mapping r : RPUB + RPEA is a functor. THEOREM 67 ([Dvure~enskijand Vetterlein, 2001; DvureEenskij and Vetterlein, 2001al). The functor I' defines a categorical equivalence between the category R P U B of unital po-groups satisfying (RDP1) and the category RPEA of pseudo effect algebras satisfying (RDP1). Now we show that as a corollary of these results we get an independent proof of the fact that every pseudo MV-algebra is an interval in a unital &group, compare with Theorem 36 and Theorem 42. PROPOSITION 68. Let E be a pseudo effect algebra fulfilling (RDPl). T h e n the following statements are equivalent. (i) (ii) (iii) (iv) (v)

E satisfies (RDP2). E i s lattice-ordered. E i s a pseudo MV-algebra. B(E) satisfies (RDP2). g ( E ) i s a n l-group.

PROPOSITION 69. A lattice effect algebra E i s can be converted into a pseudo MV-algebra if and only if a \ (aAb) = ( a V b ) \ b if and only if (aAb) / a = b / ( a V b ) for all a , b E E.

4.3 Pseudo MV-algebms and compatibility in pseudo eflect algebms As in the case of effect algebras, subsection 2.4, we now study compatibilities in pseudo effect algebras and their relation to MV-subalgebras playing the role of blocks. We introduce five kinds of compatibility of elements of a pseudo effect algebra. In the case of a lattice pseudo effect algebra four of them coincide.

Anatolij DvureEenskij

38

W e say that t w o elements a and b o f a pseudo effectalgebra E are (i) compatible (and we write a tt b) i f there are three elements a l , bl, c E E such that a = a1 c, b = bl c, and a1 bl c = bl + a1 + c E E; (ii) strongly compatible (and we write a b) if there are three elements a l l bl, c E E such that a = a1 c, b = bl c, a1 bl c = bl a1 c E E, and a1 A bl = 0 , (iii) weakly compatible, (and we write a A b) if there exist three elements a l , bl, c E E such that a = a1 c, b = bl + c, and a1 + bl + c E E and bl + a1 c E E, (iv) ultra weakly compatible (and we write a % b) if there exist three elements a l , bl, c E E such that either a = a1 bl c E El or bl + a1 + c E E, and ( v ) ultra strongly compatible (and we write a +% b) if there are three elements a l , bl, c E E such that a = al c, b = bl c, a1 A bl = 0 , a1 corn bl, and a1 bl c E E. It is evident that ( v )implies (ii),(ii)implies (i),(i)implies (iii), and (iii) implies (iv).I f E is an effect algebra, t h e n (ii)and ( v )are equivalent, and so are (i),(iii) and (iv). I f E is a lattice effect b, algebra, t h e n ( i ) and (ii) are equivalent [ ~ i e ~ a n o v20001. W e note that if E is an effect algebra with ( R I P ) , t h e n a A b i f fa tt b and ij, a A b E E , [ ~ v u r e ~ e n s k2002al.

+ + +

+ +

+ + +

+

+ +

+

+ +

+

+ +

+

PROPOSITION 70. Let a and b be two elements of a lattice pseudo effect algebra E such there are three elements a l , bl,c E E such that a = a1 c, b = bl c, a1 bl c E E , a1 bl = bl a l , and a1 A bl = 0 . Then a V b = a1 bl c, a A b = c, a1 = a \ ( a A b), and bl = b \ ( a A b). Moreover, if a = a1 + c, b = bl + c, a1 A bl = 0 , and a1 bl c E E or bl a1 c E E, then a A b = c. T H E O R E M 71. Let E be a pseudo effect algebra with ( R I P ) and a , b E E. The following statements are equivalent:

+ +

+

+

+

+ + +

+ +

+ +

(i) a + % b a n d a ~ b € ~ .

-

(ii) a - b a n d a A b E E.

(iii) a

C

b.

(iv) a + % b a n d a v b € ~ . ( v ) a t t b a n d a v b E. ~ PROPOSITION 72. Let E be a lattice pseudo effect algebra. T h e n a cr b if and onlyifa\(aAb)*b\(aAb). Inaddition, a E b i f a n d o n l y i f a \ ( a ~ b ) + % b \ ( a A b). PROPOSITION 73. Let E be a lattice pseudo effect algebra, let ai i E I , and a := V i E I a i E E. Then b tt a and

tt

b for any

New Quantum Structures

Proof. The theorem will follow from the following Claims.

Claim I . a 5 (b \ ( a A b)) /1, b \ ( a A b) + a

\

( a A b) + a A b E E.

( a V b) \ a 5 b \ ( a A b). We have ai 5 (b \ (adA b)) / 15 (b \ ( a A b)) / 1, so that a 5 (b \ ( a A b)) / 1, so that b \ ( a A b) a E E. Therefore, a V b 5 b \ ( a A b) + a \ ( a A b) + a A b and ( a V b) \ a 5 b \ ( a A b).

+

Claim 2. A,(b

\

(ai A b)) = b \ ( a A b).

b \ ( a A b) 5 b \ (ai A b) for any ai. Let d 5 b \ (ai A b) = (ai V b) \ a+ 5 ( a v b) \ ai. Then ai = ( ( a V b) \ ai) / ( a V b) 5 d / ( a V b) and a 5 d / ( a V b), so that d = (aV b) \ (d / (aV b)) 5 (aV b) \ a 5 b \ ( a A b) using Claim 1. This implies Claim 2.

By Claim 2, b \ (aAb) = Ai(b \ (&Ab)).Hence, Ai(b \ ( a i ~ b )=) Ai((aivb) \ ai). Let now x 5 (ai V b) \ ai for any i E I. Then x 5 ( a V b) \ ad, so that ai = ( ( a v b) \ ai) / ( a V b) 5 x / ( a V b). Consequently, a 5 x / ( a V b). Therefore, x = ( a V b) \ ( x / ( a V b)) 5 ( a V b) \ a, which proves Claim 3. Claim 4. a A b = V i ( a i A b). LetaiAbseforanyiEI. T h e n a i ~ b < b ~ e s o t h a t b \ ( b ~ e ) 5 b \ ( a + ~ b ) for any i E I. By Claim 2, b \ (b A e) 5 Ai(b \ (ai A b)) = b \ ( a A b). Hence ar\ b = ( b \ ( a A b)) / b 5 (b \ ( b A e ) ) / b = b A e 5 e, so that a A b = V i ( a i A b). Claim 5. a

\

( a A b) = A i ( a \ (ai A b)).

We have a \ ( a A b) 5 a \ (ai A b) for any i E I. Assume w 5 a \ (ai A b) for every i. Then ai A b = ( a \ (ai A b)) / a 5 w / a, and by Claim 4, a A b 5 w / a . Therefore, w = a \ ( w / a ) 5 a \ ( a A b) which yields Claim 5. Claim 6. a \ ( a A b) + b E E.

Claim 7. a

\

( a A b)

+ b = a V b.

It is clear hat a \ ( a ~ b ) + b2 a, b, so that a \ (aAb)+b 2 avb. Let a \ ( a ~ b ) + b2

Anatolij DvureEenskij

40

x 2 a,b. Thenx 2 ~ , b f o any r i, so that x 2 a a i V b = a i \ ( a i A b ) + b a n d

Claim 8. a e, b.

It follows from Claim 3 and Claim 7.

-

PROPOSITION 74. Let E be a pseudo effect algebra with ( R I P ) , and let ai C for i = 1 , . .. ,n,and let a := Vr=l ai E E. Then a b and

Ab

We say that a pseudo effect algebra E satisfies the countable Riesz interpolation property, ( a - R I P ) for short, if for any two sequences { x i ) and { y j ) of elements of E such that xi 5 yj for all i, j, there exists an element z E E such that xi 5 z 5 yj for all i ,j .

-

PROPOSITION 75. Let E be a pseudo effect algebra with (a-RIP), and let ai C b for i 2 1, and let a := V z l ai E E. Then a b and

-

uw b, then either PROPOSITION 76. If E i s a pseudo effect algebra. If a uw uw uw UW a-l\borb-1\a. Ifa-b, thena-l\bandb-a/l. We can ask whether ultra weak compatibility is equivalent to weak compatibility in lattice pseudo effect algebras.

A partial answer can be given by using the following notions. We say that a pseudo effect algebra E has the weak compatibility property, ( W C P ) for short, if, for a, b E El a % b implies a +% b.

New Quantum Structures

41

For example, (i) every pseudo MV-algebra, or (ii) every horizontal sum of pseudo MV-algebras, or (iii) every effect algebra, or (iv) every horizontal sum of the previous algebras has (WCP). PROPOSITION 77. Let E be a lattice pseudo effect algebra such that 1\ a = a / 1 for every a E E. Then E has (WCP). We say that a pseudo effect algebra E satisfies the difference compatibility property, (DCP) for short, if a * b, a tt c and b 5 c imply c * c \ b. Every pseudo MV-algebra, or every horizontal sum of pseudo MV-algebras, or every effect algebra, or any horizontal sum of the previous algebras has (DCP). On the other hand, there is a pseudo effect algebra having (WCP), but not (DCP). Finally, we say that a pseudo effect algebra E satisfies the compatibility complement property, (CCP) for short, if a * b implies a * 1\ b; then also a tt b / 1. We prove that in lattice pseudo effect algebras the three properties (WCP), (DCP) and (CCP) are equivalent. PROPOSITION 78. Let E be a lattice pseudo effect algebra. The following three properties are equivalent. (i) (WCP). (ii) (DCP). (iii) (CCP). We now present the main results of this subsection. THEOREM 79. Let E be a lattice pseudo effect algebra with (DCP). Then every block of E is a pseudo effect subalgebra of E which is a pseudo MV-algebra. Moreover, any such pseudo effect algebra E i s the set-theoretical union of its blocks. THEOREM 80. Let a pseudo effect algebra with (DCP) be a a-lattice. Then every block of E i s an MV-algebra, and E can be covered by commutative blocks, and in moreover, E i s a (commutative) effect algebra. The last theorem can be extended as follows. We say that a pseudo effect algebra E is Archimedean if, for an element a E E such n u := a . . . a E E for any n 2 1, we have a = 0 .

+ +

THEOREM 81. Let E be a lattice pseudo effect algebra such that every block is a pseudo effect subalgebra of E. If E is Archimedean, then E is a (commutative) effect algebra.

4 . Central elements and general comparability We introduce the concept of the center of a pseudo BGalgebra. We show that the general comparability condition entails that a pseudo effect algebra is in fact a pseudo MV-algebra having a state. Moreover, every extremal state on the center can be uniquely extended to an extremal state on the whole algebra. In particular, every maximal ideal of a pseudo MV-algebra with general comparability is normal.

42

Anatolij DvureEenskij

DEFINITION 82. A n element e of a pseudo effect algebra E is said t o b e central (or Boolean) if there exists an isomorphism

such that fe(e) = ( e ,0 ) and i f f e ( x ) = ( X I , x 2 ) , t h e n x = X I

+ 2 2 for any x E E.

W e denote b y C ( E ) t h e set of all central elements o f E, and C ( E ) is said t o b e t h e center o f E. Recall that O,1 E C ( E ) . PROPOSITION 83. Let e be a central element of a pseudo effect algebra E , and fe the corresponding mapping of Definition 82. Then

(i) f e ( e N ) = (0, e N ) . (ii) If x 5 e, then f e ( x ) = (x,O). (iii) e A e N = O . (iv) If Y I eN then f e ( y ) = ( 0 , ~ ) ( v ) eN = e-. (vi) F o r a n y x E E , x A e ~ Ea n d x A e N ~ Eand ,

+

(vii) If f e ( x ) = (x1,x2),then x = X I V x2, x1 A x2 = 0 , and xz X I = x . Moreover, i f e , f E C ( E ) , then e A f E C ( E ) , and C ( E ) is a Boolean algebra, and for the mapping pe : E + [0,el defined by pe(x) = x A e , x E E i s a homomorphism of pseudo effect algebras. T H E O R E M 84. Let a pseudo effect algebra E be monotone a-complete. Let e = V z l ei E E , where ei E C ( E ) ,i 2 1. Then e E C ( E ) , and

T H E O R E M 85. Let a pseudo effect algebra E satisfy ( R D P ) . Then e E E is central if and only zf e A eN = 0 if and only if e A e- = 0. W e say that a pseudo effect algebra E satisfies general comparability i f , given x, y E E, there is a central element e E E such that P,(x) 5 pe(y) and p e / ( x ) pet(y). T h i s means that the coordinates o f t h e elements x = ( p e ( x ) , p e / ( x ) )and y = (pe(y),pet(y))can b e compared in [0,e] and [O, e'], respectively. For example, ( i ) every linearly ordered pseudo effect algebra trivially satisfies general comparability; (ii) so does any Cartesian product o f linearly ordered pseudo effect algebras. For an "MV-analogue" o f general comparability, see t h e next result.

>

New

Quantum Structures

43

PROPOSITION 86. Let M be a a-complete MV-algebra and x E M . Then the element e := V ~ = = l ( x@l - . . @ x,), where xn = x for every n, i s a central element of M such that p e ( x ) = x . I f f i s any central element of M such that p f ( x ) = x , then e 5 f . Moreover, M satisfies general comparability. THEOREM 87 ([~vure~enskij, 2003al). Let E be a pseudo effect algebra satisfying general comparability. Then E is a lattice, and E can be converted into a pseudo MV-algebra such that the partial addition derived from E as the pseudo MV-algebra coincides with the original + taken in the pseudo effect algebra. Proof. Let x, y E E and let e E C ( E ) such that pe(x) 5 pe(Y) and P e f ( x ) 2 pef(Y). Then x = pe(x) + p e f ( x ) 2 p e ( x ) p e f ( y ) =: v E E. Claim 1. v = x A y. We have y = pe ( y ) pel ( y ) 2 pe(x) Pet ( Y ) = v , that is, v 5 x, y. Let t 5 x , y. Then p e ( t ) 5 p e ( x ) and pet(2) 5 pef(Y), that is, 2 =pe(z) pel ( t )i p e ( x ) pet ( y ) = v , that is, v = x A Y.

+

+

+

+

+

Claim 2. w :=pe(y) +pel ( x ) E E and w = x V y. Since pe(y) A p e f ( x ) = 0, then w :=p e ( y ) +pet(x) E E. We conclude now xV y = w. We have x = P,(x) +P,I(x) 5 pe ( y ) pel ( x ) = w and Y = pe ( Y ) pet ( Y ) Ipe ( Y ) pef ( x ) = W .If now L X , Y then pe ( 2 ) 2 pe ( y ) and pel ( 2 ) L pel ( x ) that is, z = pe ( 2 ) pel ( 2 ) L w.

+

+

+

Claim 3. x a: ( x A y ) = ( x V y ) \ y and y

\

+

( xA Y )= (X V Y ) \

2.

Calculate

which proves Claim 3. Finally, according to Proposition 69, Claim 3 is a necessary and sufficient condition in order to convert E into a pseudo MV-algebra ( E ;@,- ," ,0,1); we define a @ b : = (a" In such a case, the original

\

(aNAb))-,

a , b € E.

+ and the one derived from @ coincide.

It is known that every extremal state on a Boolean algebra is two-valued. In the sequel we show that every two-valued state on C ( E ) can be uniquely extended to an extremal state on a pseudo effect algebra E provided E satisfies general comparability. In particular, every pseudo effect algebra satisfying general comparability

44

Anatolij DvurGenskij

has at least one state. W e recall that there are examples o f pseudo MV-algebras ij, admitting no state [ ~ v u r e ~ e n s k20011. PROPOSITION 88. Let e be a central element of a pseudo effect algebra E and let s be a state on E. ( a ) If s ( e ) = 0 , then s o p , = 0 and s = s o p , ~ . (b) If s ( e ) = 1, then s = s o p, and s o p e , = 0 . ( c ) If s ( e ) = a, where 0 < a < 1, then the functions sl = a-ls o p e and sz = ( 1 - a ) - I s ope! are distinct states o n E such that s = as1 + ( 1 - a ) s 2 . ( d ) If s i s extremal, then s ( e ) E {O,l). Proof. ( a ) Since p,(x) = x A e , and P , ~ ( X ) = x A e', x E E, we have s ( x ) = S ( X A e ) + s ( x A e') = s ( x Ae') = s o p , ~ ( x )and s o p , ( x ) = 0 . (b)It is similar as (a). ( c ) sl and sz are states while p, and pel are homomorphisms of E onto [0,el and [O, e'], respectively. T h e n s = as1 + ( 1 - a ) s 2 , and since s l ( e ) = a - l s ( e ) = 1 and s 2 ( e ) = 0 , we have that s l # sz. rn ( d ) T h i s follows immediately from ( c ) . PROPOSITION 89. Let s be a state o n a pseudo effect algebra E and let K = C ( E ) n Ker ( s ).

(i) If s i s an extremal state, then K i s a maximal ideal of C ( E ) . (ii) If s has the property whenever t E S ( E ) such that Ker(t) K implies t = s , then s i s extremal.

>

T H E O R E M 90. Let E be a pseudo effect algebra satisfying general comparability, and let K be a maximal ideal of C ( E ) . Then there i s a unique state s o n E such that C ( E )n Ker(s) = K. This state i s extremal. I n other words, every two-valued state so on C ( E ) can be extended to a unique extremal state s o n E such that s l C ( E ) = so. C O R O L L A R Y 91. If E is a pseudo effect algebra satisfying general comparability, then E x t s ( E ) is a nonempty, compact and totally disconnected set. T H E O R E M 92. Every maximal ideal of a pseudo MV-algebra satisfying general comparability i s normal. T H E O R E M 93 ([Dvure~enskij, 2004al). Every pseudo MV-algebra M satisfying general comparability i s representable.

4.5 Ideals of pseudo eflect algebras W e consider ideals o f pseudo effectalgebras and their connection with congruences and subgroups o f t h e representing pegroups.

45

New Quantum Structures

A non-empty subset I of a pseudo effect algebra E is said to be an ideal of E if (i) x y E I whenever x, y E I and if x y is defined in E, and (ii) if x 5 y for x E E and y E I, then x E I. Then E as well as (0) are ideals of E. We denote by Z(E) the set of all ideals of E.

+

+

Let a E E, then by Io(a) we denote an ideal of E generated by a. If E satisfies (RDPo), then

~ ,m m i m a l if I is a proper subset An ideal I of E is (i) normal if a+I = ~ + a , (ii) of E and it is not included in any proper ideal of E as a proper subset, and (iii) prime if Io(a) n Io(b) c I implies a E I or b E I. We denote by N ( E ) , M ( E ) , and P ( E ) the set of all normal ideals, maximal ideals, and prime ideals, respectively, of E. Using Zorn's lemma, we see that M ( E ) is non-empty. Under certain conditions 2003b], it can be proved that M ( E ) C_ P ( E ) . on E, [~vure~enskij, We recall that {0), E E N ( E ) and if f is a homomorphism from a pseudo effect E into another one F , then

is a normal ideal of E. The following result was proved in [~vure~enskij, 2003b, Prop 3.51. PROPOSITION 94. (1) A n ideal P of a pseudo effect algebra E i s prime if and only if, for all I,J E Z(E) with I n J P, we have I C P or J C P. (2) If P is prime, then I n J = P implies I = P or J = P. If E satisfies (RDP), then an ideal P i s prime if and only ih for all I,J E Z(E) with I n J = P , we have I = P or J = P. Let P be an ideal of a pseudo effect algebra E. For a , b E E, we write a-pb iff there are two elements e, f E P such that a \ e = b \ f. We note we can define another relation, symmetric with the later, which coincides with the first in the case of the normal ideal P. PROPOSITION 95. Let E be a pseudo effect algebra with (RDP). If P i s a n ideal i s a n equivalence on E, and on EIP = { a l p : a E P } , where of E, then a / P := { b E E : b - p a } , we can define a partial ordering a / P 5 b / P if and only if there is a n element e E P such that a \ e 5 b. If a A b i s defined in E , then ( a A b ) / P = a / P A b/P. Moreover, if P is a normal ideal, then EIP can be converted into a pseudo effect algebra ( E I P ; OIP, 1 / P ) , where the partial addition is defined by a / P + b / P = c / P if and only if there are a1 E a l p , bl E b / P and cl E c / P such that al+bl = cl.

+

+,

5 ~A f is a non-empty subset of E, then a In a similar way we define A a.

+

+ A := {a + x : x E A and a + x is defined in E}.

Anatolij DvureEenskij

46

Moreover, if P is a normal ideal of E satisfying (RDP), or (RDPl), or (RDP2), then so satisfies EIP. Recall that a poset (E; 5 ) is a A-antilattice if only comparable elements of E have an infimum. Let E be a pseudo effect algebra. Then E is an antilattice iff a A b = 0 implies a = 0 or b = 0, while (a \ (a A b)) A (b \ (a A b)) = 0. PROPOSITION 96. Let P be an ideal of a pseudo effect algebra E with (RDP) and let a 5 b, a, b E E. Then a / P = b / P if and only if a = b \ s for some s E P. THEOREM 97. An ideal P of a pseudo effect algebra E with (RDP) is prime zf and only if E/P is a A-antilattice. Let G be a directed pegroup written additively, and let C(G) denote the set of all convex directed subgroups of G. In analogy with pseudo effect algebras, we say that a directed convex subgroup P of a pegroup G is a prime subgroup of G if, for all directed convex subgroups I and J of G, I n J P implies I C P or J 5 P. We denote by P(G) the set of all prime subgroups of a unital pegroup G. An equivalent definition is: C E C(G) is prime iff, for a, b E G, Go(a) n Go(b) C C implies a E P or b E P. Let C E C(G) and define x/C := {y E G : x - y E C}, and G I C := {x/C : x E G}. We order the set G/C by the usual order of left cosets of G I C via x/C y/C iff x y + c f o r some C E C. If a pseudo effect algebra E = r(G,u) satisfies (RDPl), then there exists a one-teone correspondence between the sets Z(E) of all ideals or P ( E ) of all prime ideals of E and the sets C(G) and P(G), respectively, which was established in Theorem 67.

c

<

<

THEOREM 98 ([~vure~enskij, 2003bl). Let E = r ( G , u), where (G, u) is a unital po-group satisfying (RDP1). Let I be an ideal of E. Set

Then $(I) is an o-ideal of (G, u) if and only if I is a normal ideal of E. In such the case @/I,u/I) = r'(GI#(I), u/$(I)). Moreover, if K is an o-ideal of (G,u), then its restriction to El denoted by $(K), gives a normal ideal of E, i.e., Moreover, both mappings, 4 and $, are mutually bijective and preserve set-theoretical inclusion. THEOREM 99. Let E = r(G, u), where (G,u) is a unital po-group satisfying (RDP1). Let I be an ideal of E. Set

New Quantum Structures

47

T h e n S ( I ) and & ( I ) i s the subgroup and the convex subgroup, respectively, of G generated by I , and h0(1) i s the largest directed convex subgroup of G that i s contained in & ( I ) . Let I and J be two ideals of E. Then I C J if and only if 6 ( 1 ) C_ 6 ( J ) if and only if & ( I ) 5 6,(J) i f and only i f 60(1) 5 60(J). Let K be a convex subgroup of ( G , u ) . Then

i s an ideal of E , and 6 , ( y ( K ) ) K . If K i s directed, then 6 0 ( 7 ( K ) ) = K , and ~ ( 6 ~ ( 1=) I) for any ideal I of E. I n addition, i f K1 and K Z are two directed convex subgroups of ( G , u ) , then y ( K l ) C y ( K z ) i f and only i f Kl C K 2 . If K i s a prime subgroup of ( G ,u), then 7 ( K ) := K n E i s a prime ideal of E , and if P i s a prime ideal of E , then So(P) i s a prime subgroup of ( G ,u ) . Moreover, both mappings, y and 60 are mutually bi3ective and preserving set-theoretical inclusion. Moreover, the mappings y and 60 restricted to normal prime ideals and prime o-ideals are mutually bijective.

4.6 Visualixation of pseudo MV-algebras In this subsection, we generalize results of Subsection 2.5 from effect algebras to pseudo effect algebras. The following result has been proved in [Gla, Lemma 221. THEOREM 100. A convex directed subgroup C of a directed po-group G with ( R I P ) i s prime if and only if G / C i s a A-antilattice. THEOREM 101. Every pseudo effect algebra E with ( R D P 1 ) can be represented as a pseudo effect algebra of automorphisms from A(S2)for some A-antilattice set S2 such that all finite infima and suprema existing in E are preserved. Proof. Without loss of generality, by Theorem 67, we can assume that E = I'(G,u), where ( G , u ) is a unital po-group satisfying ( R D P 1 ) . The proof will follow the following steps.

Step 1. Let P be a prime ideal of E . According to Theorem 99, h O ( P )is a prime subgroup of G , and consider the mapping q5p : E + A(S2p),where S2p = G/Go(P), defined by ( X / G ~ ( P ) ) := ~ ~( (x + ~ )a)/Go(P), a E E ( x E G ) . Then, for a , b E E, (i) a 5 b, implies 4 p ( a ) I4 p (b), (ii) 4 p ( a b) = 4 p ( a ) 0 4 p ( b ) , (iii) 4 p ( a A b) = q5p ( a ) A q5p (b) if a A b is defined in E, (iv) q5p ( a V b) = 4 p ( a ) V 4 p ( b ) if a V b is defined in E, and (v) { a E E : q5p(a) = 0 ) = n{-x+GO(P)+x : x E G } n E = P. Moreover, we have E ( P ) := 4 p ( E ) I'(A(np),q5p ( u ) ) .

+

Step 2. Let g E G and let g $ 0 and set U ( g ) := {h E G : h 2 g ) , where E = r ( G ,u). We denote by A ( g ) an ideal of E which is maximal with respect to the property U ( g ) n A ( g ) = 0. Since 0 6 U ( g ) , A ( g ) exists by Zorn's lemma. We assert A ( g ) is a prime ideal of E. Let I n J = A ( g ) , where I and J are ideals of E.

Anatolij DvureEenskij

48

Assume (as an absurdum hypothesis) that A(g) is a proper subset of I as well as of J. Take a E I r l U(g) and b E J n U (g). We have 0, g 5 a, b. By (RIP) holding in (G, u), there is an element c E G such that 0, g 5 c 5 a, b, Since 0 5 c 5 a, we have c E E, and g 5 c E I n J = A(g) which gives c E U(g)nA(g), a contradiction. Step 3. We define the Cartesian product Eo = n{A(Rg) : g E G, g $ 0 ) of the system of A-antilattices {A(Rg)}g, where R, = G/Go(A(g)), and we order Eo by coordinates. Define a mapping f : E --t Eo by f (a) = {4g(a)), (a E E), where 4, := 4,qg),and let us set Cg = GO(A(g)). We claim that f is injective. Assume f (a) = f (b). Then (x+a)/Cg = (x+b)/Cg for all x E G and g $ 0. In particular, for x = 0 which gives a/Cg = b/Cg. Hence, a - b = cg for some cg E A(g) (a - b is taken in the group G), consequently, Cg = {O}. This proves that f is an injective homomorphism of E a -b E onto f (E) Eo. Assume f (a) f (b). If g = -b + a $ 0. Then (z a)/Cg 5 (x b)/Cg for all x t: G and g $0. Consequently, this holds also for x = 0, i.e., a/Cg 5 b/C, which c$, and c$ E A(g) nU(g), means a 5 b+c$ for some c$ E A(g). Therefore, -b+a a contradiction according to Step 2. The set f (E) can be converted into a pseudo effect algebra, i.e., (f (E);o, f (0), f (1)) is a pseudo effect algebra isomorphic to E, where o is the composition of automorphisms defined by coordinates. According to Step 1, f preserves all finite infima and suprema existing in E .

nggo

<

+

+

<

Step 4. Totally order the nonnegative elements of G, say {gt : t E T), where T is a linearly ordered set. Set Rt := G/Cg,, and without loss of generality we can assume R, n Rt = 0 for all s, t E T such that s # t. Let 52 = UtETRt, and define a partial order t: on R by wl 4 w2 iff wl E R, and w2 E at and s < t or s = t and wl 5 w2 in R,. Then R is a A-antilattice with respect to 4. Define a mapping fo : E + A(R) via: let w E R, and w E Rt for a unique ) where 4g, is defined in Step 1 and Step t E T. Let (w)fo(a) = ( ~ ) ( 4 ~ , ) (Ea Rt, 3. Hence, if a E E, then fo(a)lRt maps Rt onto Rt for all t E T. Similarly as in Step 3, fo is injective from E onto fo(E), and fo(E) is a pseudo effect algebra of automorphisms of R (indeed, fo practically coincides with the function f defined in Step 3), which finishes the proof. W

As a direct consequence of Theorem 101, we show that every pseudo MV-algebra is isomorphic to a pseudo MV-algebra of automorphisms of a linearly ordered set n. COROLLARY 102. Every pseudo MV-algebra M can be represented as a pseudo MV-algebra of automorphisms from A(R) for some linearly ordered set R. For further results of the theory of pseudo effect algebras, respectively pseudo MV-algebras, see [Dvure~enskij,2001c]-[Dvure~enskijand Vetterlein, 2004b], and [Dvure~enskij,19931-[Dvure~enskij,submitted(b)].

New Quantum Structures

5 CONCLUSION In this Chapter we presented the elements of certain new algebraic quantum structures for which commutativity of (the partial) addition is not assumed. We focused mainly on pseudo effect algebras and on pseudo MV-algebras. We showed how the algebraic structure of a pseudo effect algebra can determine the whole unital pogroup. This is possible for example if E satisfies a kind of the Riesz decomposition property. This Riesz-type decomposition property is roughly speaking a sort of distributivity property connected to the refinements of decompositions of unity. Every pseudo MV-algebra is of this kind, and it is an interval in a unital E-group. The partial operation is connected with a state structure of propositions, and it corresponds to the original ideas of G. Boole [Boole, 18541, who said that for statistical information of a propositional system we only need a rule telling us how to add probabilities of events. Let us mention that in the literature we find generalizations of the above structures to more general ones, where a partial addition is not easy to define. Such i algebras are known as BL-algebras [H~jek,19981 or pseudo BL-algebras [ ~Nola et al., 2002a; Di Nola et al., 2002bl. If we have a fixed pseudo effect algebra E, we showed under what conditions it is possible to cover it by blocks. In such a case, a block is a maximal set of mutually compatible elements of E which correspond to the "classical" part of the propositional system. In contrast to orthodox quantum structures, such a block is a pseudo MV-algebra, and pseudo MV-algebras are analogues of Boolean algebras. The existence of states on pseudo effect algebras gives rise to considering an (abstract) measuring process. An important consequence is then the fact that we can substitute the whole statistical information 'contained' in pseudo effect algebras by the statistical information 'contained' in (commutative) effect algebras. This enables us to find an ideal serving as a filter which 'kills' non-commutativity and preserves that statistical information of the original statistical information which can be influenced by a noise causing non-commutativity of the propositional system. At present, no pure quantum physical experiment which can be described by non-commutative pseudo effect algebras is known yet, and it would be interesting to have such an experiment. In any case we suggested an algebraic system which describes a special kind of non-commutativity and which was inspired by our experience with quantum structures. As we saw, Hilbert's Sixth Problem inspired experts to develop a very nice theory of quantum structures which is viable independently of its original quantum mechanical motivation. Nowadays we can find situations very far from quantum mechanics where the measured data do no satisfy the axioms of the Kolmogorovian axiomatic model of probability theory. Finally, we can say that in view of the particular nature of quantum measurements the relationship between quantum mechanics and the mathematics of quantum structures is similar to that between statistical physics and classical

+

+

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probability theory. It is a fact that the efforts of mathematicians and physicists in building up probability theory and mathematical statistics were a successful enterprise for the sake of classical physics. Therefore, there is real hope that the investigation of quantum structures can bring about new and useful results for both mathematics and quantum mechanics as well as for their applications. ACKNOWLEDGEMENTS The paper has been supported by the Center of Excellence SAS - Physics of Information - 1/2/2005, the grant VEGA No. 2/6088/26 SAV and by Science and Technology Assistance Agency under the contract No. APVT-51-032002, Bratislava, Slovakia. BIBLIOGRAPHY [Alfsen, 19711 E.M. Alfsen, "Compact Convex Sets and Boundary Integrals", Springer-Verlag, Berlin 1971. [Baudot, 20001 R. Baudot, Non-commutative logic programming language NoClog, in: Symposium LICS, (Santa Barbara, 2000, Short Presentation), pp. 3-9. [Baer, 19491 R. Baer, fie sums of groups and their generalizations. A n analysis of the associative law, Amer. J . Math. 71 (1949), 706-742. [ ~ e l t r a m e t tand i Cassinelli, 19811 E.G. Beltrametti, G. Caasinelli, "The Logic of Quantum Mechanics", Addison-Wesley, Reading, 1981. [Bennett and Foulis, 19951 M.K. Bennett, D.J. Foulis, Phi-symmetric effect algebras, Found. Phys. 25 (1995), 1699-1722. [Birkhoff, 19671 G. Birkhoff, "Lattice Theory", Amer. Math. Soc. Coll. Publ., Vol. 25, Providence, Rhode Island, 1967. [~irkhoffand von Neumann, 19361 G. Birkhoff, J. von Neumann, The logic of quantum mechanics, Ann. Math. 37 (1936), 823434. [ ~ o s b a c h19811 , B. Bosbach, Concerning semiclans, Arch. Math. 37 (1981), 316-324. [Bosbach, 19821 B. Bosbach, Concerning cone algebras, Alg. Univ. 15 (1982), 58-66. [Boole, 18541 G. Boole, "An Investigation of the Laws of Thought", Macmillan, 1854, reprinted by Dover Press, New York, 1967. [Buhagiar et al., 20061 D. Buhagiar, E. Chetcuti, A. DvureEenskij, Loomis-Sikorski representation of monotone u-conzplete effect algebrm, FUZZYSets and Systems 157 (2006), 683490. [Bunce et al., 1999] L.J. Bunce, K. SaitB, and J.D.M. Wright, A Dirmier-Schaefer-Zhang theo m m for operator algebras, Proc. Amer. Math. Soc. 127 (1999), 2975-2979. [Chang, 19581 C.C. Chang, Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490. [Chovanec and KBpka, 19921 F. Chovanec, F. KBpka, O n a representation of observables i n Dposets of fizzy sets, Tatra Mt. Math. Publ. l (1992), 19-23. [Cignoli et al., 20001 R. Cignoli, I.M.L. DIOttaviano, D. Mundici, "Algebraic Foundations of Many-valued Reasoning", Kluwer Academic Publ., Dordrecht, 2000. [Di Nola, 20041 A. Di Nola, A. Dvureknskij, J. Jakubfk, Good and bad infinitesimals, and states o n pseudo MV-algebras, Order 21 (2004), 293-314. [Di Nola et al., 2002al A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo-BL algebras I, Multiple Val. Logic 8 (2002), 673-714. [Di Nola et al., 2002b] A. Di Nola, G. Georgescu, and A. Iorgulescu, Pseudo-BL algebras II, Multiple Val. Logic 8 (2002), 71S750. [Dvureknskij, 19931 A. DvureEenskij, llGleason's Theorem and Its Applications", Kluwer Academic Publ., Dordrecht, and Ister Science Press, Bratislava, 1993. [Dvureknskij, 2000] A. DvureEenskij, Loomis-Sikorski theorem for o-complete MV-algebras and e-groups, J . Austral. Math. Soc. Ser. A 68 (2000), 261-277.

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[Dvure~enskij,20021 A. DvureEenskij, Pseudo MV-algebras are intervals in e-groups, J . Austral. Math. Soc. Ser. 72 (2002), 427-445. [Dvure~enskij,20011 A. Dvureenskij, States o n pseudo MV-algebras, Studia Logica, 68 (2001), 301-327. [Dvure~enskij,2001aI A. Dvurdenskij, States and idempotents of pseudo MV-algebras, Tatra Mt. Math. Publ. 22 (2001), 79-89. [Dvureknskij, 20021 A. DvureEenskij, Measures o n quantum stmctures, In: Handbook of Measure Theory, E. Pap (Editor), Elsevier Science, Amsterdam, 2002, Vol. I1 pp. 827-868. [Dvureknskij, 2002al A. Dvurdenskij, O n effect algebras which can be covered by MV-algebras, Inter. J. Theor. Phys. 4 1 (2002), 221-229. [DvureEenskij, 2002b] A. DvureEenskij, Test spaces, pseudo-effect algebras, and tensor product of pseudo-effect algebras, Demonstr. Math. 35 (2002), 689-715. [Dvure~enskij,2002c] A. DvureEenskij, Visualization of effect algebras by autonaorphisms, Inter. J. Theor. Phys. 4 1 (2002), 2311-2318. [Dvure~enskij,2002dl A. Dvureknskij, Product effect algebras, Inter. J. Theor. Phys. 4 1 (2002), 1827-1839. [Dvure~enskij, 20031 A. DvureEenskij, Blocks of pseudo-effect algebras with the Riesz interpolation property, Soft Computing 7 (2003), 441-445. [Dvureknskij, 2003al A. Dvurdenskij, Central elenaents and Cantor-Bernstein's theorena for pseudo-effect alegbras, J. Austral. Math. Soc. 7 4 (2003), 121-143. [Dvureknskij, 2003bl A. DvureEenskij, Ideals of pseudo-effect algebras and their applications, Tatra Mt. Math. Publ. 27 (2003), 45-65. [Dvure~enskij,20041 A. DvureEenskij, Lorenzen's theorem for pseudo-effect algebras, Math. Slovaca 5 4 (2004), 23-42. [ ~ v u r e k n s k i j2004aI , A. DvureEenskij, States o n pseudo-effect algebras with general comparability, Kybernetika 4 0 (2004), 397-420. [Dvureknskij, 20051 A. Dvureenskij, Loomis-Sikorski theorem for nzonotone o - w m p l e t e effect algebras, J. Austral. Math. Soc. 79 (2005), 305-318. [Dvure~enskij,20061 A. DvureEenskij, Holland's theorem for pseudo-effect algebras, Czechoslovak Math. J. 56 (2006), 47-59. [ ~ v u r e ~ e n s k i2006al j, A. DvureEenskij, Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups, J. Austral. Math. Soc. (2006) [Dvure~enskij, submitted(a)] A. DvureEenskij, O n n-perfect GMV-algebras, submitted. [Dvure~enskij, submitted(b)] A. DvureEenskij, O n Konaori's characterization of varieties of GMV-algebras, submitted. [ ~ v u r e ~ e n s kand i j Holland, to appear] A. DvureEenskij, W.C. Holland, Top varieties of generalized MV-algebras and unital lattice-ordered groups, t o appear. [DvureEenskij and Kalmbach, 2002] A. DvureEenskij, G. Kalmbach, States o n pseudo M V algebras and the hull-kernel topology, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 131-146. [Dvure~enskijand Pulmannovi, 20001 A. DvureEenskij, S. Pulmannovi, "New Trends in Quant u m Structures, Kluwer Acad. Publ., Dordrecht, and Ister Science, Bratislava, 2000. [Dvure~enskijand Vetterlein, 20011 A. DvureEenskij, T. Vetterlein, Pseudo-effect alegbars. I. Basic properties, Inter. J. Theor. Phys. 40 (2001), 685701. [~vuretenskijand Vetterlein, 2001a] A. DvureEenskij, T . Vetterlein, Pseudo-effect algebras. II. Group representation, 40 (2001), 703-726. [DvureEenskij, 2001c] A. Dvureenskij, T . Vetterlein, Congmences and states o n pseudo-effect algebras, Found. Phys. Letters 1 4 (2001), 425446. [DvureEenskij and Vetterlein, 2001b] A. DvureEenskij, T . Vetterlein, Generalized pseudo-effect algebras. In: Lectures o n Soft Conaputing and Fuzzy Logic, Di Nola, A., Gerla, G., (Eds.), Physica-Verlag, Springer-Verlag Co., Berlin, 2001, pp. 89-111. [DvureEenskij and Vetterlein, 20021 A. DvureEenskij, T. Vetterlein, Algebras i n the positive cone of po-groups, Order 19 (2002), 127-146. [Dvureknskij and Vetterlein, 2003a] A. DvureEenskij, T . Vetterlein, O n pseudo-effect algebras which can be covered by pseudo MV-algebras, Demonstr. Math. 36 (2003), 261-282. [Dvure~enskijand Vetterlein, 2004al A. DvureEenskij, T. Vetterlein, Archimedeanicity and the MacNeill wnapletion of pseudoeffect algebras and po-groups, Algebra Universalis 50 (2004), [Dvureknskij and Vetterlein, 2003bl A. Dvurdenskij, T . Vetterlein, Infinitary and Riesz properties for pseudoeffect algebras and po-groups, 3. Austral. Math. Soc. 75 (2003), 295-311.

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[Dvurehnskij and Vetterlein, 2004bl A. DvureEenskij, T . Vetterlein, Non-commutative algebras and quantum structures, Inter. J . Theor. Phys. 43 (2004), 1599-1612. [Foulis and Bennett, 19941 D.J. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325-1346. [Foulis and Randall, 19721 D.J. Foulis, C.H. Randall, Operational statistics. I. Basic concepts, J. Math. Phys. 13 (1972), 1667-1675. [Fuchs, 19631 L. Fuchs, "Partially Ordered Algebraic Systems", Pergamon Press, OxfordLondon-New York-Paris, 1963. [Georgescu and Iogulescu, 20011 G . Georgescu, A . Iorgulescu, Pseudo-MV algebras, Multiple Val. Logic 6 (2001), 95-135. [Giuntini and Greuling, 19891 R. Giuntini, H. Greuling, Toward a formal language for unsharp properties, Found. Phys. 19 (1989), 931-945. [Glass, 19721 A.M.W. Glass, Polars and their applications i n directed interpolation groups, 'llans. Amer. Math. Soc. 166 (1972), 1-25. [Glass, 19811 A.M.W. Glass, "Ordered Pernautation Groups", London Math. Soc. Lecture Note Series, vol. 55 Cambridge Univ. Press, Cambridge, 1981. [Glass, 19991 A.M.W. Glass "Partially Ordered Groups", World Scientific, Singapore, 1999. [Gleeason, 19571 A.M. Gleason, Measures o n the closed subspaces of a Hilbert space, J . Math. Mech. 6 (1957), 885-893. [Goodearl, 19861 K.R. Goodearl "Partially Ordered Abelian Groups with Interpolation", Math. Surveys and Monographs No. 20, Amer. Math. Soc., Providence, Rhode Island, 1986. [Greechie, 19681 R. Greechie, O n the structure of orthonaodular lattice satisfying the chain condition, J . Comb. Theor. 4 (1968), 210-218. [Greechie, 19711 R. Greechie, Orthonaodular lattices admitting no states, J . Comb. Theor. 10 (1971), 119-132. [Gudder, 19791 S. Gudder, "Stochastic Methods i n Quantum Mechanics", Elsevier, North Holland, 1979. [HAjek, 19981 P. HAjek, Basic fuzzy logic and BL-algebras, Soft Computing 2 (1998), 124-128. [Heisenberg, 19301 W . Heisenberg, ' T h e Physical Principles of Quantum Theory", Dover, New Y o r k , 1930. [ ~ o l l a n d19631 , C . Holland, The lattice-ordered group of automorphism of a n ordered set, Michigan Math. J . 10 (1963), 399-408. [Kalmbach, 19831 G . Kalmbach, "Orthomodular Lattices", Academic Press, London, New Y o r k , 1983. [Kolmogorov, 19331 A.N. Kolmogorov, "Grundbeghffe der Wahrscheinlichkeitsrechnung", Berlin, 1933. [KGpka and Chovanec, 19941 F. KGpka, F. Chovanec, D-posets, Math. Slovaca 44 (1994), 21-34. [Mackey, 19631 G.W. Mackey, aMathenaatical Foundations of Quantum Mechanics", Benjamin, New Y o r k , 1963. [Maeda, 19801 S. Maeda, "Lattice T h w r y and Quantum Logic", Maki-Shoten, Tokyo, 1980 (in Japanese). [Mundici, 19861 D. Mundici, Interpretation of A F C*-algebras i n Lukasievicz sentential calculus, J . Funct. Anal. 65 (1986), 1 5 6 3 . [Mundici, 19951 D. Mundici, Averaging the truth-value i n Lukasiewiu logic, Studia Logica 55 (1995), 113-127. [Mundici, 1999) D. Mundici, Tensor products and the Loonais-Sikorski theorem for MValgebras, Advan. Appl. Math. 22 (1999), 227-248. [Navara, 19761 M. Navara, Construction of quantuna structures, this Handbook. [Piron, 19761 C. Piron, "Foundations of Quantuna Physics", Addison-Wesley, Reading, Mass., 1976. [Pt& and PulmannovA, 19911 P. PtAk, S. PulmannovA, "Orthomodular Structures as Quantum Logics", V E D A and Kluwer Acad. Publ., Bratislava and Dordrecht, 1991. [Rachiiek, 20021 J . Rachiiek, A non-cornmutative generalization of MV-algebras, Czechoslovak Math. J . 52 (2002), 255-273. [Ravindran, 19961 K. Ravindran, O n a structure theory of effect algebras, PhD thesis, Kansas State Univ., Manhattan, Kansas, 1996. [ R i e k n and Mundici, 20021 B. %&an, D. Mundici, Probability o n MV-algebras. In: Handbook o f Measure Theory, E. Pap (Editor), Elsevier Science, Amsterdam, 2002, Vol. 11, pp. 869-909.

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[RieEanovd, 20001 2. Ri&anovd, A generalization of blocks for lattice effect algebras, Inter. J. Theoret. Phys. 39 (2000), 231-237. [~ikorski,19641 R. Sikorski, "Boolean Algebras", Springer-Verlag, Berlin, Heidelberg, New York, 1964. [Varadarajan, 19681 V.S. Varadarajan, "Geometry of Quantum Theory", Vol. 1, van Nostrand, Princeton, New Jersey, 1968. [von Neumann, 19321 J. von Neumann, "Mathematische Grzlndlagen der Quantenmechanik", Springer-Verlag, Berlin, 1932. [Wright, 19731 J.D.M. Wright, On approximating concave functions by convex functions, Bull. London Math. Soc. 5 (1973), 221-222. [Wyler, 19661 0. Wyler, Clans, Compos. Math. 17 (1966), 172-189.

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HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

55

QUANTUM STRUCTURES AND FUZZY SET THEORY

Jaroslaw Pykacz

1 INTRODUCTION

At first glance, quantum structures, which are abstract algebraic objects, do not seem to have very much in common with fuzzy sets. However, already more than 30 years ago Maciej Mqczyliski proved a functional representation theorem [1973a; 19741 which allows to represent any a-orthomodular poset with an ordering set of states ('quantum logic') in the form of a family of [O, 11-valued functionals, which in turn can be treated as membership functions of fuzzy sets. Of course, the mere representation of elements of a quantum structure by fuzzy sets is not very interesting for its own sake. However, further investigations showed that not only can elements of a structure turn out to be expressible in terms of standard relations and operations on fuzzy sets but also physically interesting relations and operations on them. It turned out that the general relationship between various quantum structures and fuzzy sets is as close as the general relationship between Boolean algebras (which are specific examples of quantum structures) and traditional 'crisp' sets (which are specific examples of fuzzy sets) so that the 'representation theorems' of this Chapter can be viewed as generalizations of the Loomis-Sikorski theorem. The representation of quantum structures by families of fuzzy sets allows us to place the quantum probability calculus on an equal footing with the classical (Kolmogorovian) probability calculus. The only difference between them boils down to the fact that in the latter probability measures are defined on a-Boolean algebras of crisp sets while in the former they are defined on specific families of fuzzy sets which are quantum structures. Finally, since fuzzy sets bear the same relation to infinite-valued logic as do traditional crisp sets to classical two-valued logic, the representation of abstract quantum structures by families of fuzzy sets allows us to identify quantum structures, in particular 'quantum logics', with a specific kind of infinitevalued Lukasiewicz logics endowed with partially defined operations. This problem, however, will be discussed in the present Chapter only very briefly.

2 BASIC NOTIONS AND DEFINITIONS

2.1 The archetypal quantum structure: Birkhofl-von Neumann's quantum logic We begin with the most basic 'quantum structure', i.e. the quantum logic in Birkhoff-von Neumann's [1936] sense. Since the term 'quantum logic' has slightly different meanings in various books and papers, let us agree that throughout this Chapter it denotes an orthocomplemented a-orthocomplete orthomodular poset with an ordering set of states, i.e., a partially ordered set L which contains the smallest element 0 and the greatest element I, in which the orthocomplementation map I:L + L satisfying the conditions (a)-(c) exists: (a) ( a l l L = a. (b) If a 5 b, then bL 5 aL. (c) The greatest lower bound (meet) a A aL and the least upper bound (join) a V aL with respect to the given partial order exist in L and a A aL = 0, avaL=I. Moreover, the a-orthocompleteness condition holds: (d) If ai 5 a t for i # j (such elements are called orthogonal usually denoted by ai Iaj), then the join Viai exists in L, and so does the orthomodular identity: (e) I f a s b, then b = a v ( a L ~ b ) = a v ( a v b L ) l . Two elements a, b E L are called compatible iff there exist in L pairwise orthogonal elements al, bl,c such that a = a1 V c and b = bl V c. Probability measure on a quantum logic L is a mapping s : L + [0, 11 such that (i) s(I) = 1, (ii) ~ ( ~ i a=i Cis(ai) ) for any sequence of pairwise orthogonal elements of L. If the elements of a quantum logic L represent experimentally verifiable propositions about a physical system, then the probability measures defined on L represent states of a physical system and, therefore, are often themselves called states on L. Compatible elements of a quantum logic represent propositions that are simultaneously verifiable. In the case of the standard 'Hilbertian quantum logic' consisting of the projection operators on a Hilbert space associated to a physical system compatible elements are projection operators that commute and states on this logic are, by a famous theorem, namely Gleason's theorem, represented by density operators.

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A set of probability measures (states) S on a quantum logic L is called ordering (full, order determining) 8 s(a) 5 s(b) for all s E S implies a 5 b. Let us note that the only way in which one can experimentally establish the partial order relation between various propositions about a physical system is to conduct experiments on a system prepared in various states, which means that only families of propositions allowing for ordering sets of states can be given a physical interpretation. This is the reason why we decided, contrary to certain other authors, to require that a 'quantum logic' is not merely a a-orthocomplete orthomodular poset but such a poset together with an ordering set of states.

2.2

The hierarchy of quantum structures: from efect algebras t o Boolean algebras

The most general algebraic structure encountered in investigations into the logicoalgebraic foundations of quantum mechanics has as its generic example the family of self-adjoint operators on a Hilbert space that belong to the interval [d,i], the so called 'effects'. Therefore, it bears the name 'effect algebra'. Nowadays, the most popular definition of an effect algebra is probably the following [~oulisand Bennett, 19941: An effect algebra is a system (El @, z,u) consisting of a set E with two distinguished elements z, u called the zero and the unit and equipped with a partially defined binary operation @ satisfying the following conditions for all a, b, c E E : (i) Commutative Law: If a @ b is defined, then b @ a is defined and a @ b = b @ a. (ii) Associative Law: If b @ c is defined and a @ (b @ c) is defined, then a @ b is defined, (a @ b) @ c is defined, and a @ (b @ c) = (a @ b) @ c.

(iii)Orthosupplementation Law: For every a E E there exists a unique a' E E such that a @ a' is defined and a @ a' = u. (iv) Zero-Unit Law: If u @ a is defined, then a = z. This way of defining the concept of an effect algebra is very convenient since it permits us to see easily that effect algebras are proper generalizations of other less general quantum structures. Foulis and Bennett [1994] (see also [~ulmannovi, 19941 where, however, another terminology is used) showed that if condition (iv) is replaced by the stronger: (iv') Consistency Law: If a @ a is defined, then a = z, then we obtain an orthoalgebra. If to the conditions (i)-(iv') we add the following: (v) Coherence Law: If a defined,

@

b, b @ c, and a @ c are defined, then (a @ b) @ c is

then we obtain an orthomodular poset, and finally, if we add:

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Jarodaw Pykacz

(vi) Law of Compatibility: For all a, b E El there exist a l , bl, c E E such that b l @ c and a1 (bl c) are defined, a = a1 @ c and b = bl a c, then E can be turned into a Boolean algebra in a unique way. It should be noticed that in all structures mentioned above the operation @ allows us to define a partial order relation as follows: a 5 b iff there is c E E such that a c is defined and a c = b. On the contrary, in the case of orthomodular posets and less general quantum structures (orthomodular lattices and Boolean algebras) the operation CB coincides with the join of orthogonal elements.

2.3 Fuzzy sets According to the original idea of Lotfi A. Zadeh [1965],who is generally recognized as the founding father of fuzzy set theory, membership of an element x to a fuzzy set A, denoted pA(x) or simply A(x), can vary from 0 (full non-membership) to 1 (full membership) i.e. it can assume all values in the interval [0, 11. Therefore, a membership function PA : x H pA(x) E [ O , l ] completely characterizes the fuzzy set A and it is an obvious generalization of the characteristic function xA(x) of a traditional set: 0 for x !$ A 1 for x E A As soon as membership functions of fuzzy sets are established we can define all relations and operations known from traditional set theory on them. The definitions of the basic relations and operations on fuzzy sets were put forward early on by Zadeh in his historic paper [1965], and these are still the most frequently used in all contributions to fuzzy set theory as well as its applications. Zadeh's basic relations and operations are defined with the aid of membership functions as follows (we assume, as is usually done in fuzzy set theory, that all fuzzy sets considerd are in fact fuzzy subsets of a fixed universe of discourse U): Equality of fuzzy sets: A = B iff for all elements x in the universe U

Inclusion of fuzzy sets: A C B iff for all elements x in the universe U

Complement (negation) of a fuzzy set: A' is a complement of A iff for all elements x in the universe U

Union (sum) of fuzzy sets: A U B is a union of A and B iff for all elements x in the universe U

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Intersection (product) of fuzzy sets: A n B is an intersection of A and B iff for all elements x of the universe U (6)

HAnB{x) = m\n[nA(x),fJ,B(x)}.

However, although Zadeh's original union and intersection of fuzzy sets are still the most frequently used, there are many other operations on fuzzy sets (even infinite families of them parametrized by real numbers) which constitute proper generalizations of traditional set-theoretic union and intersection of crisp sets. For the sake of building fuzzy set models of quantum structures the following operations, called bold union and intersection by Giles [1976], who was the first to study them within fuzzy set theory (other names: Giles, truncated, bounded, arithmetic, Lukasiewicz operations)1 proved to be the most useful: Lukasiewicz union (sum) of fuzzy sets: A LJ B is a Lukasiewicz union of A and B iff for all elements x of the universe U (7)

VAUB(X) = min[^(z) + HB{X), 1].

Lukasiewicz intersection (product) of fuzzy sets: A n B is a Lukasiewicz intersection of A and B iff for all elements x of the universe U (8) 3

/Mns (x) = max \fiA (x) + fj.B (x) - 1,0]. MACZYNSKI'S FUNCTIONAL MODEL OF B-VN QUANTUM LOGIC

In 1973, thirty seven years after the introduction of the very notion of quantum logic by Birkhoff and von Neumann [1936], Maciej Maczyriski published [1973a] a theorem which twenty years later turned out to be a milestone in building fuzzy set models of Birkhoff-von Neumann quantum logics. This theorem, expressed by Maczyriski in [1974] in an equivalent form better suited for our purposes, reads as follows: THEOREM 1. Let S be a non-empty set and let L be a set of mappings from S into [0,1] having the following three properties: (i) (ii) (in)

0 (the null function) belongs to L a £ L implies 1 — a € L for any (finite or countable) sequence ai,
1 It seems that these operations were rediscovered many times by various authors, which explains the multiplicity of their names. Although these operations did not appear explicitly in any Lukasiewicz paper, the name Lukasiewicz operations seems to be the most popular nowadays and will be used throughout this Chapter.

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Then L i s a quantum logic with respect to the natural partial order of real functions, with orthocomplementation aL = 1 - a. Every point u E S induces a where mu(a) = a(u) for all a E L, and the probability measure mu o n (L, <,I), family of measures {mu : u E S) is ordering. Conversely, if (L, < , I ) is a quantum logic with an ordering set S of probability measures, then each a E L induces a function E : S -+ [ O , l ] where E(m) = m(a) for all m E S. The set of all such functions Z = {E : a E L) has properties (2)-(iii) and 5,'- ) is isomorphic to (L, <,I).

(r,

Mgczyriski's functional representation theorem provides a very useful tool for studying quantum logics since it allows abstract algebraic and order-theoretic notions that appear in the original definition of a quantum logic t o be expressed in the more convenient language of real functions. Moreover, since any [O, 11-valued function may be viewd as a memebership function of a fuzzy set, Mqczyriski's theorem allows any quantum logic to be represented in the form of a family of fuzzy sets. Indeed, if we think of the Mqczyliski functionals as membership functions of fuzzy sets, then condition (i) means that the empty set belongs to L, while condition (ii) means that if a fuzzy set belongs to L, then its standard fuzzy complement (4) also belongs to L. Only the third of Mgczyriski's conditions cannot be directly expressed in terms of fuzzy set notions. However, we shall show in the next section that this obstacle can be overcome at the expense of adding one more, very natural, condition. 4 THE GENERAL FUZZY SET MODEL OF B-VN QUANTUM LOGIC In the late eighties of the twentieth century, the formal similarity of some operations on fuzzy sets to order-theoretic operations on quantum logics gave rise to the idea of constructing fuzzy set models of quantum logics [pykacz, 19871. It was obvious from the very beginning that orthocomplementation should be modelled by the standard fuzzy set complementation (4) which, however, excludes the possibility of representing meets and joins by Zadeh's unions (5) and intersections (6) [P~kacz,19921. Indeed, the formal fuzzy set counterparts of the law of excluded middle and the law of contradiction which are assumed in point (c) of the definition of the Birkhoff-von Neumann quantum logic:

are not satisfied by any genuine fuzzy (i.e., non-crisp) set, but they are satisfied if we replace Zadeh's operations (5), (6) by Lukasiewicz's operations (7), (8). Moreover, the fact that Zadeh's operations are distributive means that they cannot be used in building fuzzy set models of generally non-distributive quantum logics. The attempts at building fuzzy set models of quantum logics made by the author since 1987 were, initially, flawed by the necessity of using, together with genuine

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fuzzy set operations (4),(7), and ( a), pointwise algebraic sums of membership functions, which are alien to fuzzy set theory because they may yield outcomes z , at the expense bigger than 1. This drawback was overcome in 1994 [ ~ ~ k a c19941 of a slight modification of the third of the original Mgczyliski [1973a; 19741 conditions described in the previous section, and the addition of the natural requirement that an empty set is the only fuzzy set that is (weakly) disjoint with itself. Mgczyliski's functional representation theorem quoted in the previous section, expressed in the language of fuzzy sets, and combined with the author's modification yields the following theorem proved in [Pykacz, 19941: THEOREM 2. Any quantum logic L with an ordering set of probability measures S can be isomorphically represented in the form of a family L(S) of fuzzy subsets of S satisfying the following conditions:

(a) L(S) contains the empty set 0 , i.e. such set that ~ ~ (=s0)for all s E S. (b) L(S) is closed with respect to the standard fuzzy set complementation (4), i.e., if A E L(S), then A' E L(S). (c) L(S) i s closed with respect to the countable Lukasiewicz unions (7) of pairwise weakly disjoint [ ~ i l e s ,19761 sets, i.e., such sets that their Lukasiewicz's intersection is the empty set. I n symbols: i f AAin Aj = 0 for i # j , then UiAi E L(S). (d) The empty set 0 is the only set in L(S) that is weakly disjoint with itself, i.e., for any A E L(S), if A n A = 0, then A = 0. Conversely, any family of fuzzy subsets of an arbitrary universe U satisfying conditions (a)-(d) i s a quantum logic partially ordered by the inclusion of fuzzy sets (3), with the fuzzy set complementation (4) as orthocomplementation, the orthogonality of the elements coinciding with their weak disjointness, and an ordering set of probability measures generated by points of the universe U according t o the formula (11) s,(A) = p ~ ( x )for all x E U. Because of the second part of Theorem 2, any family L(U) of fuzzy subsets of an arbitrary universe U satisfying conditions (a) - (d) of Theorem 2, i.e., such that (a) 0 E C(U), (b) if A E L(U), then A' E L(U), (c) if Ai n Aj = 0 for i

# j, then UiAi E L(U),

(d) if A n A = 0, then A = 0, will be called a quantum logic of fuzzy sets or simply a fuzzy quantum logic. Of course the second part of Theorem 2 implies that any quantum logic of fuzzy sets L(U) has an ordering set of probability measures generated by points of the universe U according to the formula (11).

5 TWO PAIRS OF BINARY OPERATIONS Since any fuzzy quantum logic is a a-orthomodular poset with respect to the standard fuzzy set inclusion as a partial order, it is endowed with two pairs of binary operations: the Lukasiewicz union U and intersection n, with the help of which it is defined, and the ordinary join V and meet A with respect to the partial Of course, since by Theorem 2 any quantum logic with an ordering set of order probability measures can be isomorphically represented as a fuzzy quantum logic, one can also think of it as being equipped with these two pairs of binary operations. Therefore, it is of the utmost importance to study the relations between these operations and also to investigate their logical interpretation. The theorems of this section were announced without proofs in the author's talk given at the 10th International Congress of Logic, Methodology, and Philosophy z, However, due to an unexpected of Science (Firenze, August 1995) [ ~ ~ k a c19991. delay, the proofs were published only five years later in [pykacz, 20001.

s.

THEOREM 3. Let L(U) be a quantum logic of fuzzy subsets of a universe U and let A, B E L(U). Then A fl B E L(U) ifl A A B E L(U), and in this case A and B are compatible, A fl B = A A B, and A U B = A V B. Theorem 3 cannot be 'reversed' in the sense that for arbitrary A, B E L(U) neither the existence of A A B, A V B in L(U), nor the compatibility of A and B implies the existence of A fl B, A U B in L(U). To justify the first part of this statement let us note that, since by Theorem 3 the existence of a Lukasiewicz union or intersection of any two elements of L(U) implies their compatibility and since Boolean algebras are quantum logics in which all elements are compatible k PulmannovB, 1991]),Theorem 3 yields the following corollary: (see, e.g., [ ~ t B and COROLLARY 4. If A fl B or A U B erists in a fuzzy quantum logic for all A, B E L(U), A # B, then L(U) is a Boolean algebra. Therefore, if the existence of the meet or join of two elements A, B E L(U) were sufficient for the existence in L(U) of their Lukasiewicz union and intersection, every orthomodular lattice would be a Boolean algebra, which is obviously not true. The fact that compatibility of two elements of a fuzzy quantum logic is not sufficient for the existence of their Lukasiewicz union and intersection is shown in the following example given already in [Pykacz, 19991. This example actually shows too that Corollary 5 cannot be 'reversed', i.e., that even if L(U) is a Boolean algebra, A fl B and A A B do not necessarily belong to L(U) for all possible pairs A , B E L(U), A # B. EXAMPLE 5. Let X be a triangle with vertices a, b, c, and let L(X) consist of all fuzzy subsets of X whose membership functions are affie functions assuming on all vertices of X the values 0 or 1only. It was proved in [pykacz, 19831 that L(X) is a Boolean algebra and, therefore, all its elements are compatible. Let A, B be elements of L(X) such that A(a) = 1, A(b) = A(c) = 0, B(b) = 0, B(a) = B(c) = 1,and let us construct their Lukasiewicz union and intersection. Since (AuB)(a) = min(1 1 , l ) = 1, (A U B)(b) = min(0 0 , l ) = 0, (A U B)(c) = min(0 1 , l ) = 1,

+

+

+

Quantum Structures and fizzy Set Theory

+

63

+

and (A n B)(a) = max(1 1- 1,O) = 1, ( A n B)(b) = max(0 0 - 1,O) = 0, (An B) (c) = max(0 1- 1,O) = 0, AU B and A n B would belong to L(X) iff their membership functions are affine. But this is impossible: Let d be the midpoint of the edge ab. Since the values taken by affine functions at the point d should be the arithmetic means of their values taken at the vertices a and b , it might be expected that (A U B)(d) = ( A n B)(d) = However, from A(d) = B(d) = we have that (AUB)(d) = min(4+4, 1)= 1,and (AnB)(d) = m a x ( + + i -1,O) = 0. Therefore, the membership functions of the sets A U B and A n B are not affine functions and the sets A U B and A n B do not belong to C ( X ) .

+

4.

i,

Another difference between Lukasiewicz's and order-theoretic operations on the quantum logics of fuzzy sets concerns idempotency: Order-theoretic operations of meet and join are idempotent, i.e. A A A = A V A = A for any A E L(U). The following theorem, being straightforward consequence of Theorem 3 and general non-idempotency of Lukasiewicz's operations, shows that neither A MA, nor A n A belongs to L(U) for any 'genuine fuzzy', i.e., non-crisp element of L(U): THEOREM 6. Let L(U) be a quantum logic offuzzy subsets of a universe U and let A E L(U). Then A U A and A n A belong to L(U) iff A is a crisp subset of U, i.e., ifj A(x) E {O,1) for all x E U. In this and only in this case Lukasiewicz's operations coincide with ordinary set-theoretic union and intersection, and they are idempotent. The results of this section may provide new arguments for solving the longstanding problem of whether meet and join are proper models of conjunction and disjunction in quantum logics. Birkhoff and von Neumann already noticed in their historic paper [1936] that modelling conjunction and disjunction by meet and join is, in the case of noncompatible (i.e., not simultaneously verifiable) propositions, problematic. Therefore, the unrestricted existence of meets and joins in quantum logics that are orthomodular lattices (e.g., in Hilbertian quantum logics) is a serious drawback. The fuzzy set representation of quantum logics in which the Lukasiewicz's operations, which are only partially defined, play a crucial role suggests that the Lukasiewicz's operations may be better suited for this purpose. If this hypothesis is true, Theorem 3 explains why meet and join could also, in some cases, be good: they simply coincide with Lukasiewicz's operations if only the latter are defined. However, the fact that Lukasiewicz's operations are not idempotent and that they are not necessarily defined on compatible elements of a logic still raises problem, so this issue deserves further investigation. Before closing this section let us consider the frequently studied case of a so called concrete logic (see, e.g., [Ptbk and Pulmannovb, 19911) which is a family A of crisp subsets of a fixed set R such that (a)

0EA

(c) if {Ai : i E M) is a countable family of mutually disjoint sets, then UiAi E A.

Jarodaw Pykacz

64

Since the traditional set-theoretic operations can also be viewed as Lukasiewicz's operations applied to this 'degenerate' (from the point of view of fuzzy set theory) case, we see that all three conditions appearing in the definition of a concrete logic are actually identical with the first three conditions of the definition of a quantum logic of fuzzy sets. Since the remaining condition (d) of the definition of a fuzzy quantum logic (if A n A = 0 then A = 0) is always satisfied by any crisp set A, we see that our definition of a quantum logic of fuzzy sets is a straightforward generalization of the definition of a concrete logic or, the other way round, that every quantum logic of fuzzy sets automatically becomes a concrete logic if all its elements are crisp. Let us, however, note that concrete logics, although providing nice mathematical examples of the general notion of a quantum logic, are not very interesting from the physical point of view, since they admit ordering families of twevalued (i.e., dispersion-free) states while, by Gleason's theorem, Hilbertian quantum logics do not possess any twevalued state if the dimension of the Hilbert space is strictly greater than two (see, e.g., [Ptiik and Pulmannovb, 19911). Therefore, abandoning crisp models and working with genuine fuzzy sets is an indispensable step if one wants to produce set-theoretic models of quantum logics which are of any value to quantum physics. 6 GENERAL QUANTUM STRUCTURES AND FUZZY SETS Recent trends in the study of the logicealgebraic foundations of quantum mechanics have led to the exploration of more general structures such as orthoalgebras or effect algebras rather than orthomodular posets or lattices, and it is natural to study the possibility of representing these structures also by families of fuzzy sets or many-valued propositional functions equipped with suitably chosen operations. This was partially done by DvureEenskij in [1996] and [1997], where algebraic sums of membership functions, easily expressible in terms of Lukasiewicz's operations, were used.' The aim of the present chapter is to generalize these results to families of fuzzy sets endowed with operations that are pointwisely defined by dual pairs of triangular norms and conorms, and to consider the role of Lukasiewicz's operations within these models. In a preprint [1996] DvureEenskij noticed3 that a system of fuzzy sets L [0, 1IX,X # 0 satisfying the following conditions (in what follows we identify fuzzy sets with their membership functions, in particular the universum X with its characteristic function lx): (Dl) X E L

(D2) A' = lx - A E L whenever A E L 2~urmeisterand Mwzyriski [1994], and Pulmannovi 119941 obtained a large number of his results two years earlier working, however, within the framework of [O, 11-valued functions, not fuzzy sets. 3see, however, footnote 2, esp. [Pulmannovi, 19941

Quantum Structures and Fuzzy Set Theory

(D3) A

+ B E L whenever A + B I lx

can be organized into an effect algebra (L, a , 0, X ) if we assume that A @ B is defined iff A + B 5 lx and in this case we put A @ B = A + B. It is easily seen that condition (D3) can be expressed in terms of Lukasiewicz operations (7) and (8) as follows: (D3') A U B E L whenever A n B

= 0,

+

+

since A B 5 lx iff A n B = 0 and in this case A B = A U B. Since condition (D2) actually states that L is closed under the standard fuzzy set negation (4), the effect algebra of fuzzy sets defined by DvureEenskij is a family of fuzzy sets L containing the universe X (Dl), closed under the standard fuzzy negation (D2), and containing Lukasiewicz's unions of pairs of weakly disjoint sets, in which case an effect-algebraic operation coincides with Lukasiewicz's union. In the preprint [I9961 DvureEenskij also noticed that if we add to the conditions (Dl)-(D3) the following condition (D4) A

+ A 5 lx for A E L implies A = 0,

then L becomes an orthoalgebra. Again this condition can be expressed in terms of Lukasiewicz's intersection: (D4') A n A = 0 for A E L implies A = 0, and it means that the empty set is the only set in L that is weakly disjoint with itself (which, actually, coincides with our condition (d) of the definition of a fuzzy quantum logic). Therefore, (D4) actually expresses the Consistency Law (iv') which turns an abstract effect algebra into an orthoalgebra. The least general quantum structure of fuzzy sets studied by DvureEenskij in [1996] is an orthomodular poset which has to satisfy (Dl), (D2), and

Actually, this fact was formulated (in the language of [O, 11-valued functions, not fuzzy sets) and proved by Mgczyliski and Traczyk in [1973] (see also [Burmeister and M3czyliski, 1994]), and the proof given by DvureEenskij is essentially the same as in [ ~ ~ c z ~ l and i s k Traczyk, i 19731. Today, making use of the previously quoted results, we can give another proof since (D5) yields (D3) if we put C = 0, and if we assume that A A 5 lx,then (D5) applied to the triple A, A, lx - A yields (D4). Therefore, condition (D5), which is essentially the Coherence Law (v) of [Foulis and Bennett, 19941, combined with the fact that L C [0, 1IXthat fuifills (Dl)-(D4) is an orthoalgebra, turns it into an orthomodular poset. Let us notice that condition (D5) cannot be expressed in terms of Lukasiewicz's operations by a simple replacement of the algebraic sum by Lukasiewicz's union U in the triple sum A + B + C since it may happen that for some A, B, C E L and x E X , A(x) B(x), B(x) C(x), A(x) C(x) I 1 but A(x) B(x) C(x) > 1. This difficulty can be solved as in Section 4 by assuming

+

+

+

+

+

+

+

and, independently, (D4'), since (D4') is not implied by (D5'), contrary to (D4) which is implied by (D5). When such a modification is made, all the conditions defining an orthomodular poset of fuzzy sets are expressed entirely in terms of Lukasiewicz's operations and the standard fuzzy set negation. It should be noticed that in this way we have actually obtained a 'converse' part of Theorem 2. However, the form obtained is weaker than that presented in Section 4 since the condition (D5') concerns only triples of pairwise weakly disjoint sets, while in the condition (c) of Theorem 2 countable sequences of such sets are allowed. Therefore, a family of fuzzy sets obtained from conditions (a)-(d) of Theorem 2 is a a-complete orthomodular poset with an ordering set of probability measures (i.e. a quantum logic), while conditions (Dl), (D2), (D4'), and (D5') do not guarantee that the obtained poset is a-complete. Finally, let us study the problem when a family of fuzzy sets L c [0, 1IX,which is an orthomodular poset, i.e. it satisfies conditions (Dl), (D2), and (D5), or (Dl), (D2), (D4'), and (D5'), becomes a Boolean algebra. We know from the results of Foulis and Bennett that in this case it is enough to guarantee that a suitable form of the Law of Compatibility (vi) holds in L. If we are satisfied to work with algebraic sums of membership functions, then (vi) can be written as follows4. (D6) for all A, B E L, there exist A1, B1, C E L such that A1 A1+C
+ B1, B1 + C,

Actually, the fact that L C [0, 1IXsatisfying (Dl), (D2), (D5), and (D6) can be turned into a Boolean algebra was noticed and proved (in the language of [O,l]-valued functions) by Mgczyriski [1973b] as early as 1973. The Lukasiewic~operationsversion of condition (D6) looks as follows: (D6') for all A, B E L, there exist Al, B1, C E L such that Al n B1 = Bl l l C = A1nC=0,A=AlUC,andB=Bl~C. It should be noticed that conditions (D6) or (D6') actually mean that all elements of an orthomodular poset L 5 [0, 1IXare mutually compatible - a fact also noticed by Mqczyriski in [1973b]. Before this section closes, it can be shown why the most popular union of fuzzy sets - Zadeh's (A U B)(x) = max[A(x), B(x)], although commutative and associative, cannot be combined with any pointwisely defined negation as orthosupplementation in order to construct an effect algebra L [O, 1IXcontaining genuine fuzzy (i.e. non-crisp) sets and lx as the unit. Indeed, replacing @ by U in the Orthosupplementation Law would imply that for every A E L there exists a unique A' E L such that A U A' = lx. This would mean that A'(x) = 1 at least for all x E X such that A(x) # 1. Let us take x E X such that A(x) # 0 , l . Then 4 ~ denote e this condition (D6) for the sake of the uniformity of notation, although DvureEenskij did not study Boolean algebras of fuzzy sets in [1996].

Quantum Structures and Fuzzy Set Theory

67

in order to recover A(x) = AN(x),the orthosupplementation operation should distinguish between 1' = (At(x))' = AN(x)= A(x) # 0 and 1' = (lx(x))' = 0, i.e. it should depend not only on the value A(%) of a membership function A at a point x but also on the whole membership function A, i.e. it could not be pointwisely defined. At the end of the next section we shall give an example of an effect algebra containing a genuine fuzzy set with the Zadeh union as an effect-algebraic operation @ and non-pointwisely defined orthosupplementation, although in fact we do not expect such 'exotic' structures to be useful in physics. 7 EFFECT ALGEBRAS OF FUZZY SETS GENERATED BY NILPOTENT TRIANGULAR NORMS In 1994, Radko Mesiar [1994] generalized the notion of a quantum logic of fuzzy sets based on Lukasiewicz's operations and the standard fuzzy set negation to the case when operations are pointwisely defined by nilpotent triangular norms generated, together with the negation, by a common generator. The aim of the present section is to generalize the results of both Mesiar [I9941 and DvureEenskij [1996] by showing that the same can be done in the case of effect algebras. It should be recalled that a triangular norm (shortly: t-norm ), already introduced by Menger [1942] in 1942 and since then intensively studied (not only in fuzzy set theory: see, e.g. [Schweizer and Sklar, 1958; Frank, 1979; Schweizer, 1991]), is a function T : [O, 11 x [O, 11 + [O, 11 that is commutative, associative, non-decreasing in each argument, and such that T(x, 1) = x for all x E [0, 11. In the case of a triangular conorm (shortly: t-conorm) S : [O,1] x [O,l] -t [0, 11 the last condition is replaced by S(x, 0) = x for all x E [0, 11. Triangular norms and conorms define, pointwisely, intersections and unions of fuzzy sets:

We will in the sequel be interested in Archimedean t-norms ( t-conorms) which are continuous5 and such that T(x, x) < x for all x E (0,l) (resp. S(x, x) > x for all x E (0,l)). According to Ling [1964] any Archimedean t-norm T (t-conorm S) is generated by a continuous function f (resp. g), termed the additive generator of T (resp. S ) such that:

where f : [O, 11 -t [0, m ] is strictly decreasing with f (1) = 0 and f(-')(z) f (min[f (O), z]), respectively:

Saccordingto the convention used in [ ~ e b e r19841 ,

=

68

Jaroslaw Pykacz

where g : [O, 11 --t [0,KI] is strictly increasing with g(0) = 0 and g(-')(z) = g-l(min[s(l>, 4 ) . An Archimedean t-norm T (t-conorm S) is termed strict iff it is strictly increasing on (0,l) x (0,l) and nilpotent otherwise, and this difference is significant also for their additive generators since T is strict iff f (0) = KI (S is strict iff g(1) = co). Since additive generators are unique up to multiplication by positive numbers we can assume without any loss of generality that additive generators of nilpotent t-norms (t-conorms) are normed, i.e. that f (0) = 1 and g(1) = 1. The standard fuzzy set complementation A' = 1-A is the simplest example of a vast family of fuzzy complementations AC(x) = c(A(x)) pointwisely defined by strictly decreasing continuous mappings c : [O, 1 --t [O, 11, c(0) = 1, c(1) = 0 and such that c(c(x)) = x for all x E [0, 11. Trillas [1979] proved that any such mapping is generated by a continuous and strictly increasing function g : [O, 11 --+ [O, 11, g(0) = 0, g(1) = 1 such that c(x) = g-l(l - g(x)) for all x E [O,l]. If the same function is chosen simultaneously to be a normed generator of a nilpotent t-conorm S and a t-norm T is defined by T(x, y) = c(S(c(x), c(y))), then the pair (S, T) is called a c-dual pair and it generates, together with c, a de Morgan triplet of fuzzy operations (Us, nT,C ). Mesiar [1994] generalized the notion of a quantum logic of fuzzy sets by replacing the Lukasiewicz operations by arbitrary operations pointwisely defined by a c-dual pair of nilpotent t-(co)norms. In [Pykacz, 19971, the same procedure was applied t o effect algebras of fuzzy sets: THEOREM 7. Let (Us, nT,') be a de Morgan triplet of fuzzy operations defined pointwisely by a c-dual pair (S, T) of nilpotent t-(co)nonns, where S and c are generated by the same normed generator, and let L C [0, 1IX,X # 0, satisfy the following conditions: (i) lx E L (ii) ACE L whenever A E L (iii) A US B E L whenever A nT B = Ox.

Then (L, @, Ox, lx), where A @ B is defined ifl A A @ B = A Us B, is an effect algebra.

B = Ox and then we put

Let us notice that nilpotent (i.e. Archimedean but non-strict) t-(co)norms that appear in this theorem cannot be replaced by strict t-(co)norms. Indeed, the formula c(x) = g-l(l - g(x)) is a special case of a more general formula c(x) = g-l(g(l)-g(x)), valid for an additive generator of a nilpotent t-conorm which is not necesarily a normed generator [Weber, 19841. However, any attempt at using this formula for an additive generator g of a strict t-conorm would be unable to yield desirable results since in this case g(1) = KI so one would obtain Ac(x) = g-l(m g(A(x))) = g - l ( ~ )= 1 for all points x E X in which A(%) # 1 but in all points x E X such that A(x) = 1,Ac(x) would not be defined at all. Even if one excluded the case A(x) = 1 and in this case defined AC(x) = 0,such 'complementation'

Quantum Structures and Fuzzy Set Theory

69

would not be involutive, so it could not be the orthosupplementation in an effect algebra. The considerations given above do not imply that it is impassible to construct effect algebras of fuzzy sets in which effect-algebraic sums are generated by strict t-conorms, provided that we impose some special constraints on the definition of effect-algebraic sums: EXAMPLE 8. Let X = {xl,x2}, x l # x2, and let L = {Ox,lx, F,G} where F(xl) = 0, F(x2) = G(xl) = 1, and G(x2) = 112. It can be easily tested that L equipped with a probabilistic union (16) (AH B)(x) = A(x)

+ (B(x) - A(x)B(x)

as an effect-algebraic operation @, with the constraint that A @ B = A H B is defined whenever A B 5 (1/2)x, where (17) (A

B) (x) = A(x) B(x)

is a probabilistic intersection, is an effect algebra with Ox and lx as the zero and the unit. However, although the probabilistic operations H and El are generated by a strict t-conorm S(x, y) = x y - xy and a strict t-norm T(x, y) = xy respectively, which form a de Morgan triplet together with the standard fuzzy set negation, the orthosupplementation operation in L generated by @ does not coincide with the standard fuzzy set complementation. In fact, it can be versed that the orthosupplement of F is G and the orthosupplement of Ox is lx (and vice versa, of course), so the orthosupplementation operation is not defined pointwisely by a single formula valid for all elements of L.

+

Similar objections can also be raised against using fuzzy set unions generated by non-Archimedean t-conorms in attempts to build effect algebras consisting of fuzzy sets. We have already seen that if one tries to use the Zadeh union as an effect-algebraic operation @, then the orthosupplementation operation cannot be defined pointwisely. Similar arguments can be applied to other fuzzy set unions generated by non-Archimedean t-conorms. Again this does not mean that it is impossible to construct effect algebras of fuzzy sets using such unions as effect-algebraic operations, provided that a family L [0,1IXis suitably chosen, A @ B = A Us B is defined only for carefully selected pairs of fuzzy sets, and the orthosupplementation operation is not defined pointwisely. For example, the Celement effect algebra L = {Ox,lx, F, G, } described in the previous example is equally well defined when probabilistic operations are replaced by Zadeh operations with the same constraint in defining the effect-algebraic partial sum @. We should remember, however, that the prototype of all effect algebras: the family E(7-l) of all self-adjoint operators between 6 and i acting on a Hilbert space 'FI and its subalgebra P(7-i)consisting of all projection operators, which is the prototype of all quantum logics, when these structures are represented as families of fuzzy subsets of the set of density operators, are endowed with pointwisely defined Lukasiewicz's operations and the standard fuzzy complementation which form a

de Morgan triplet generated by the identity function. According to the results of Mesiar [1994; 19951, any quantum logic of fuzzy sets based on a de Morgan triplet of fuzzy operations generated by a common generator g is actually isomorphic to a logic based on Lukasiewicz's operations and the standard fuzzy complementation, and this result can in a straightforward way be generalized to effect algebras of fuzzy sets as defined by Theorem 17. Therefore, if one wants to obtain fuzzy set or many-valued models of quantum structures endowed with pointwisely defined o p erations, then the Lukasiewicz operations are unique up to an isomorphism given by a generator g. This implies that 'exotic' examples of effect algebras of fuzzy sets based on strict or non-Archimedean triangular norms, such as those described above, though mathematically interesting and even desirable, cannot be used to describe actually existing physical systems. 8 FUZZY SET MODELS OF QUANTUM PROBABILITY In certain experiments on quantum systems the relative frequencies of obtaining certain results, interpreted as probabilities, do not satisfy the numerical constraints imposed by classical (Kolmogorovian) probability theory. Such instances, usually connected with the violation of Bell's inequalities, strongly indicate the necessity of modifying the probability calculus used in quantum mechanics. There are several approaches to the subject that can generally be termed "quantum probability", and even a brief review of all of these would lead us far beyond the scope of this section. Therefore, we shall concentrate on the quantum-logical treatment of this subject. In the quantum logic approach to the foundations of quantum mechanics the Kolmogorovian triple (R,T, P ) consisting of a space of elementary events R, a Boolean a-algebra .F of selected subsets of R (random events), and a probability measure P, is replaced by a pair (L,p) consisting of a a-orthocomplete orthomodular poset (i.e. quantum logic) L and a probability measure (state) p defined on L. It follows from the very definition (see Section 2.1) that probability measures on quantum logics satisfy all numerical constraints imposed on Kolmogorovian probability measures: they are nonnegative, normalized, and a-additive on families of pairwisely disjoint (in the language of the 'orthodox' quantum logic: pairwisely orthogonal) elements. However, this clearly does not mean that Kolmogorovian probability calculus, which is based on Boolean a-algebras, is an adequate tool for quantum mechanics6. This is particularly evident in the quantum logic approach where several theorems were proved showing that various versions of Bell-type inequalities are satisfied by probability measures defined on a quantum logic if this logic is a Boolean algebra (see, e.g. papers by Santos [1986], PulmannovL and Majernik [1992], or Beltrametti and Mgczyliski [1993a; 1993bl).

allen en tine's [1986]conviction that he has "refuted any and all claims that 'classical' probability theory is not valid in quantum mechanics" seems to be based on a superficial analysis in which he took into account neither Bell-type inequalities, nor the differences in algebraic structures on which classical and quantum probability measures are defined.

Quantum Structures and Fuzzy Set Theory

71

Besides these numerical and 'structural' differences between classical and quantum probabilities there is another important difference: quantum random events are not subsets of the space of elementary events but mathematical objects of another kind. In the Hilbert space model they are represented by closed subspaces of a Hilbert space (or orthogonal projections onto such subspaces), while in an abstract model they are simply elements of an orthomodular poset. This does not allow quantum random events to be treated as subsets of a phase space of a physical system. In the quantum logic approach the states of any physical system are represented by probability measures on the logic of this system, and they form a convex set whose extreme points represent the pure states of the system. In the particular case of the phase space description of a classical statistical system its logic is identifled with a Boolean a-algebra of Borel subsets of a phase space and pure states are Dirac measures concentrated on one-point subsets of a phase space so they may be identified with points of a phase space of a system. On the other hand the elements of a logic, i.e. Borel subsets of a phase space may be identified with random events since each random event is in an obvious way defined by a property of the physical system: it consists of those pure states for which the given property holds. Therefore, traditional set-theoretic unions and intersections of random events are generated by disjunctions and conjunctions of propositions about the physical system under study in full accordance with the spirit of Kolmogorovian probability theory. This is no longer true for a quantum system. Properties of a quantum system, represented by the elements of a logic, can no longer be represented by crisp subsets of the set of pure states. However, there is still a possibility of representing the elements of a logic, even of a 'genuine' quantum system, by fuzzy subsets of the set of its pure states. It should be noted that the conditions (a)-(d) that define a quantum logic of fuzzy sets show remarkable similarity to the conditions that define Boolean a-algebras of random events in Kolmogorovian probability theory. The difference between condition (c) and the Kolmogorovian requirement that a a-algebra of random events be closed under countable unions of arbitrary, not only pairwise disjoint, sets seems to be unimportant since this requirement of Kolmogorov is superfluous: probability measures are assumed to be a-additive on pairwise disjoint, not arbitrary sequences of sets and it is possible to construct a reasonable 'classical' probability theory with this requirement being suitably modifled (for a detailed discussion of this problem see the book of T. L. Fine [1973]). Since condition (d) in the domain of crisp sets is trivially satisfied, we conclude that the notion of a quantum logic of fuzzy sets is in a sense a 'minimal' generalization of the notion of a a-algebra of random events to a family of fuzzy sets endowed with Lukasiewicz's operations, which permits to construct a reasonable probability calculus. Replacing the abstract quantum logics that appear in the foundations of quantum probability calculus by families of fuzzy subsets of sets of pure states of quantum systems one obtains, at the price of allowing fuzzy sets to come into

72

Jaroslaw Pykacz

play, a perfect parallelism between Kolmogorovian probability calculus applied to classical statistical systems and quantum probability calculus applied to quantum systems: in both cases random events are represented by subsets of sets of pure states of physical systems and they are defined by the properties of these systems. Conjunctions and disjunctions of properties of physical systems define intersections and unions of respective subsets. However, in quantum physics, in contrast t o the situation encountered in classical statistical physics, the results of these operations do not always belong to a quantum logic of fuzzy sets, even if this logic is a lattice. Therefore, making use of joins and meets in order to construct 'compound' quantum random events - a common practice in quantum probability - instead of Lukasiewicz's unions and intersections, can be a source of serious difficulty. Finally, it should be mentioned that it is possible to build a fuzzy probability theory using, instead of Lukasiewicz's operations, other operations chosen from the vast family of fuzzy unions and intersections. This has in fact been done in a number of papers (see, e.g. [Zadeh, 1968; Klement et al., 1981; Piasecki, 1985; Mesiar, 1992; Mesiar and Navara, 19961 to mention a few) in which a fullfledged fuzzy probability theory was developed. However, in the majority of these papers the authors use the original Zadeh operations which cannot be used to build fuzzy set models of quantum logics since, as was earlier noticed (in the realm of a many-valued logic) by Gonseth [1938], when combined with the standard fuzzy set complementation, they do not satisfy the excluded middle law and the law of contradiction for any genuine fuzzy set. At the end of Section 7 we also gave arguments why only Lukasiewicz's operations (or other isomorphic to them pointwisely defined by nilpotent triangular norms and conorms operations on fuzzy sets) can reasonably be used to build fuzzy set models of quantum logics and, therefore, to construct fuzzy set models of quantum probability spaces. Of course, because of the results presented in Sections 6 and 7, probability measures can be defined on suitable families of fuzzy sets that represent quantum random events also in more general approaches to quantum probability e.g. in those based on effect algebras instead of quantum logics. 9 SUMMARY

In this Chapter we described relations between abstract quantum logics and other more general abstract quantum structures and some special families of fuzzy sets endowed with partially defined Lukasiewicz's union and intersection. These relations are generally the same as the relations between abstract Boolean algebras and Boolean algebras of traditional ('crisp') sets endowed with the traditional settheoretic operations. In particular, this approach allows us to see the full parallelism between the classical Kolmogorovian probability calculus, where probability measures are defined on Boolean algebras consisting of crisp sets of random events, and the quantum probability calculus, in which probability measures are defined, depending on the level of generality, on various quantum structures consisting of fuzzy sets of quantum random events.

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Quantum Structures and Fuzzy Set Theory

It was argued that Lukasiewicz's union and intersection of fuzzy sets (or 'is* morphic' to them operations on fuzzy sets pointwisely generated by by nilp* tent t-(co)norms) are, within the uncountable family of possible union-like and intersection-like operations on fuzzy sets, the only operations that may be used to construct quantum structures of fuzzy sets that might be of interst to physicists. Since fuzzy set models of quantum logics are endowed with two pairs of binary operations, namely by Lukasiewicz's union and intersection and order-theoretic meet and join, it was necessary to consider relations between them and decide which pair is better suited to serve as an algebraic model of logical conjunction and disjunction. Some arguments were given that, contrary to the 60-year old tradition, not the order-theoretic operations of meet and join, but Lukasiewicz's operations should be used for this purpose, but this problem still deserves further investigation. ACKNOWLEDGMENTS This work was carried out within the project G.0335.02 of the Flemish Fund for Scientific Research. BIBLIOGRAPHY [Ballentine, 19861 L. A. Ballentine. Probability theory in quantum mechanics American Journal of Physics, 54 (1986) 883-889. [Beltrametti and Mgczyfiski, 1993al E. G. Beltrametti and M. J. Mzjczyfski. On the characterization of probabilities: A generalization of Bell's inequalities, Journal of Mathematical Physics, 34 (1993) 4919-4929. [Beltrametti and Mgczyfiski, 1993bl E. G. Beltrametti and M. J. Mqczyfiski. On some probabilistic inequalities related t o the Bell inequality, Reports on Mathematical Physics, 33 (1993) 123-129. [Birkhoff and von Neumann, 19361 G. Birkhoff and J. von Neumann. The logic of quantum mechanics, Annals of Mathematics, 37 (1936) 823-843. [Burmeister and Mpczyriski, 19941 P. Burmeister and M. Mpczyliski. Orthomodular (partial) algebras and their representations, Demonstratio Mathenzatica, 27 (1994) 701-722. [Dvure~enskij,19961 A. Dvureknskij. Fuzzy set representations of some quantum structures, Preprint Series of the Mathematical Institute of the Slovak Academy of Sciences, 2/1996. [Dvureknskij, 19971 A. DvureEenskij. Fuzzy representation of effect algebras and the moment problem, in: [IFSA, 19971, Vol. IV,pp. 146-151. [Fine, 19731 T. L. Fine. Theories of Probability (Academic Press, New York, 1973). [Foulis and Bennett, 19941 D. J. Foulis and M. K. Bennett. Effect algebras and unsharp quantum logics, Foundations of Physics, 24 (1994) 1331-1352. [Frank, 19791 M. J. Frank. On the simultaneous associativity of F ( z , y) and x y - F ( x , y), Aequationes Mathematicae, 19 (1979) 194-226. [Giles, 19761 R. Giles. Lukasiewicz logic and fuzzy set theory, International Journal of ManMachine Studies, 67 (1976) 313-327. [Gonseth, 19381 F. Gonseth. Les entretiens de Zurich sur les fondenzents et la mDhode des sciences mathematiques 6-9 de'cembre 1938 (Ziirich, 1941). [Klement et al., 19811 E. P. Klement, R. Lowen, and W. Schwychla. Fuzzy probability measures, h z y Sets and Systems, 5 (1981) 21-30. [Ling, 19641 C. H. Ling. Representation of associative functions, Publicationes Mathematicae Debrecen, 12 (1965) 189-212.

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[Mpczyfiski, 1973al M. J. Mpczyfiski. The orthogonality postulate in axiomatic quantum mechanics, International Journal of Theoretical Physics, 8 (1973) 353-360. [Mpczyhki, 1973bl M. J. Mpczyxkki. On some numerical characterization of Boolean algebras, Colloquiuna Mathematicum, 27 (1973) 207-210. [Mpczyfiski, 19741 M. J. Mpzyriski. Functional properties of quantum logics, International Journal of Thwretical Physics, 11 (1974) 149-156. [Mpczyfiski and Traczyk, 19731 M. J. Mpzyfiski and T . Traczyk. A characterization of orthomodular partially ordered sets admitting a full set of states, Bull. Acad. Polon. Sci. Str. Sci. Math. A s k . Phys . 21 (1973) 3-8. [Menger, 19421 K. Menger. Statistical metrics, Proceedings of the National Academy of Sciences of the USA, 28 (1942) 535-537. [Mesiar, 19921 R. Mesiar. Fuzzy sets and probability theory, Tatra Mountains Mathenaatical Publications, 1 (1992) 105-123. [Mesiar, 19941 R. Mesiar. h-fuzzy quantum logics, International Journal of Theoretical Physics, 33 (1994) 1417-1425. [Mesiar, 19951 R. Mesiar. Do fuzzy quantum structures exist?, International Journal of Theoretical Physics, 34 (1995) 1609-1614. [Mesiar and Navara, 19961 R. Mesiar and M. Navara. T,-tribes and T,-measures, Journal of Mathematical Analysis and Applications, 201 (1996) 91-102. [Piasecki, 19851 K. Piasecki. Probability of fuzzy events as denumerable additivity measure, Fuzzy Sets and Systems, 17 (1985) 271-284. [IFSA, 19971 Proceedings of the Seventh International fizzy Systems Association World Congress, Prague, June 25-29, 1997 (Academia, Prague, 1997). [Ptik and Pulmannovi, 19911 P. P t i k and S. Pulmannovi. Orthomodular Structures as Quant u m Logics (Kluwer Academic Publishers, Dordrecht, 1991). [Pulmannovii, 19941 S. Pulmannovi. A remark on orthomodular partial algebras, Demonstratio Mathenzatica, 27 (1994) 667-699. [Pulmannovi and Majernik, 19921 S. Pulmannovi and V. Majernik. Bell inequalities on quantum logics, Journal of Mathematical Physics, 33 (1992) 2173-2178. [Pykacz, 19831 J. Pykacz. Affine Mpzyriski logics on compact convex sets of states, International Journal of Theoretical Physics, 22 (1983) 97-106. [Pykacz, 19871 J. Pykacz. Quantum logics as families of fuzzy subsets of the set of physical states, Preprints of the Second International Fuzzy Systems Association Congress, Tokyo, July 20-25, 1987, vol 2 , pp. 437-440. [Pykacz, 19921 J. Pykacz. Fuzzy set ideas in quantum logics, International Journal of Theoretical Physics, 31 (1992) 1767-1783. [Pykacz, 19941 J. Pykacz. Fuzzy quantum logics and infinite-valued Lukasiewicz logic, International Journal of Theoretical Physics, 33 (1994) 1403-1416. [Pykacz, 19973 J. Pykacz. Triangular norms - based quantum structures of fuzzy sets, in: [IFSA, 19971, Vol. IV, pp. 159-164. [ ~ ~ k a c19981 z, J. Pykacz. Fuzzy quantum logics a s a basis for quantum probability theory, International Journal of Thwretical Physics, 37 (1998) 281-290. [Pykacz, 1999] J. Pykacz. Attempts a t the logical explanation of the wave-particle duality, in: M. L. Dalla Chiara et al. (eds) Language, Quantum, M w i c (Kluwer, Dordrecht, 1999) 269-282. [ ~ y k a c z20001 , J. Pykacz. Lukasiewicz operations in fuzzy set and many-valued representations of quantum logics, Foundations of Physics, 30 (2000) 1503-1524. [Santos, 19861 E. Santos. The Bell inequalities as tests of classical logics, Physics letters A , 115 (1986) 363-365. [Schweizer and Sklar, 19581 B. Schweizer and A. Sklar. Espaces m&riques albatoires, Conaptes Rendus Acad. Sci. Paris, 247 (1958) 2092-2094. [~chweizer,19911 B. Schweizer. Thirty years of copulas, in: G. Dall'Aglio et al. (eds) Advances in Probability Distributions with Given Marginals (Kluwer, Dordrecht, 1991) 13-50. [Trillas, 19791 E. Trillas. Sobre functiones de negacion en la teoria de conjuntos difusos, Stochastica, 3 (1979) 47-59. [Weber, 1984) S. Weber. I-decomposable measures and integrals for Archimedean t-conorms I, Journal of Mathematical Analysis and Applications 101 (1984) 114-138. [Zadeh, 19651 L. A. Zadeh. Fuzzy sets, Infornzation and Control, 8 (1965) 338-353. [Zadeh, 19681 L. A. Zadeh. Probability measures on fuzzy events, Journal of Mathenaatical Analysis and Applications, 23 (1968) 421427.

HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

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ALGEBRAIC AND MEASURE-THEORETIC PROPERTIES OF CLASSES OF SUBSPACES OF AN INNER PRODUCT SPACE David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

1 INTRODUCTION In the classical (Kolmogorov) probability theory [Kolmogorov,19331, the set of experimentally verifiable events assigned to a physical systems (i.e. the event space) can be identified with a measurable space (X, C). The order relation 5 induced by the lattice operations V and A is logically interpreted as the implication relation, whereas the complement operation ' is understood as the negation operation of events. A probability measure is a countably additive, normalized and nonnegative function p on C. Random variables are the C-measurable real-valued functions on X. Ever since the celebrated paper The logic of quantum mechanics by G . Birkhoff and J. von Neumann [Birkhoff and von Neumann, 19361, there has been an intense research in the mathematical foundations of quantum mechanics. The algebraic study of quantum logics that generalize Boolean a-algebras has given rise to the theory of orthomodular posets, and the study of states to non-commutative measure theory. One of the most important quantum logic is the projection lattice of a Hilbert space H. The basic axiom of the so called Hilbert space model is that the events of a quantum system can be represented by projections on a Hilbert space or, equivalently, the collection L(H) of closed subspaces of a Hilbert space H. When endowed with the inclusion relation C and orthocomplementation I,L(H) is a complete orthomodular lattice. The transition from the Boolean a-algebra C to the projection lattice L(H) consists in replacing the disjointedness of sets by a geometric concept of the orthogonality of subspaces. The notion of a random variable generalizes to that of an observable. Let B(W) denote the Borel a-algebra of B. An observable is a map a : B(R) + L(H) such that: (i) a(W) = H , (ii) a(A)Ia(B) for any disjoint pair of Borel sets and (iii) a(UiEnAi) = ViCNa(Ai) for any pairwise disjoint sequence {Ai I i E M) of Borel sets. An observable is bounded if there exists a compact subset C of B such that a(C) = H. In view of the spectral theorem, the set of bounded observables can be identified with the self-adjoint part of B(H), the algebra of bounded operators on H. A probability measure on L(H) is a map s : L(H) + [O, 11 such that

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David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

s(H) = 1 and s(ViENMi) = CiENs(Mi) for every sequence (Mi 1 i E N) of pairwise orthogonal elements of L(H). Every positive trace-class operator T with unit trace (density matrix) on H defines a probability measure on L(H) as follows: M E L(H) H trTPM, where PMdenotes the projection of H onto M. It turns out that if 3 5 dim H 5 No, then every probability measure has this form. Thus every probability measure on L(H) extends to a linear functional on B(H). This characterization is due to A. Gleason leason on, 19571. Instead of restricting ourselves to Hilbert spaces, we can consider general inner product spaces (= pre-Hilbert spaces). A natural question can be raised about the role of the purely mathematical concept of (metric) completeness of H in the Hilbert space model with respect to the physical interpretation the model bears. Can we retrieve a similar model if we drop the assumption of completeness? A considerable amount of work has been done in this line, for example [Amerniya and Araki, 1966-67; Gross and Keller, 1977; Hamhalter and PtLk, 1987; Holland, 19691 (and many others). In this connection we also mention the very important results of H. A. Keller and M. P. Solkr. In [Keller, 1980],H. A. Keller exhibited an example of a nonarchimidean orthomodular quadratic space such that the corresponding projection logic resembles the projection logic L(H) for a Hilbert space H and M. P. Solkr showed that if an orthomodular quadratic space admits at least one infinite orthonormal sequence, then this space has to be a Hilbert space [Solkr, 19951. We also remark that as it was the case with Boolean algebras, the algebraic structure of the projection lattices of Hilbert spaces and the set of positive o p erators lying under the identity (effects) were thoroughly generalized and today there is a whole hierarchy of quantum structures, for example: orthomodular posets and lattices as quantum logics [Varadarajan, 1985; Kalmbach, 1983; PtLk and PulmannovL, 19911, effect algebras [Giuntini and Greuling, 1989; Bennett and Foulis, 19971, D-posets [Kbpka and Chovanec, 19941 and many others (refer to the monograph New Trends in Quantum Structures by A. DvureEenskij and S. PulmannovL [ ~ v u r e ~ e n s kand i j PulmannovB, 20001 for a complete survey).

2 FAMILIES OF CLOSED SUBSPACES OF AN INNER PRODUCT SPACE S In this section we define six different families of closed subspaces of an inner product space S and investigate their interrelations. Let S be an inner product space over D (where D is the real field R or the complex field C1)and let (. , .) be the inner product on S. For each subset A S we denote the (metric) closure of A in the completion 3 of S by 2 and therefore, is the (metric) completion of A. We write A'-s to denote the orthogonal complement of lhlany results can be generalised to the quaternion case but our aim here is maximal clarity and simplicity rather than maximal generality.

Algebraic and Measuretheoretic properties

A in S so that

. ..

A~= " {b E S I (a, b) = 0 for all a E A ) .

We will consider the following families of closed subspaces of S. W(S) = { M is a subspace of S I M is closed in S) ; F ( S ) = { M is a subspace of S I MI"'" = M) ; Eq(S) = { M is a closed subspace of S I M @ M I S is dense in S) ; E(S) = { M is a subspace of S I M @ M I " = S) ; C(S) = { M is a subspace of S I M is complete or cocomplete ) ; P ( S ) = { M is a subspace of S I M is finite or cofinite dimensional) .

M is cocomplete (cofinite dimensional) means that M = ALs for some complete (finite dimensional) subspace A of S . A subspace M C S is said to be orthogonally closed if M E F(S); splitting if M E E(S); quasi-splitting if M E Eq(S). Let us begin by showing that

Proof. F ( S ) C W(S). This follows immediately from continuity of inner product.

Eq(S) C F(S) . Let M E Eq(S); then

where the third equality follows from Proposition 37. Consequently, M E F(S).

E(S) G Eq(S).This follows directly from definition. C(S) g E(S). Suppose that M E C(S) and let N E { M ,M I " ) be complete. Then N is splitting in 3,that is 'S = N @ N l 3 . Let s E S be arbitrary. We have s = n n' for some n E N and n' E N I T . Moreover, n' = s - n E SnN ~ = T N l s . Hence S = N N's and therefore M E E(S) .

+

P ( S ) C(S). This follows from the fact that every finite dimensional space is complete.

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78

It is not difficult to see that:

.

Each of the above families is closed under orthocomplementation. {0), S E P(S).

When S is complete, C(S) = E(S) = Eq(S) = F ( S ) = W(S). By taking various subspaces of the Hilbert space 12,we now show that in general the six families are distinct. In all the following examples we let ei = {bij I j E N), i E N, be an orthonormal basis of 12.For a vector u E S, where S is an inner product space, we let [u] be the one-dimensional subspace span{u). EXAMPLE 1. This is to show that in general W(S) # F(S). Let f = CiEN i e i and let S = span{f, e2, e3, . . . ). It is not difficult to see that M = span{ez, e3, . . .) is a closed subspace of S. On the other hand, M I s = (0) and therefore, M I S I S= S. Consequently, M $! F(S). EXAMPLE 2. This is to show that in general F(S) # Eq(S). 1 03 1 Let f = el 4ez Cz2&ezi and g = el - ;ie2 Ci=l -ezi+l. Define S = span{ei I i = 3,4,5,. . . ) [f] [g], and let M = span{ei I i = 4,6,. . . ) and N = span{ei I i = 3,5,. . .). It is not difficult to verify that M I S = N [g] s N. Observe that and N L s = M + [f]. Therefore MIS-'-" = M and N ~ s - ' -= el E ii?I7, but el 4 [g] = That M $! Eq(S) now follows from Proposition 37.

+

+

+

+ + +

+

n.

EXAMPLE 3. This is to show that in general Eq(S) # E(S). Let f = CiEN iei. Define S = span{ei I i E M) [f],M = span{ej I i = 2,4,. . . ) and N = span{ei I i = 1,3,. . . ). It is not difficult to verify that M I S = N and M E Eq(S); but it is clear that M @ E(S).

+

EXAMPLE 4. This is to show that in general E(S) # C(S). Let S = Cf = span{el, e2,. . . ) C 12. If M = span{ei I i = 2,4,. . . ) then M I S = span{ei I i = 1,3,. . . ). Clearly S = M @ M I Sand therefore M E E(S). M E C(S) implies either M or M I S is complete, i.e. a Hilbert space. But any Hilbert space is either finite dimensional or else its linear dimension is uncountable; therefore neither M nor M I S can be complete. This means that M $! C(S). Finally, the last example which is trivial. Note that in this example, S is a complete inner product space. EXAMPLE 5. This is to show that in general C(S) # P(S). Let S = l2and M = span{ei I i = 2,4,. . . ). Then M E C(S) but evidently, M $ P(S). Before we analyze the algebraic structure of the above families we give a result that we would need in later sections. PROPOSITION 6. Let u E S be arbitrary.

(1) If M E P(S), then M

+ [u] E P(S).

Algebraic and Measure-theoretic properties

...

+ [u] E C(S). (3) If M E E(S), then M + [u] E E (S). (4) If M E Eq(S), then M + [u] E Eq(S). (5) If M E F(S), then M + [u] E F(S). ( 6 ) If M E W(S), then M + [u] E W(S). (2) If M E C(S), then M

Proof. We give the proof for the case of F(S). If u E M, then result is trivial. Assume that u 4 M such that llull = 1. Since M E F(S) we have u 4 MISIS. Therefore there exists b E MIS such that (b, u) = 1. Now, for all y E MIS, we have

because y - (y,u)b E Take any x E ( M

+[

MI" and

(y - (y,u)b,u) = (y,u) - (y,u)(b,u) = 0.

u ] ) ~For ~ any ~ ~ y. E M I S , we have

Therefore x - (x, b)u E M'-S-'-S = M. This implies that x E M

+ [u].

rn

We end this section by giving some definitions. DEFINITION 7. An orthocomplemented poset is a bounded poset (L, 5 , 0 , I), where 0 is the smallest element, 1 is the largest element and 0 # 1, with orthocomplementation I: L -t L satisfying

(OCP3) a V a l exists ~ and is equal to 1.

As a consequence to the above axioms we have a A a l exists ~ and is equal to 0. Here by V (respectively, A) we mean supremum (respectively, infimum) with respect to the partial order 5. An orthomodular poset is a bounded poset (L, <,0, I), where 0 is the smallest element, 1is the largest element and 0 # 1,with orthocomplementation I: L + L satisfying

David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

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(OMP3) a

< bLL & a v b exists in L.

Any ort homodular poset is an orthocomplemented poset . An orthocomplemented poset which is at the same time a lattice is called an ortholattice while an 01-thomodular lattice is an orthomodular poset which is at the same time a lattice. DEFINITION 8. A quantum logic is a a-(ortho)complete orthomodular poset. That is, an orthomodular poset (L, <, 0,1, I)with the following property: for any family {a,I n E M) in L satisfying a, a* for every n , m E M,n # m, the supremum VnENa, exists in L.

<

c

Let us endow the above families with the partial ordering of set-inclusion and orthocomplementation I. Our next step is to analyze the algebraic structure of the said families. 3 ALGEBRAIC STRUCTURE OF P(S), C(S), E(S), E,(S), F(S) AND W(S) It is a well known fact that (P(S), c , {0), S, V, A) is a (bounded) modular lattice, where a (bounded) lattice (L, <,0,1, V, A) is said to be modular if for every a, 6, c E L we have (ML) a < b = + a V ( b ~ c ) = ( a v b ) ~ c . Let us now show that (C(S), C, {0), S,I)and (E(S), C, {0), S,I)are orthomodular posets. We will see later that in general both families need not be lattices. PROPOSITION 9. (1) If A, B E E(S) such that A

c B'-S, then A@B E E(S) and A @ B = AVE B.

(2) I ~ A , B E C(S) such thatA 2 BIS, t h e n A @ B C(S) ~ andA@B=AVcB. Proof.

(1) We show that for all d E S, we have:

Since A and B are splitting subspaces of S we have:

Algebraic and Measure-theoretic properties

...

Then

= (a

+ b) + (6 - b) = (a + b) + ( i - a);

and therefore

Hence A @ B E E (S) .

(2) We note that since A BIS, then A is complete when B is cocomplete. (Conversely, B is complete when A is cocomplete.) So we need only consider two cases. A and B are both complete; it then evidently follows that A @ B is complete. A is complete and B is cocomplete. (The case when A is cocomplete and B is complete is identical). By part (1) we know that A @ B E E(S) F(S) and therefore we have

Since ALs n BIS is complete, (AL" n BLs)'-s is cocomplete.

As already mentioned above, for a general inner product space S, E(S) and C(S) need not be lattices (see Section 3.2). In Proposition 9 we have shown that if A and B are two orthogonal elements in E(S) (respectively, in C(S)), then A V E B (respectively, A V c B) exists in E(S) (respectively, in C(S)) and is equal to A @ B. Moreover, since (A @ B ) I s = ALs f l BLs, we have ALs AE BLs (respectively, ALs A c B ~ S exists ) in E(S) (respectively, in C(S)) and is equal to A l s nB ~ S . We can now show that the two families are orthomodular posets. PROPOSITION 10. E(S) and C(S) are orthomodular posets.

Proof. We must verify the axioms (OMP1) - (OMP4) of Definition 7. S and (0) are in C(S) and E(S). If A & B are subspaces of S, then A1s BLs. Since C(S) & E(S) & F(S), A'-"Is = A for any subspace A in E(S) or C(S). Axiom (OMP3) follows immediately from Proposition 9. It remains to show that C(S) and E(S) satisfy (OMP4). Starting with E(S), let A G B be two splitting subspaces in S. We must prove that the infimum B AE ALs and the supremum A V E ( B AE A1s) exist and, moreover, we have the equality B = A V E ( B AE ALs). Since A 5 B , by Proposition 9 we have A @ BLs E E(S) and therefore,

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82

(A @ B ~ = ALs ~ nB )E E(S). ~ Thus ~ B AE ALs exists. Since B AE ALs G ALs , we have A@( B AE ALs) E E(S) and therefore, AVE ( B AE ALs) exists. Si~llilarly one can show that if A c B are in C(S), then AVc ( B Ac A'-") exists and is equal to A @ ( B n ALs). Finally we verify that B = A @ ( B n ALs). Since A B, we have B = B n (A@A1s) G ( ~ A)n@ (B n ALs) = A @ ( ~ rALs). l The converse inclusion is trivial. H Let us now show that (Eq(S),C, {0), S,I)need not be an orthomodular poset, in fact we have the following result. THEOREM 11. Eq(S) is orthomodular iJ and only if, E(S) = Eq(S). Proof. Let S be an inner product space. If E(S) = Eq(S) then Eq(S) is orthomodular since E(S) is always orthomodular by Proposition 10. Now suppose that Eq(S) is orthomodular and let M E Eq(S) \ E(S) for a contradiction. Then there exists a vector x E S which is not in M @ MIS. By Proposition 6, M [XI E Eq(S) and by hypothesis we have

+

+ [XI = M V ((M + [x]) A M I S ) . Since M + [XI # M , we cannot have ((M + [XI)r\ M I S ) = (0) and hence, there exists 0 # y E ( M + [XI)n MLs. Therefore, y = a x + m E MLs for some m E M . M

This implies that x E M @ MIS unless a = 0. The two implications lead to a contradiction and thus E(S) = Eq(S). H

c,

c,

We next show that (F(S), {0), S,V, A) and (W(S), {0), S,V, A) are complete lattices with smallest element (0) and largest element S. F(S) is also an ortholattice with respect to Ibut in general W(S) is not orthocomplemented. We will also see later that in general both families need not be orthomodular. PROPOSITION 12. W(S) and F(S) are complete lattices, where for the meet we = AF and i s given by have A =

Aw

while for the join we have

V M, = span (U M,) iEI

v

iEI

in w(S),

iEI

Mi = (span

(2

Mi))

lsls

in F(s).

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...

83

Proof. It is not difficult to show that W(S) is a lattice with the meet and join defined above. Now let {Mi I i E I) F(S). First we show that (niEI M i ) l s l s = Mi. For any j E I, (niEI Mi)ls MjLs, and therefore

c

niEI

>

Mj ,

for all j E I

iEI

Consequently

and hence

niEI

niEI

Mi E F(S) and therefore AiEI Mi = Mi. Thus Now (span(UiEI M i ) ) l s l s E F(S) and surely it is an upperbound of the collection {Mi I i E I). Let A E F ( S ) be such that A _> Mi for all i E I . Then

so that A=

2 span (U Mi)

lsls

.

iEI

Hence

Vie, Mi = (span(UiEIMi))lSLS.

rn

Although as already mentioned above, the structure on E(S) and C(S) need not be a (complete) lattice, the following propositions show that the infima and suprema in E(S) (respectively, in C(S)) are the same as in F ( S ) provided that these exist in E(S) (respectively, in C(S)). PROPOSITION 13. Let {Ai I i E I) C E(S). The following statements hold:

AiEIAi If ViEI Ai

(1) If

(2)

AiEI Ai = niEIAi. exists i n E(S), then ViEI Ai = (span(UiEI Ai))lSLS. exists i n E(S), then

Proof.

niEI

(1) Let C = AiEI Ai. Then C C_ Ai. If there is a vector v E S which belongs to (niEI Ai) \ C, then C + [v] E E(S) by Proposition 6. Since C [v] Ai for all i E I , we have that C # AiEI Ai which is a contradiction.

+ c

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

84

Then by De Morgan's laws, (2) Suppose that ViEIAi exists in E(S). AiEI A": exists in E(S) and moreover we have

=

niEIA":

and therefore

where for the last equality merely note that

njEIA":

=

By the first part, we have that

AiEIA":

(span(UiEI Ai))ls

PROPOSITION 14. Let {Ai I i E I) 5 C(S). The following statements hold:

AiEI Ai exists i n C(S), then AiEI Ai = niEIAi. ~f Vier Ai exists i n C(S), then Vier Ai = (span(UiErA ~ ) ) ~ s ~ s .

(1) If (2)

rn

Proof. The proof is analogous to the proof of Proposition 13.

How much does the algebraic structure of the above classes tell us about the inner product space S itself? Despite the fact that not every inner product space possesses an orthonormal basis (see [Gudder, 1975]), it is still possible (and useful) to define the orthogonal dimension of an inner product space as the cardinality of any maximal orthonormal system in S. However, in contrast to Hilbert spaces, this orthogonal dimension reveals very little information about the properties of the space itself. We recall the well-known fact that two Hilbert spaces are isomorphic if, and only if, they have the same orthogonal dimension. This is not the case with incomplete inner product spaces. Unless we assume completeness, isomorphism between two inner product spaces can only be proved by exhibiting a bijective unitary transformation between the two spaces. Let us recall that P(S1) is said to be isomorphic to P(S2) when there exists a bijective mapping : P(S1) t P(S2) such that:

+

(11) $(AL%) = (~,b(A))-'-~z, for all A E P(S1); (111) +(A v B) = +(A) v +(B), whenever A, B E P(S1) and A 5 ~ (IV)

+-l

satisfies (I), (11) and (111).

~

~

1

;

Algebraic and Measure-thwretic properties

.. .

85

In [ ~ u h a ~ iand a r Chetcuti, 20041 the following question was analyzed: Suppose that we are given two separable inner product spaces S1 and S2over W, such that P(S1) is algebraically equivalent to P(S2) as modular lattices. What can be said about S1 and S2? Using Gleason theorem, it was proved that in this case, S1 and S2are unitarily equivalent. This result was then extended by S. Pulmannovii [~ulmannovii,to appear] to non-separable inner product spaces over the complex field C or the division ring of quaternions W. The proof is based on results in projective geometry. One can look at this as a generalization of the classical Wigner's theorem [Wigner, 19591. THEOREM 15. Let S1 and S2be two inner product spaces over K,where K = W, cC or W. Then the following statements are equivalent: (1) S1 is isomorphic to S2 (as inner product spaces); (2) P(S1) is isomorphic to P(S2) (as modular lattices); (3) C(S1) is isomorphic to C(S2) (as orthomodular posets); (4) E(S1) is isomorphic to E(S2) (as orthomodular posets);

(5) Eq(S1) is isomorphic to Eq(S2) (as orthocomplemented posets); (6) F(Sl) is isomorphic to F(S2) (as complete lattices).

4 ALGEBRAIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES In this section we characterize complete inner product spaces by the algebraic structures of C(S), E(S), Eq(S), F(S) and W (S). In Subsection 4.1 we develop a completeness criterion which is a modification, given by Ptiik and Weber [Ptiik and Weber, 19771 of the original Amerniya-Araki theorem [Amemiya and Araki, 1966671. In Subsection 4.2, we consider the subsequent and successive strengthening of H. Gross and H. Keller's result that S is complete if, and only if, E(S) is a s Keller, 19771. The lattice properties of E(S) and complete lattice [ ~ r o s and C(S) are also investigated in Subsection 4.2 with respect to the completeness of S [Ptiik and Weber, 20011. In Subsection 4.3 we investigate the lattice properties of Eq(S) and also prove that for an inner product space S with d i m y / ~< CCJ we have Eq(S) = E(S) if, and only if, S is complete [Buhagiar and Chetcuti, to appear]. In Subsection 4.4, we define a modular/dual-modular pair in a lattice L and show that an inner product space S is complete if, and only if, in the lattice W(S) of closed subspaces of S, modularity and dual-modularity are equivalent [Holland, 19691.

86

1

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

The Amemiya-Aralci Theorem.

Inspired by a paper of Piron iron, 19641, Amemiya and Araki proved that if F(S) is orthomodular, then S is complete [Amemiya and Araki, 1966-671. This result initiated further research on the interplay between the algebraic and metric structure of inner product spaces. By introducing the notion of a weakly atomic orthomodular lattice, Ptkk and Weber arrived to the same conclusion of and Wethe Amemiya-Araki Theorem with a somewhat milder assumption [ ~ t & ber, 19771. DEFINITION 16. Let L be a poset. An element a E L is said to be an atom if the inequality b 5 a for b E L implies either b = 0 or b = a. A lattice L is called atomic if for every c E L, c # 0, there exists an atom a E L such that a 5 c. DEFINITION 17. Let L = (L, V, A, I,0 , l ) be an atomic orthocomplemented lattice. L is said to be weakly atomic orthomodular if the following condition is satisfied: If a , b L~a n d a i s a n atom, t h e n a V b = ( ( a ~~ ) A ~ - ! - Lb. ) v Obviously, any orthomodular lattice is weakly atomic orthomodular. However the notion of being weakly atomic orthomodular is in general strictly weaker than the notion of being orthomodular even for finite atomic orthocomplemented lattices. It is proved in [ ~ t b kand Weber, 19771 that an inner product space is complete if, and only if, F(S) is weakly atomic orthomodular. Let us prove this result. PROPOSITION 18. A n inner product space S i s complete if, and only iJ for every x, y E 3 satisfying s l y there exists A E E(S) such that x E 2 and y E -. Proof. We show that the condition is sufficient for the completeness of S. Let x E S\ S be a unit vector. There exists h E S such that (h, a) = 1. Put y = x - h. Then y l x and therefore by hypothesis, there exists a subspace A E E(S) such that x E 2 and y E G.Thus

h = hl + h 2 , where hl E A and h2 E ALs; and

which implies that hl completes the proof.

=x

and h2 = -y, and therefore x E S. The contradiction

PROPOSITION 19. Let M be a complete subspace o f 3 such that dim M T h e n M n S i s dense in M .

<

~ F 00.

Proof. We first prove the result for dim M 1 = ~ 1; that is MI^ = [el, where llell = 1. We can find h E S such that llhll = 1 and I(h,e)l > 112. Let x be any element of M . For any E > 0 there exists y E S such that ))x- yll < €13. Since x E M, we have I(Y,~)I = I(Y-x,e)l < €13.

Algebraic and Measure-theoretic properties

...

Let u = y - u h . Since (u, e) = 0, we have u E S n M. Now (be)

To prove the result for the general finite dimensional case, we apply the process . that M n S is dense in M when of induction on the dimension of M ~ YSuppose dim MLy 5 n - 1. Consider the case when M'-T = span{el, e2, . . .,en). We have

Let N = M @ [el]. Then 3 = N @ span{e2, e3,. . . ,en) and by the induction hypothesis, N r l S = N. Since M is a complete subspace of N and codimension of M in N is one, we have (NnS)nM=M. Consequently, S n M

= M,

that is S n M is dense in M .

rn

20. Let {vl, v2, . . . ,v,) be a finite set of orthonormal vectors in S. Then there exist sequences {upi I i E N), p = l , 2 , . . . ,n, in S such that

PROPOSITION -

lim upi = up ; 2

(upi, vq) = 0 , for each i E N when p # q ; (upi, uqj) = 0 , for each i,j E N when p # q

Proof. Let Hll = span{v2, 213,. .. ,vn)lg. By Proposition 19, there exists a unit vector ull E HllnSsuch that IIv1-u1111 < 112. Set Hzl = span{u11, VI, v3,. . . , v n ) l r and hence, again by Proposition 19, we can find unit vector u21 E H21n S such that llv2 - 2121 11 < 112. Put H31 = span{ull, 211,1121,212, v4, . . . ,v , ) l ~ and let u3l E H31n S be a unit vector such that llv3 - ~ ~ < 112. ~ 1 After ) repeating this step for n times we obtain an orthonormal family {upl 1 1 5 p 5 n) such that llvp - up1II < 112 and ( ~ p lup) , = ( ~ p luql) , = 0 for P # 9 . We proceed by induction. Suppose that the finite sequences of unit vectors {up!, up2, . . . ,upk), 1 I pI n, have been constructed in S such that llvp - 'llpjll < 1/23 and (upi, uqj) = (upi,vq) = 0 for all 1 5 i, j 5 k when p # q. Let

and let u1 k + l be a unit vector chosen from HI k + l 19, such that llvl - u1 k+lll < 1 / 2 ~ + l .Next, put

n S, in virtue of Proposition

Since v2 E Hz k + l , there exists a unit vector u2 k + l E H2 k + l r l S such that 1 1 ~ 2u2 k+lll < 112"'. After repeating this for n times, we obtain the orthonormal

88

David Buhagiar,

Emmanuel Chetcuti and Anatolij Dvurehnskij

family {up k + l l 1 5 p 5 n ) in S such that (up k + l , tiqj) = (up k + l , vq) = 0 for all 1 5 j 5 k + 1 when p # q and llvp - zip k + l l l < 1/2~+'. Thus, the required sequences can be constructed by induction. PROPOSITION 21. Let {x, y} be an orthogonal pair in M E F(S) such that x E 3 and y E

m.

S.

Then there exists

Proof. By Proposition 20, there exist sequences {xi I i E N} and {yi I i E N} in S such that limi xi = X, limi yi = y and satisfy x i l y j , s l y i and y l x i for all i,j E N. Then M = {yi I i E M}" E F(S) satisfies

With these results in hand, it becomes trivial to show that: THEOREM 22. An inner product space S is complete if, and only if, F(S) = E(S). THEOREM 23. If F(S) is weakly atomic orthomodular, then F(S) = E(S). Proof. Supposing it is not the case, let M E F ( S ) \ E(S). Then there exists x E S \ (M @ M I S ) ; [XI is an atom in F(S). Since x 6 M , we have [XI A M = (0) in F(S). By hypothesis we have

(([x] V M) A MI") V M = [x] V M = [XI

+ M,

where for the last equality we simply refer to Proposition 6. Since [x]+M # M , we cannot have (([x]vM)AM'~) = {O}, and therefore there exists y E ( [ X ] V M ) A M ~ ~ and y # 0. We then have

and hence y = a x + m E M I " for some m E M. This implies x M @ M I s unless a = 0. The two conclusions lead to a contradiction and thus E(S) = F(S).

rn We end this subsection with the following corollary which will be used in the sequel. COROLLARY 24. An inner product space S is complete if, and only if, for any orthogonally closed subspace M of S and any maximal orthonormal system {xi I i E I} of M we have M = span{xi I i E I}'-sLs. Proof. We only need to show sufficiency. Let A C B be elements of F(S) and suppose that {ai I i E I} and {cj I j E J} are maximal orthonormal systems in and ALs n B = A and ALs n B, respectively. Then A = span{ai I i E I}'-"~"

Algebraic and Measuretheoretic properties

.. .

89

span{cj I j E J ) ~ " ~One " . can easily verify that {ai, cj I i E I, j E J) is a maximal orthonormal system in B and therefore,

=AV (ALs n B). This implies that F(S) is orthomodular and therefore S is complete by Theorem rn 23.

4.2 Splitting Subspaces Criterion and the Lattice Properties of C(S) and E (S) . Unless S is a Hilbert space, neither C(S) nor E(S) can be a complete lattice. Indeed, H. Gross and H. Keller [Gross and Keller, 19771 proved that S is complete if, and only if, E(S) is a complete lattice. This was subsequently and successively strengthened as follows: S is complete if, and only if, (A) E(S) is a a-lattice [Cattaneo and Marino, 19861; (B) E(S) is a a-orthomodular poset [Dvure~enskij,19881;

(C) E(S) is atomic weakly a-complete [ ~ t &and k Weber, 19771. Recall that a poset L is said to be atomic weakly a-complete if for any sequence {a,I n E N) of atoms in L there is an infinite subsequence {hi I i E N) of {a,( n E N) such that the supremum ViENa,, exists in L. Since the atoms of E(S) are the one dimensional subspaces of S, it follows that S is complete if, and only if, C(S) is atomic weakly a-complete. 20011 that S is complete In addition to the above, it was shown in [~vure~enskij, if, and only if, E(S) satisfies the a-Riesz interpolation property. Let us elaborate on this and give a proof to the said result. A poset (P, 5 ) is said to satisfy (i) the Riesz interpolation property (RIP) if, for every al, az, bl, b2 E P with % 5 b j for i, j = 1,2, there exists an element a E P such that % 5 a 5 bj for i,j = 1,2; (ii) the a-Riesz interpolation property (a-RIP) if, for all sequences {R I i E N), {bj I j E N) in P satisfying a.j 5 bj for all i,j E N, there exists an element a E P such that a+ 5 a 5 bj for all i,j E N. It is clear that if P is a lattice (a-lattice), then P satisfies (RIP) ((a-RIP)) . THEOREM 25. An inner product space S is complete if, and only if, E(S) satisfies (a-RIP). Proof. With the aim of using Proposition 18, let x, y E 3 satisfy x l y . By Proposition 20 there exist sequences {xi I i E N) and iyiI i E N) in S such that limi xi = x, liw yi = y and satisfy x i l y j , x l y i and y l x i for all i, j E N. for all i,j E N Let Xi = [xi] and % = [yj]Lsfor all i, j E M. Then Xi for all and therefore, by (a-RIP), there exists A E E(S) such that Xi C A

s

s

90

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureeenskij

i, j E N. Thus xi E A and yj E ALs and hence, x E then follows from Proposition 18.

3 and y E ALs. The result rn

It is further noted in [~vure~enskij, 20011 that the proof of Theorem 25 can be used to prove the following criterion. THEOREM 26. An inner product space S is complete i;f, and only .I;f, for every two sequences {x, I n E N) and {y, I n E N) of vectors in S such that [x,] [ym]Ls for all n , m E N, there exists M E E(S) satisfying [x,] C M C [ym]Ls for all n , m E N. This result was further strengthened in [ ~ t and ~ kWeber, 20041 by proving that an inner product space S is complete if, and only if, E(S) satisfies the atomistic subsequential interpolation property. One can recall that an orthomodular poset P has the atomistic subsequential interpolation property if for any two sequences of atoms in P, {a,I n E N) and {b, I n E N), such that a, 5 bkp for all m, n E N, there exist infinite subsequences {a,, ( k E N) and {b,, I k E N) such that one can find an element a E P satisfying a,, 5 a 5 b k for each k E N. A natural question arising from the above results is whether E(S) can be a k Weber, 20011, lattice for S incomplete. This question was answered in [ ~ t h and by a careful analysis of the lattice properties of E(S) and C(S). An inner product space S was constructed such that S is a hyperplane of and E(S) = P(S), and thus E(S) (= C(S)) is a modular lattice. However, in [Gross and Keller, 19771, it was shown that for any inner product space having a countable (infinite) linear dimension, E(S) is not a lattice. Thus, the lattice properties of E(S) do not seem to have an explicit bearing on the metric completeness of S. Let us elaborate on this. We begin by showing that for any inner product space S possessing a countable linear basis, the orthomodular poset E(S) is not a lattice. Moreover, for such an inner product space, C(S) consists of all finite and cofinite dimensional subspaces of S and therefore C(S) is a modular lattice.

+

PROPOSITION 27. Let S = [f] lf C 12,where f = CiEN (see examples in Section 2). Then E(S) is not a lattice whereas C(S) consists of all finite and cofinite dimensional subspaces of S and therefore C(S) is a modular lattice.

Proof. Let Fl = span{e2, es, es, . . . ) and Fz = span{el, es, es, . . . ). It can be easily seen that Fl @ F2= lf and that F ": = F2and F$" = Fl. The rest can be established in a series of statements. CLAIM 28.

F2

+ [f]$ E(S).

CLAIM 29. Let

+ 2)e2,+2 1 n E N, n is odd) + 2)ezn + (2n)e2,+2 I n E N, n is odd).

A =span{(2n)ezn - (2n

B =span{(2n Then

Algebraic and Measure-theoretic properties

...

(i) f E ALs , (ii) A C BLs and moreover, A 63 B = Fl, (iii) A 63 ([f]

+ B + Fz) = S.

CLAIM 30. Let

C = span((2n + 2)e2,+2 - (2n + 4)e2,+4 I n E N,n is odd) D =[ez]

+ span((2n + 4)e2,+2 + (2n + 2)e2,+4

I n E N,n is odd).

Then again (i) f E CIS, (ii) C C DI" and moreover C 63 D = Fl, (iii) C63 ( [ f

+ D + Fz) = S.

CLAIM 31. B n D

=

(0).

CLAIM 32. Let K KnL=F2+[f].

=

[f] D

+ + F 2 and L = [f]+ B + Fz. Then K , L E E(S) and

Let us give a proof to this statement. Suppose that z E K n L. Then z = d x = b y, where x, y E F2 [f]. Then d - b = y - x E F2 [f]. From the definition of B, D and F2we immediately see that d - b = 0 and therefore b = d. Since B n D = (0) we have d = b = 0 and hence z E F2 [f]. Surely we have that F2 [f] K n L and therefore K n L = F2 [f]. In the light of Claim 1 above and Proposition 13 we conclude that K A L does not exist in E(S) and therefore E(S) is not a lattice. And finally,

+

+

+

+

+

+

+

CLAIM 33. C(S) consists of all finite and cofinite dimensional subspaces of S . In view of the fact that the linear dimension of a Hilbert space is either finite or uncountably infinite, the complete subspaces of S must be finite dimensional and hence C(S) consists of all the finite and cofinite dimensional subspaces of S . H Since any two inner product spaces Sl and S2with a countable linear dimension are isomorphic (as inner product spaces), and therefore E(S1) and E(S2) are isomorphic as orthomodular posets, it follows that E(S) is not a lattice for any inner product space S with a countable linear dimension. Moreover, C(S) consists of all finite dimensional and cofinite dimensional subspaces of S and therefore C(S) is a modular lattice. We next show that E(S) = C(S) for any linear hyperplane S in 3. PROPOSITION 34. If S is an inner product space which is a linear hyperplane in 3,then E (S) = C(S).

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvurehnskij

92

Proof. Let A E E(S) \ C(S). We show that if such a subspace exists, then d i m ( z / ~ )2 2. We haveS = A @ ALs, where neither A nor ALs is complete. Let a E 2 \ A and a' E ALs \ ALs. We show that {a + S,a' + S) is a linearly independent set in 3 1 s . Indeed, if cr(a+S)+p(af+S) = S, then aa+pa1 = s E S. This implies that a a + pa' = sl + s2 where 31 E A and s 2 E ALs, and therefore rn we have a a = sl and pa' = s z which is a contradiction unless a = P = 0. It is shown in [ ~ t i and k Weber, 20011 that for an infinite dimensional Hilbert space there is a dense hyperplane S in H such that E(S) (= C(S)) is not a lattice. On the other hand, in the same paper the authors show that in any separable Hilbert space H there is a dense linear hyperplane S such that E(S) is a modular lattice. We shall need the following lemma. LEMMA 35. Let H be a separable Hilbert space. Then (1) Card(H) = 2"0 . (2) Card{{vi)

I sequence in H ) = 2"o .

(3) Card{A

HI A is a complete infinite dimensional subspace of H )

= 2"o

.

THEOREM 36. Let H be a separable Hilbert space. Then H possesses a dense hyperplane S such that E(S) consists of finite or cofinite dimensional subspaces of S. Thus, E(S) is a modular lattice.

Proof. By Lemma 35 we know that the collection

IU = {A

H I A is a complete infinite dimensional subspace of H),

has cardinality 2"o. By the well-ordering principle, we can well-order the family IU. Let w be the initial ordinal number of cardinality 2"o. Then IU = { A , I a < w). We will construct a linearly independent family of vectors in H, {v, I a < w), such that for each a < w we have v, E A,. We proceed by transfinite induction. Let vl E A1 be an arbitrary non-zero vector. Suppose that the collection {up I P < a) has been constructed. Since the linear dimension of A, is 2"0 and since Card{@I P < a) < 2"0, there must be a vector v, E A, such that the family {up I P 5 a) is linearly independent. Thus we can construct the family V = {up I P < w) of linearly independent vectors of H with up E Ap. Extend this to a Hamel basis U for H. We can well-order the unbounded interval [I,co) in R to get {pa I a < w). Then define the linear map f on H into R by f (v,) = p, for all v, E V and f (u) = 0 for all u E U \ V. Then f is an unbounded linear functional on H and S = N(f) is a dense hyperplane in H. By construction, S does not contain a complete subspace of infinite dimension. rn

4.3 Quasi-Splitting Subspaces Criterion and the Lattice Properties of E4 (S)' The family Eq(S) was introduced and investigated in [Buhagiar and Chetcuti, to appear] as an intermediate between E(S) and F(S). We recall that whereas F(S)

Algebraic and Measure-theoretic properties ...

93

is always amply endowed with Boolean a-subalgebras, E(S) need not contain any Boolean a-subalgebra at all. The members of Eq(S) should behave somewhat similar to the splitting subspaces, however Eq(S) (at least when S is assumed to be separable) is always furnished with a "good supply" of Boolean a-subalgebras. Indeed, by the Gram-Schmidt orthonormalization procedure, we can convert any countable dense subset of S into an orthonormal basis {xi I i E N). Clearly, the image of the map

I' : ( I E 2")

t ,

( S n span{xi I i E I) E Eq(S))

is a Boolean a-subalgebra of Eq(S). Let us begin by giving a characterization of Eq(S) for an inner product space S.

37. Let Ml and Nl be orthogonal subspaces of 3 satisfying M1 @ PROPOSITION Nl = S. If Mo and No are subspaces of S such that Mo G MI and No 2 Nl, then Mo @ N o i s dense in S .if, and only if, Mo and No are dense in Ml and Nl, respectively. Consequently, for any closed subspace M of S , M E Eq(S) if, and only if, -Ig M I S = M (= ~ ~ 3 ) . Proof. Suppose that Mo @Nois dense in S and let us assume (for contradiction) that Mo is not dense in MI. Then we can find a nonzero vector z E M1 such that (z, x) = 0 for all x E Mo. Since z E MI, and M I I N l , we have that z is orthogonal with any vector in Mo @No,which implies that z E SIT= (0). The case when No is not dense in Nl is exactly the same and the converse is trivial. is dense in S if and only if MI" is So, for any subspace M of S, M @ MI" -IF and thus a closed subspace M of S is a quasi-splitting subspace if dense in M - -17 and only if MLs = M . rn Observe that the two inner product spaces considered in Examples 2 and 3 both have a countable linear dimension. Since, as already mentioned above, any two such inner product spaces are isomorphic (as inner product spaces), we have that E(So) g Eq(So) F(So) for any inner product space So with a countable linear dimension. We have already shown in Theorem 11that Eq(S) is orthomodular if, and only if, it is equal to E(S). Let us now investigate the lattice properties of Eq(S). It turns out that the lattice structure of Eq(S)is strongly dependent on the particular choice of the inner product space itself; Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. This is in contrast with both E(S) and F(S). THEOREM 38. If S be a dense hyperplane of a Hilbert space H , then Eq(S) = F ( S ) and hence, Eq(S) i s a complete lattice.

Proof. Suppose that Eq(S) F(S) and let M E F ( S ) be an orthogonally closed subspace which is not quasi-splitting in S. Thus, there exists 0 # x E H such

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

94

that x l M and XIMI". Since M is not in Eq(S) then both M and MI"are \ M . Observe that (x, y) = 0. Since S is a hyperplane of incomplete. Let y E H, there exist nonzero scalars a and ,B such that a x ,By E S. Then a x ,By E MI"'-" = M and therefore a x = (ax + py) - ,By E M.This contradicts the fact that x l M . Hence, Eq(S) = F(S).

+

+

In Proposition 34 it is shown that if S is an inner product space which is a linear hyperplane in 3,then E(S) = C(S). Thus, in this case Eq(S) is much larger than E(S). However, as shown in [~uhagiarand Chetcuti, to appear], Eq(S) need not be a lattice. In fact we have the following result. THEOREM 39. I f S has a countable linear dimension, then Eq(S) i s n o t a lattice. So the size of Eq(S) fluctuates drastically; can Eq(S) ever coincide with E(S) for an incomplete inner product space? The authors of [Buhagiar and Chetcuti, to appear] cite the following problem. PROBLEM 40. If Eq(S) = E(S), does it follows that S is complete? We end this subsection with a partial result to this problem. THEOREM 41. Let S be a n inner product space such that d i m s / ~< co. T h e n Eq(S) = E(S) if, and only if, S i s complete. Proof. Let {yo, yl, . . . ,ynPl) be orthonormal vectors in 3\ S such that

Let K = span{yl, yz, . . . ,Y,-~}. Since K I T n S is dense in KIT (by Proposition , exists h E K L n~S such that (h, yo) # 0. Let x = yo 19) and yo E K ~ Fthere h Then x E span{yo, yl, . . . ,y , - ~ ) ~ aand yo - x = h,vo By Proposition 20 we can construct sequences {xi I i E N) and {yki I i E N) (where k = 0,1, . . . ,n- 1) in S such that limi xi = x, limi yki = yk for each k = 0,1, . . . ,n - 1 and (xi,ykj) = ( ~ k i~, k ' j )= 0 for each i,j E N and k # kt. Let us show that M = {xi I i E N ) I "is ~aSquasi-splitting subspace of S. Suppose that this is not the case, i.e. M $ Eq(S). Now yk E MLs and yki E MIS for each k = 0,1,. . . ,n - 1 and i E N. There exists a unit vector z E 3 such that and z E =lS. Since d i m S / ~= n we can find scalars a,h,. . . ,,Bn-1 such that a # 0 and w = a z + ,Boyo+ . + ,Bn-lyn-l is in S. Then w E MI" and since yk E MLs for each k, we get z E MLs which contradicts the fact that z E *l". Now suppose that M is a splitting subspace of S. Then h = hl + hz where and yo E hl E M and hz E M I s . But h = (h, yo)yo- (h, yo)x where x E which implies that hl = -(h, yo)%and hz = (h, yo)yo which is a contradiction rn since x, yo $ S .

h.

m.

-

m,

COROLLARY 42. Let S be a n inner product space such that dirn;T/S = n < co. T h e n Eq(S) is orthomodular if, and only if, S i s complete.

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4.4 Closed Subspaces Criterion. It has already been shown that for an inner product space S, the family W(S) of closed subspaces of S, forms a complete lattice when ordered by set inclusion. Moreover, we have already seen that

V Mi

=span(u

Mi)

and

/\ Mi = nMi

for any family {Mi I i E I}C W(S). The completeness criterion of S determined by W(S) is dictated by the modularity property of the lattice [Holland, 19691. Let us start by showing that unlike the other families, W(S) is not complemented in general. PROPOSITION 43. An inner product space S is complete if, and only if, W(S) is complemented. Proof. If S is a Hilbert space, then every M E W(S) satisfies M @ M I s = S and thus I is a complementation on W(S). For the converse, suppose that S is not complete. Let x E 3 \ S and M = [ x ] l n~ S E W(S). By Proposition 19 we have M = [xlL3. Thus M I S = (0) and M V M'-s = M # S. Following the work of S. Holland [1969], we inquire modular and dual-modular pairs in W(S). We recall that the pair (a, 6) where a and b are elements of some lattice L is a modular pair (expressed as (a, b)M in L) when (cVa) A b = c V (aA 6) for all c 5 6. We say that (a, b) is a dual-modular pair (and write (a, b)M* in L) when

(c A a) V b = c A (a V 6) for all c 2 6. The dual-modular pairs of W(S) can be characterized as follows. The proof of Proposition 44 relies on the fact that for any closed subspace M of S and any vector x of S, M [XI is also closed in S (Proposition 6).

+

PROPOSITION 44. Let M and N be two closed subspaces of an inner product space S. Then (M, N)M* in W(S) if, and only if, M N is closed in S.

+

To characterize the modular pairs of W(S) we need to introduce the notion of completely disjoint subspaces. Two subspaces M and N of W(S) are said to be completely disjoint if there exists a > 0 such that for all x E M , IIxII = 1 and y E N we have Ilx - yll 2 a. LEMMA 45. Let S be an inner product space and let M, N be closed subspaces of S. The following conditions are equivalent:

(1) M and N are completely disjoint;

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David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

(2) There exists a

> 0 such that Ilxll + llyll 5 a11x + yll for

all x E M, y E N;

(3) M n N = (0) and the (well-defined) linear operators P(X+Y) = x, T(x+y) = y are both bounded o n the subspace M N .

+

PROPOSITION 46. If the closed subspaces M , N of a Hilbert space H satisfy M n N = (0) then M and N are completely disjoint if, and only .if, M N is closed.

+

Proof. Suppose that M and N are completely disjoint. Then there exists a such that for all x E M, y E N , we have

Let (xi

>0

+ yi) be a Cauchy sequence in M + N. Then

which implies that {xi) and {yi) are Cauchy sequences in M and N respectively. Therefore, there exist x E M and y E N such that limi xi = x and l i w yi = y. For the converse, suppose that M N is complete. Then the operators P and T defined by P ( x + y ) = x and T(x + y) = y are closed operators and therefore, by the Closed Graph Theorem, are bounded.

+

Note that the proof of Proposition 46 relies very strongly on the assumption that H is a Hilbert space. In fact, as we will see later on, the statement proved above holds only for complete inner product spaces. In the following proposition modularity is characterized in terms of complete disjointness. PROPOSITION 47. Let M and N be closed subspaces of a n inner product space S such that M n N = (0). Then (M, N ) M in W(S) if) and only if, M and N are completely disjoint subspaces of S. As a consequence of Propositions 44, 46 and 47 we have that when M , N are closed subspaces of a Hilbert space such that M n N = (0) then the pair (M, N ) is modular if, and only if, it is dual-modular. Using the orthomodular property of the projection logic of a Hilbert space it is possible to do without the assumption that M n N = (0) and hence deduce the following result. COROLLARY 48. Let M and N be closed subspaces of a Hilbert space H . T h e n (M, N ) M if, and only if, (M, N)Mf in the lattice of closed subspaces of H . In general, modularity and dual-modularity in the lattice W(S) are not equivalent. In fact, the main result of this subsection states that modularity is equivalent to dual-modularity in W(S) if, and only if, S is a Hilbert space. We shall prove the following lemma before stating the main result. LEMMA 49. An inner product space S is complete zf, and only if, M @ N is a closed subspace of S for any M, N E W ( S )such that M C N'-".

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Proof. We only need to show that the condition is sufficient for the completeness of S. Let x be a unit vector in S\ S. There exists h E S such that ( h , x ) = 1. Put y = x - h. Note that y E 3 \ S and (y,x ) = 0. By Proposition 20 there exist sequences {xi I i E N) and {yi I i E N) in S such that limi xi = x, limi yi = y and satisfy x i I y j , x i l y and y i l x for all i,j E M. Let M = s p a n s i n and N = M I s . Each of the vectors zi = xi - yi is in M @ N and limi zi = x - y = h E S. By hypothesis M @ N is closed in W ( S ) and therefore h E M @ N , i.e. h = m n where m E M and n E N. But h = x y, where x E M and y E Then x - m = - ( y - n ) , which implies that x = m and y = n and therefore x E S which is the required contradiction.

+

x.

+

THEOREM 50. An inner product space S is complete if, and only if, every modular pair is dual modular in W ( S ) .

Proof. Let M and N be any two orthogonal subspaces of W ( S ) . It is easy to check that M and N are completely disjoint. By Proposition 47 it follows that ( M ,N)M in W ( S )and hence (by hypothesis) ( M ,N)M* in W ( S ) .By Proposition 44 it follows that M @ N E W ( S ) .Result then follows from Lemma 49. 5 MEASURES ON P ( S ) , C ( S ) ,E ( S ) ,E,(S), F ( S ) AND W ( S ) If S is an incomplete inner product space, then the structure of closed subspaces resolves itself into different substructures. The aim of the rest of this survey is to study the measure-theoretic properties of these structures. A charge on any of the respective families introduced in Section 1is an additive real-valued function. Formally, if we let L be any of P ( S ) ,C ( S ) ,E ( S ) ,Eq(S)or F ( S ) ,a charge m on C is a map m : L + R, such that

m(AV B ) = m ( A )+ m ( B ) , whenever A, B E C and A I B .

(1)

We denote by Q(L)the topological linear space of all charges on C endowed with the topology of pointwise convergence on the elements of L. A charge m is said to be:

completely-additive, if equation (1) holds for any collection {Ai I i E I ) of pairwise orthogonal subspaces in C such that the supremum VierAi exists in L. Let R,,(C) be the subspace of all completely-additive charges on L. a-additive, if equation (1) holds for any sequence {Ai I i E N) of pairwise orthogonal subspaces in C such that the supremum Vie, Ai exists in C. Let Q,(C) be the subspace of all o-additive charges on C.

> 0 such that Im(A)I 5 k for each A in C; regular, if for every A in C and every E > 0 there exists a finite dimensional subspace M , contained in A, such that Im(A) - m(M)I 5 E . Let Q,(L) be bounded, if there exists k

the subspace of all regular charges on L.

David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

free or singular, if m(A) is zero for every finite dimensional subspace A in L. Let !2,(C) be the subspace of all free charges on L. A measure is a positive charge on C and a state s on L is a normalized (i.e. s(S) = 1) positive charge. The set of all states on C, denoted by S(C), is a convex subset of the cube [O,1IL. When endowed with the product topology, [O, 11' is a compact space, and since S(L) is closed in [O, I]', it follows that the state space of any of the respective families described above is a compact, convex topological space. 6 GLEASON AND DOROFEEV-SHERSTNEV THEOREMS The entire theory discussed in this section depends heavily on the classical result of A. M. Gleason leason on, 19571. By solving the problem that was originally posed ~ , A. M. Gleason did not only succeed in describing by G. Mackey [ ~ a c k e19631, all the a-additive states on L(H), where H is a separable Hilbert space - he has also revealed the very delicate interplay that exists between the measuretheoretic properties of L(H) and the geometric structure of H. Gleason Theorem for bounded a-additive charges was independently proved by A. N. Sherstnev [~herstnev,19741 and A. DvureEenskij [A. DvureEenskij, 19781, and extended for bounded completely-additive charges on non-separable Hilbert spaces by M. Eilers and E. Horst [1975], T. Drisch (19791 and S. Maeda [1980] (see also [~vure~enskij, 19921). THEOREM 51 (Gleason Theorem). Let H be a Hilbert space of dimension greater than 2. For any bounded completely-additive charge m on L(H) there exists a unique bounded self-adjoint operator T on H of trace class such that

REMARK 52. Observe that in the formulation of Gleason Theorem, we can r e place 'completely-additive' with 'a-additive' if, and only if, the dimension of H is 19861). a non-measurable2 cardinal (see [~vure~enskij, Observe that Gleason Theorem shows that every bounded completely-additive charge on L(H) can be extended to a (normal) linear functional on the algebra of bounded operators on H, here denoted by B(H). We recall that B(H) is the dual space of the Banach algebra consisting of all the trace class operators endowed with the trace-norm. This duality is determined by (T, A) = tr TA, where T is a trace class operator and A E B(H). That is to say, Gleason Theorem is in fact a theorem of non-commutative integration. It can be said that Gleason 2We recall that a cardinal a is said to be non-measurable if for any set X with cardinality a, there is no 0-additive probability measure on 2 X which vanishes identically on each singleton of X. For example, No and 2 N are ~ non-measurable cardinals.

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Theorem is one of the most far-reaching mathematical results of the 20th Century. Apart from the extensive impact on the mathematical foundations of quantum mechanics, Gleason Theorem provides a solid ground for non-commutative integration theory. The basis of the proof of Gleason Theorem lies in the treatment of the real three-dimensional Hilbert space. The original proof is rather complex. Many mathematicians and physicists tried to simplify the original proof of Gleason Theorem and to extend it to more general lattices and measures (for example, measures taking values in a Banach space). For a complete documentation and citation we refer the reader to the monographs [Dvure~enskij,19921 and [Hamhalter, 20031 that treat Gleason Theorem in full detail. The most elementary proof of this Theorem is due to R. Cooke, M. Keane and W. Moran [Cooke et al., 19851, which was first presented in 1984 at a conference in Cologne, Germany. It is worth noting that if we eliminate the assumption of completeadditivity (or a-additivity when H is separable), the technique developed by A. M. Gleason cannot be used. That is why Gleason Theorem for charges on the projection lattice of a Hilbert space has been established much later by E. Christensen [christensen, 19821 for positive charges and by L. J. Bunce and J. D. M. Wright [Bunce and Wright, 19941 for the general case. (For basic definitions and terminology regarding von Neumann algebras we refer to [ ~ a d i i o nand Ringrose, 1983; Sokr, 19791 whereas for a complete treatment of the Generalized Gleason Theorem we refer to [Hamhalter, 20031.) THEOREM 53 (Generalized Gleason Theorem). Let M be a von Neumann algebra with no direct summand of Type 12. Then each bounded charge on P ( M ) extends to a bounded linear functional on M. One of the main and widely used consequences of Gleason Theorem can be easily exhibited here. DEFINITION 54. Let S(H) be the unit sphere of a Hilbert space H . A function f : S(H) + [O, 11 is called a frame-function on H if (1) f(Xx) = f (x) for all x E S ( H ) and X E D,(2) there is a constant W(f) such that

for any orthonormal basis {xi I i E I) of H. It is clear that the completely-additive charges on L(H) are in one-to-one correspondence with the frame-functions on H and therefore, in view of Gleason Theorem, for every frame function f on S(H), there exists a unique bounded trace class operator T on H, such that f (u) = (Tu, u), for all u E S(H). This means that

100

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvuretenskij

which implies that f is uniformly continuous on S(H). As an immediate conse quence, observe that L(H) (dim H 3) does not admit any two-valued states. (This is related to the dispute on hidden variables in quantum theory [ ~ o c h e n and Specker, 19671.) We remark that Gleason Theorem does not hold for the case when H is two-dimensional - it is straightforward to check that L(R2) admits plenty of two-valued states. The Hahn-Jordon Decomposition Theorem is mathematical folklore in classical measure theory [Halmos, 1988, Section 291. Consequently, every completelyadditive (finite) measure on a sigma-field C is bounded. This is not the case if we replace C by the lattice of projections of a finite dimensional Hilbert space H. (See [Dvure~enskij,1992, Proposition 3.2.41.) However, S.V. Dorofeev and A. N. Sherstnev proved the deep and surprising result that if the Hilbert space is infinite dimensional then every completely-additive charge on L(H) is bounded [~orofeevand Sherstnev, 19901. We shall need the following notion.

>

+

DEFINITION 55. Let S be an inner product space. A mapping f : S(S) -+ R is said to be a frame-type function on S if: (1) for any orthonormal system {xi I i E I}in S, the series CiEI f (xi) converges; (2) for any finite dimensional subspace K of S, f 1 B(K) is a frame function on K. A frame-type function f on S is bounded if 1f (u)I 5 k for some k > 0 for all u E B(S). We can now present the remarkable result of S. V. Dorofeev and A. N. Sherstnev. THEOREM 56 (Dorofeef-Sherstnev). Any frame-type function on an infinite dimensional inner product space S is bounded. Clearly a frame function on a Hilbert space H is a frametype function. From Theorem 56 it follows that any completely additive charge m on L(H) (where dim H = co)is bounded on the atoms. By restricting m to L(N), where N is a three-dimensional subspace of H, we get a bounded charge m 1 L(N) on L(N). By Gleason Theorem, there exists a bounded symmetric bilinear form t~ on N x N such that m([x]) = tN(x, x) holds for all x E B(N) . Since every symmetric bilinear form is uniquely determined by its quadratic form, we can define a bilinear form t on H x H as follows: for any x, y E H , let N be any threedimensional subspace containing x and y, then put t(x, y) = tN(x,y). It is clear that t(x,x) = m([x]) for all x E S(H). Since m is bounded on the atoms, t is also bounded and therefore there exists a unique bounded self-adjoint operator T on H such that m([x]) = (Tx, x) for all x E B(H). The complete additivity of m entails that T is a trace class operator and m(M) = tr TPM for any M E L(H). Since T can be decomposed as the difference of two positive trace class operators, it follows that m can be expressed as the difference of two positive charges and thereby m is bounded.

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COROLLARY 57. Let H be a n infinite dimensional Hilbert space. Every completelyadditive charge o n L(H) i s bounded. In view of Corollary 57, the requirement of boundedness in the formulation of Gleason Theorem becomes superfluous for infinite dimensional Hilbert spaces. THEOREM 58. (Gleason-Dorofeev-Sherstnev) Let H be a n infinite dimensional Hilbert space and let m be a completely-additive charge o n L(H). T h e n there exists a unique bounded self-adjoint operator T o n H of trace class such that

7 IS EVERY REGULAR CHARGE ON L(H) COMPLETELY-ADDITIVE? We now investigate the relation between R,,(L(H)) and n,(L(H)). Of course, n,,(L(H)) S2,(L(H)) and if H is finite dimensional, then S2(L(H)) = O,,(L(H)) = S2,(L(H)). This is not the case when dim H is infinite. To exhibit a regular charge on L(H) which is not completely-additive we make use of the following theorem. (For the proof the reader can refer to [Chetcuti and DvureEenskij, 2003a, Theorem 3.81 and [ ~ h e t c u tand i DvureEenskij, 2004b, Theorem 3.11.) THEOREM 59. Let H be a n infinite dimensional Hilbert space. ( 1 ) Let m be a bounded charge o n L(H). Denote by X and p the infimum, and supremum of {m(A) I A E L(H)}, respectively. Then, Range(m) i s a convex set containing (A, p) .

(2) L e t s be a regular state o n L(H). For every infinite dimensional closed subspace A of H , w e have [O, 4 4 ) ) C {s(M) I M C A, dim M

< m ) C [O, s(A)].

REMARK 60. (1) Observe that in particular, if s is a state on L(H) and dimH = m , then Range(s) = [O, 11 (see [Dvure~enskijand Pt&k, 20021). If H is finite dimensional, the range of a state s on L(H) is not necessarily convex. Indeed, if H is a Hiibert space with dimension n < m , the state s(M) = dim(M)/n, M E L(H), is a state on L(H) with a discrete range. This is the only such state. (2) It is worth recalling that the range of an unbounded charge on L(H) need not be necessarily convex (see the charge constructed in Theorem 61). (3) Whereas part (1) of Theorem 59 remains valid if we replace L(H) with F(S), it fails in the case of E(S). Indeed, let S be an inner product space such that 0 < d i m s / ~= n < m. Then, the mapping s on E(S) defined by

David

Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

d i m p / ~ s ( M )= 7 , is a state taking at most n+ 1 values and vanishing on each complete subspace of S . (We note that it would be interesting to describe all states on E ( S ) when 0 < d i i s / ~< oo in a Gleason's type formula, see [ ~ h e t c u tand i DvureEenskij, 2004b, Exer. 8.51.)

THEOREM 61. Let H be a n infinite dimensional Hilbert space. Then, nc,(L(H)) i s a proper subset of n , ( L ( H ) ) . Proof. Let 93 = { x , I s E C) be a Hamel basis in R over the field of rational numbers. We can assume that x , > 0 for each s E C. Fix an element x,, E 93. Then every real number x E W can be uniquely expressed in the form

where a is a finite subset of C \ { s o ) and all p's are rational numbers. We define a ,!3,xs) = Pso discontinuous function 4 : W -+ Q by letting 4 ( x ) = ~ ( ~ , , x , , CsEa whenever x E W is expressed in the form of equation ( 2 ) . Let s be any regular state on L ( H ) . We show that 4 o s is a regular charge. Let E > 0 and A E L ( H ) be given. If 4 ( s ( A ) ) = 0, we take M = {0), which CsEu PSx,, where yields 14(s(A))- 4(s(M))I < E. So let 0 # s ( A ) = P,,xs0 p,, # 0. There is an integer n 2 1 such that l / n < E and x,,/n < s ( A ) . Then 0 < (p,, - l/n)x,, CsEa Psxs < s ( A ) . By (1) we can find a finite dimensional subspace M of A such that s ( M ) = (p,, -l/n)x,, +CsEu ,B8x,. Hence, \ @ ( s ( A )) 4(s(M))I= l / n < E which proves that 4 0 s is a regular charge on L ( H ) . However the charge 4 0 s is not bounded since by Theorem 59 the range of a bounded charge on L ( H ) is a convex subset of W and Range(4 o s) Q. Therefore, by Corollary 57, q5 o s is not completely-additive.

+

+

+

We conclude by giving a sufficient condition for a regular charge on L ( H ) to be completely-additive. First we prove the following elementary lemma. We prove it for E ( S ) as we will need to refer to it later on. LEMMA 62.

(1) Let m be a bounded charge o n E ( S ) . For any M E E ( S ) and 6 > 0 there exists a finite dimensional subspace Mo of M such that Im(N)I < E for every finite dimensional subspace N C M ~ M . (2) Let ml and mz be bounded regular charges o n E ( S ) such that m l ( A ) = m z ( A ) for every finite dimensional subspace A of S . T h e n ml = ma. Proof. (1) If the statement is false, then a sequence {Ni I i E N) of pairwise orthogonal finite dimensional subspaces of M can be found with the property Im(Ni)1 2 E (for each i ). This would lead to a contradiction in view of the fact that m is bounded and that one of the sets { i E N I m ( N i ) 2 E) and { i E N I m ( N i ) 5 - E ) is infinite.

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(2) Fix an arbitrary M E E ( S ) . By part (1) we can find finite dimensional subspaces Ml and M2 of M such that for any finite dimensional subspace A of M

A G M,IM =+- m l ( A ) < 614, and ACM ~ M m z ( A ) < 1214.

*

+

Let B = M1 M2. Since ml and m2 are regular it follows that Iml(MkM)l5 €14, lm2(MkM)15 ~ / 4 ,I ~ ~ ( M ; B5)€14 I and ~ m z ( M k5~ €14. ) ~ Then

Iml(M) - mz(M)I I Iml(M) - ml(B)I

+ Iml(B) - m2(B)I + Im2(B) - m2(M)I

+ Imz(M) - mz(M2) - m2(MkB)I

= I m l ( M ) - m l ( M 1 )- m l ( M f B ) I

5 l m l ( M k M ) l +I m l ( ~ f ~+)I Im 2 ( ~ ; " ) l +

Im2(~,'~>l

5 6

rn THEOREM 63. Let H be a n infinite dimensional Hilbert space and let m be a regular charge o n L ( H ) bounded o n the atoms (i-e. there exists K > 0 such that m([u]) < K for all u E B(H)). Then there exists a unique self-adjoint trace class operator T o n H such that m ( M ) = trTPM for all M E L ( H ) . Hence m i s completely-additive.

Proof. We can restrict m to finite dimensional subspaces of H and since m is bounded on the atoms we can use Gleason Theorem to obtain a bounded selfadjoint operator T on H satisfying that m ( [ x ] = ) ( T x ,x ) for all x E S ( H ) . We show that T is a trace class operator. We recall that T can be expressed as the difference of two positive operators T I and T2, and H can be split into two orthogonal subspaces H1 and Hz such that T1H2 = T2H1 = (0). Since T I is positive, to show that it is a trace class operator, it is sufficient to verify that C i E I ( T 1 x ixi) , is summable for just one orthonormal basis {xi 1 i E I ) in H . Let {xi I i E I o ) and { y j I j E Jo) be orthonormal bases of HI and H2 respectively. Then

Since m is regular, and because m is positive on all the finite dimensional subspaces of H I , it follows that m is positive (and therefore monotone) on L ( H l ) . Hence, for any finite subset I; of lo,we have 0 5 m ( [ x i ] )5 m ( H l ) . This m([xi]< ) w, and therefore T I is a trace class operator. The implies that 0 I CiEIo same can be shown for T2, and therefore it follows that T is a trace class operator. We can now apply part (2) of Lemma 62 to deduce that m ( M ) = tr T P M . H THEOREM 64 (Aarnes). L e t s be a state o n L ( H ) , where H has dimension greater than 2. Then s i s a convex combination of a completely-additive state sl and a free state s2, i.e. s = As1 (1 -X)s2.

+

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

8 MEASURETHEORETIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES The main objective of this section is the study of the interplay that exists between the linear topology of inner product spaces, ordered properties of specifk subspace structures and continuity of measures on them. It will be shown, for example, that for incomplete inner product spaces there is no hope of finding completely-additive charges on any of these subspace structures. This direction of research owes its beginning to J. Hamhalter and P. P t i k ha am halter and P t a , 19871 who proved that a separable inner product space S is complete if, and only if, S(F(S)) contains a c~-additivestate. This surprising result was after generalized for non-separable inner product spaces and completely-additive charges - see [~vure~enskij, 1992; Hamhalter, 20031 for a complete survey. Here we obtain these results as corollaries of Theorem 66. We remark that in the proof of this theorem [~uhagiarand Chetcuti, submitted], in contrast with the original proofs in [Hamhalter and Ptik, 1987; DvureEenskij, 1992; Chetcuti and DvureEenskij, 2003b], no use is made of Gleason Theorem. PROPOSITION 65. Let m be a charge on F(S). For any A, B E F(S), A C B, we have m(B) = m(A) + m(AL n B).

Proof. This follows directly from the following equalities:

As a consequence of the previous proposition observe that whenever A and B are elements of F(S) such that A C_ B and A I B = (0) then m(A) = m(B) for every charge m on F(S). THEOREM 66. Let S be an incomplete inner product space. Then, every charge o n F(S) i s free.

Proof. By Corollary 24, if S is incomplete, there exists a subspace M in F(S) and a maximal orthonormal system {xi I i E I}in M such that span{xi I i E I } ~ " ~ - " M . Let X = span{xi I i E I}lsLs. Then MIS 2 XIS. Let u be a unit vector in X I S \ M I S . Observe that u $! M @I M I s and that

s

Suppose that m is a charge on F(S) such that m([v]) # 0 for some unit vector v of S. Let U be a unitary operator on S such that U(u) = v and let ii.l be the charge on F(S) defined by k ( M ) = m(UM). Then ii.l([u]) = m([v]) # 0. By Proposition 65 we obtain ii.l(M) + fi([u]) = %(X) + ii.l([u]) = k ( X @I [u]) = k ( M which is a contradiction.

+ [u]) = fi(M), rn

Algebraic and Measure-theoretic properties . . .

105

COROLLARY 67. Let S be an inner product space. The following statements are equivalent: (1) S is complete; (2) F(S) admits a non-singular charge; (3) F(S) admits a regular non-zero charge; (4) F(S) admits a completely-additive non-zero charge.

When the Hamhalter-Ptbk Theorem was published in 1987 the problem of whether F(S) can admit a charge for an incomplete inner product space S, became eminent (see [Ptbk, 19881 and [Dvure~enskij,1992, Problem 4.3.121). Recall that if H is a Hilbert space, then 'S(L(H)) is the closure (in the topology of O(L(H))) of the convex hull of the vector states. However, the restriction of every vector state on F(S) is regular and therefore, for incomplete inner product spaces F(S) does not admit any vector state. This would somehow suggest that F(S) has no state restricting from L(H) and that F(S) could be a naturally-born example of a stateless complete lattice. Only recently it has been shown that contrary to one's intuition, the existence of a state on F(S) is not sufficient for S to be complete [Chetcuti and DvureEenskij, 2004a; Chetcuti and DvureEenskij, 2005a]. A wide class of incomplete inner product spaces for which the lattice of orthogonally closed subspaces admit a charge restricting from the projection logic of the completion was exhibited. DEFINITION 68. An inner product space S is said to be strongly dense (in its completion) if for every infinite dimensional closed subspace M of 3, we have M n s # (0). Every inner product space S satisfying d i r n s / ~< 2"0 is strongly dense (see [Chetcuti and DvureEenskij, 2005a, Proposition 21). In particular, every dense hyperplane of a Hilbert space is strongly dense. on the For every inner product space S we can define a binary relation B if there exist finite dimensional subspaces elements of F(S) as follows: A Nl, N2 of S such that A Nl = B N2. It is not difficult to check that is reflexive, symmetric and transitive, i.e. defines an equivalence relation on F(S). Let 5(S) denote the partition F(S)/-. For any A E F(S), let (A) denote the equivalence class containing A. Then ((0)) and (S) represent the classes of finite and cofinite dimensional subspaces of S, respectively. For any (A), (B) E 5(S), we say that (A) is less than (B) and write (A) 5 (B), if for every C E (A) there exists D E (B) such that C C D. Observe that (A) 5 (B) if, and only if, there exists C E (A) and D E (B) such that C C_ D. It is easily verifiable that 5 is reflexive and transitive. Moreover, if (A) 5 (B) and (B) 5 (A), then we can find subspaces C E (A) and D E (B), such that A D 5 C . Since dim CIA < oo, it follows that A D, and therefore D E (A). This implies that (A) = (B), i.e. 5 defines a partial order on 5(S). Indeed, it can be shown that 5 induces on 5(S)

+

-

-

-

+

-

-

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a lattice structure homomorphic to the lattice structure of F ( S ) . Moreover, for strongly dense inner product spaces we have the following theorem [Chetcuti and DvureEenskij, 2005a, Theorem 11. THEOREM 69. Let S be an inner product space such that it is strongly dense in its completion. Then 5 ( S ) is an orthomodular lattice. Furthermore, 5 ( S ) is isomorphic to (as orthomodular lattices).

5(s)

Since when S is incomplete, every charge on F ( S ) must vanish on the finite dimensional subspaces, the charges on F ( S ) are in one-to-one correspondence with the charges on Z ( S ) and hence we have the following theorem [ ~ h e t c u t i and DvureEenskij, 2005a, Theorem 21. THEOREM 70. Let S be an incomplete, strongly dense inner product space. There is a linear homeomorphism q5 : m o m @between R ( F ( S ) )and R f ( L ( H ) )(where H = 3). Each charge m on F ( S ) is the restriction of m@ in the sense that m ( M ) = m+(%), for all M E F ( S ) . The original problem (see [ ~ t & k19881 , and [~vure~enskij, 1992, problem 4.3.121) is now reduced as follows: PROBLEM 71. Does there exist an incomplete inner product space S for which F ( S ) is stateless ? Although F ( S ) does not admit any regular charge for incomplete S , E ( S ) always allows for a separating set of regular bounded charges. For any self-adjoint traceconsider the map class operator T on

s,

r n :~E ( S ) + [O, 11,

where

M

ct

tr T%.

(1)

It is clear that equation (1)defines a bounded charge on E ( S ) . We prove regularity. There exists a Let M be in E ( S ) and {yi I i E I ) be an orthonormal basis of finite subset I0 of I such that

z.

Let K E N be the cardinality of lo.By Proposition 20 there exists a finite orthonormal system {xi I i E I o ) in M such that llxi - yill < E/(~KIITII) for each

Algebraic and Measure-theoretic properties

.

i € I o Then

i.e. mT is regular. In the following theorem we show that every regular bounded charge on E(S) arises in this way. THEOREM 72. Let S be an inner product of dimension at least 3 and let m be a regular bounded charge on E(S). There exists a unique self-adjoint trace class operator T on 3 such that m(M) = trT* for all M E E(S). Proof. Restrict m to E(N), where N is any subspace of S with a finite dimension greater than 2. By Gleason Theorem, there exists a bounded Hermitian conjugatebilinear form t N on N x N such that m([x]) = tlv(x,x) for every unit vector x of N . Define a conjugatebilinear form t on S x S as follows: t(x, y) = tN(x, y), where N is any subspace of S with a finite dimension greater than 2 that contains x and y. In view of the polarization identity of Hermitian conjugatebilinear forms, it is clear that the definition of t depends only on x and y, i.e. t is well defined. Since m is bounded, t is also bounded and therefore t extends continuously to a unique Hermitian conjugate-bilinear form, again denoted by t, on 'S x 'S. Let T denote the unique self-adjoint operator on 'S such that t(x, y) = (Tx, y) for all x, y E 'S. We claim that T is a trace class operator. Let {yi I i E I ) be any orthonormal basis of 3.Let I. be a finite subset of I . By Proposition 20, for any preassigned positive number 6 we can find a finite orthonormal system {xi I i E 10) in S such that

This implies that

.

Since m is bounded, it follows that CiEI(Tyi, yi) is summable, i.e. T is a trace class operator. From part (2) of Lemma 62 it follows that m(M) = tr T*.

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We remark that Theorem 70 can be viewed as the generalization of Gleason Theorem for incomplete inner product spaces. It is straightforward to check that the E ~ ( 3defines ) ) an embedding (as orthomodular mapping I? : (M E E(S)) I+ posets) of E(S) into ~ ( 3 )Theorem . 70 implies that every regular bounded charge on E(S)can be lifted to a normal functional on the algebra B@) of bounded o p erators on 3 . However, it is still not known whether this holds for every bounded charge on E(S) .

(z

PROBLEM 73. Is it possible to extend every bounded charge on E(S) to a linear functional on ~ ( s ) ? In Theorem 63 we have seen that if S is a Hilbert space then, the bounded regular charges on E(S) are precisely those that are completely-additive. This is not the case when S is incomplete. Although E(S) has always an ample supply of bounded regular charges, no completely-additive charge can exist on it unless S is complete. THEOREM 74 (DvureEenskik, 1995, Theorem 3.3). A n inner product space i s complete if, and only if, there exist a frame-type function f : B(S) -+ W and a constant k > 0 such that C l f ( ~ i )Ll k iEI

for every maximal orthonormal system {xi I i E I ) in S. COROLLARY 75. A n inner product space S i s complete i f , and only if, there i s a unit vector x E S and k > 0 such that, for any maximal orthonormal system {xi I i E I ) in S, 0 < k
holds. I n particular, S i s incomplete if, and only if, for any unit vector x,

where the infimum i s taken over all m a i m a l orthonormal systems {xi I i E I ) of S. COROLLARY 76. Let S be a inner product space. The following statements are equivalent: (1) S i s complete;

(2) Eq(S)admits a completely-additive non-zero charge; (3) E(S) admits a completely-additive non-zero charge; (4) C(S) admits a completely-additive non-zero charge;

Algebraic and Measure-theoretic properties

. ..

(5) P(S) admits a completely-additive non-zero charge. Proof. It is straightforward to verify that (1) (2), (2) & (3), (3) =+ (4) and (4) =+ (5). Suppose that m is a completely-additive charge on P(S). First we assume that m(S) # 0. Define f : S(S) + W by f (x) = m([x]). Then f is a frame-type function on S and for every maximal orthonormal system {xi I i E I ) in S we have

and therefore, by Theorem 74, S is complete. If m(S) = 0, find a unit vector u in S such that m([u]) # 0 and let So= [ u ] l s .Then m(S0) # 0 and by the previous argument, Soand therefore S, is complete. 9 CONVERGENCE OF CHARGES The aim of this section is to study some of the basic principles of measure theory in the set-up of the projection logic E(S). Namely, we will investigate possible generalizations of the Nikodfm Boundedness and Convergence Theorems and of the Vitali-Hahn-Saks Theorem. THEOREM 77. (Nikodfm Boundedness Theorem) Let H be a n infinite dimensional Hilbert space. Let {mi I i E I ) be a collection of completely-additive charges o n L(H) such that SUP Imi(M)I < iEI

for each M E L(H). T h e n {mi I i E I } i s uniformly bounded o n L ( H ) . Proof. For every i E I there exists a trace class operator Ti on B ( H ) such that mi = t r T j P ~for every M E L(H). Let fi denote the normal functional A H trT,A on B(H). Clearly, result would follow immediately from the BanachSteinhaus Theorem if we show that {fiI i E I ) is pointwise bounded on B(H). Let 0 5 A 5 I . By the Spectral Theorem A = CnEN +pn for a sequence of commuting projections in L(H). By the classical Nikodjrm Boundedness Theorem there exists k > 0 such that

and hence supiEI I fi(A)I < k. Since every operator can be decomposed as a linear combination of four positive elements, it follows that {fi 1 i E I ) is pointwise bounded on B(H) . The situation in the case that S is incomplete is still not clear - indeed the following problem is still open. Of course, when S is incomplete we replace the

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completely-additive charges with the bounded regular ones. Observe that in view of Theorem 72 every such charge is determined by a trace class operator. PROBLEM 78. Let S be an incomplete inner product space. Suppose that a sequence {mi I i E N) of regular bounded charges on E ( S ) converges to some charge m. Is m bounded? We recall that Nikodfm Convergence Theorem asserts that the pointwise limit of a sequence of a-additive measures {pi I i E N) on a a-field C is a a-additive measure and the set {pi I i E N) is uniformly countably additive (i.e. for any disjoint sequence { X k : k E N) in C, and for any E > 0, there exists K E N such that 1 C k > Kpi(Xk)l < E for each pi). We further remark that convergence theorems of completely-additive charges on L ( H ) , were first studied by R. Jajte [Jajte, 19721. THEOREM 79 (Nikodfm Convergece Theorem). Let H be a n infinite dimensional Hilbert space. Let {mi I i E N) be a sequence of completely-additive charges o n L ( H ) converging pointwise to some charge m E S1(L(H)). T h e n m i s completelyadditive. Moreover, the sequence {mi I i E N) i s uniformly completely-additive. Proof. Let {Mk I k E I ) be a collection of mutually orthogonal closed subspaces

of H. We can find a countable subset fi of I such that m i ( M k ) = 0 for every k E I \ Il and i E N. Each mi induces a a-additive measure on 2'1 as follows: pi : (J C I l ) mi(VkEJ Mk). Clearly, {pi 1 i E N) converges pointwise on 2'1 and hence, by the classical Nikodfm Convergence Theorem, for every E > 0 we can find a finite subset I. f Il such that

for every mi, i.e. the set { m i ( i E N) is uniformly completely-additive and therefore, the limit charge m is completely-additive. We shall now investigate the same problem for the case when S is incomplete. As we shall see, this case does not allow for such a clear answer. We say that a sequence {mi I i E N) of regular bounded charges on E ( S ) is uniformly regular if for any M E E ( S ) and for any E > 0, there exists a finite dimensional subspace Mo of M such that Imi(M) - mi(Mo)l < E for each mi. As a consequence of Theorem 63 and Theorem 79 we have the following version of the Nikodfm Convergence Theorem. COROLLARY 80. Let H be a n infinite dimensional Hilbert space. Let { m i I i E N) be a sequence of regular bounded charges o n L ( H ) converging pointwise t o some charge m E S1(L(H)). T h e n m i s bounded and regular. Moreover, the sequence {mi / i E N) i s uniformly regular. In the following theorem it is shown that the set of regular bounded charges on E ( S ) need not be sequentially closed in S1(E(S))for an incomplete S - not

Algebraic and Measure-theoretic properties

...

111

even if we restrict ourselves to states. For incomplete inner product spaces, the projection logic can be extremely poor and consequently the pointwise topology on n(E(S)) can be very coarse. This provides a negative answer to the problem posed by A. DvureEenskij (see [Dvure~enskij,1992, Problem 4.3.151). THEOREM 81. Let H be a n infinite dimensional, separable Hilbert space. There exists a dense hyperplane S of H such that the set of regular states on E(S) i s not sequentially closed in S(E(S)). Proof. In Theorem 36 a dense hyperplane S of H was constructed such that E(S) = P(S). Let {ei I i E N} be an orthonormal basis of S and let si : E(S) + [O, 11, M H l + i 1 12 (i E N) be the associated vector states on E(S). We claim that {si I i E N} converges (pointwise) to the unique twevalued state s in S(E(S)) assigning 0 to each complete subspace and 1 to every cocomplete subspace of S . Let u be a unit vector of S. Then limi si([u]) = liw I(u, ei)I2 = 0. This implies that limi si(M) = 0 for every finite dimensional subspace M of S, i.e. s is the unique free state in S(E(S)).

We now give a sufficient condition under which the limit of a convergent sequence of regular bounded charges on E(S) is regular and bounded. First of all, observe that the limit of a convergent, uniformly regular sequence of regular charges on E(S), is necessarily regular. The following proposition will be used in the proof of Theorem 83. It was first proved by R. Jajte [Jajte, 19721. Here we give a simpler proof without using Schur's Theorem. PROPOSITION 82 (Jajte). Let H be an infinite dimensional Hilbert space. The sequence {mi I i E N} of regular bounded charges o n L(H) converges pointwise if, and only zf, the following two conditions hold: (1) {mi([x]) I i E N} converges for each unit vector x of H; ( 2 ) for any orthonormal sequence {xk I k 6 N} in H , the series CkEnmi([xk]) converges uniformly with respect to i.

I n such case, the limit charge m i s regular and bounded, and the sequence {mi I i E N} i s uniformly regular. Proof. The necessity follows directly from Nikodfm Convergence Theorem (Theorem 79). Suppose that (1) and (2) are true. Fix Y in L(H) and let {yk I k E I} be an orthonormal basis of Y. We can find a countable subset Il of I such that for each mi, mi([yk])= 0 for all k E I \ Il. In view of condition (2), for any given positive E , there exists a finite subset loof 4, such that for all i E N, we have

In view of condition (I), let Q E N satisfy

112

for any p

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

> q 2 Q.

Then, for any p

> q 2 Q, we have

This implies that {mi I i E N) is pointwise convergent on L ( H ) . Given a positive regular charge mo on E(S), we say that the bounded regular charge m is absolutely continuous with respect to mo if, for every E > 0 there exists 6 > 0 such that Im(M)I < E, whenever M E E(S) and mo(M) < 6. Furthermore, we say that m is atomic absolutely continuous with respect mo if, for every E > 0 there exists 6 > 0 such that Im([x])l < E, whenever x E S(S) and mo([x]) < 6. A sequence {mi I i E N) of bounded regular charges on E(S) is uniformly absolutely continuous with respect to mo if, for every E > 0 there exists 6 > 0 such that Imi(M)I < E for every mi, whenever M E E ( S ) and mo(M) < 6.

THEOREM 83. Let S be a n inner product space and let {mi I i E N) be a sequence of bounded regular charges o n E(S) converging pointwise t o m. Suppose that there exists a positive regular charge mo o n E(S) such that every mi i s atomic absolutely continuous with respect to mo. T h e n m i s bounded and regular and the sequence {mi ( i E N) i s uniformly regular. Proof. Denote by Ti (i = 0,1, . . . ) the corresponding bounded self-adjoint trace for all M E E(S). class operator on 3 associated with mi, i.e. mi(M) = trTj* First we prove that the sequence {('I'z, z) I i E N) converges for each unit vector z of 3. From the following inequalities

it follows that llTill 5 (lTollfor all i = 1,2,. . . . Fix a unit vector z in 3 and let 6 > 0 be arbitrary. There exists a unit vector x of S such that 11% -211 < (llTo11c)/6. In addition, we are guaranteed that for some I E N, I((Tp- Tq)x,x)I < €13 for all p > q 2 I. Thus, for p > q 2 I, we have

and therefore {(Tiz, z) I i E N) is convergent.

Algebraic and Measure-theoretic properties .. .

Let {zk I k E N) be an arbitrary orthonormal system in (TOzk,zk)l < 00,there exists a positive integer K such that

113

S.

Since

CkEN 1

for any 6 > 0 and i E N. If we denote by riii the the charge on E(S)defined by fii(N) = tr TipN,for N E E(S),then it is clear that the sequence {riii I i E N) satisfies conditions (i) and (ii) of Proposition 82. This implies that {riii I i E N) converges pointwise on E(S) to some bounded regular charge r72 on ~ ( 3 ) It . is clear that m is the restriction of f i , i.e. m(M) = r72(M) for all M E E(S). The uniform regularity of {mi I i E N) follows from that of {riii I i E N). Indeed, for any M E E(S) and E > 0 there exists a finite dimensional subspace Mo of % such that Iriii(M) - riii(Mo)l < E for each i. Using Proposition 20 a finite dimensional subspace N of M can be found with the property that Imi(M) - mi(N)( 5 6 for rn each i. Let C be a a-algebra of subsets of a set X and denote by ca(C) the linear space of all a-additive measures on C endowed with the topology of pointwise convergence. Suppose that p is a positive a-additive measure on C. Then, the Vitali-Hahn-Saks Theorem asserts that if K ca(C) is a relatively compact set such that each element of K is absolutely continuous with respect to p on C, then K is uniformly absolutely continuous with respect to p (see [ ~ u n f o r dand Schwartz, 19581). In [chetcuti and Hamhalter, to appear, Example 4.51 it was shown that this cannot be carried over directly to the non-commutative case. In fact, the Vitali-Hahn-Saks Theorem fails for L(H) when dim H is infinite. This demonstrates a different character of non-commutative measure theory as far as charges are concerned. However, in [Chetcuti and Hamhalter, to appear, Theorem 4.61 it is proved that if we restrict to positive charges then the Vitali-Hahn-Saks Theorem holds for L(H). We thus have the following theorem. THEOREM 84 (Vitali-Hahn-Saks Theorem). Let H be a n infinite dimensional Hilbert space and let {mi I i E N) be a pointwise convergent sequence of positive regular charges o n L(H) such that such that each mi i s absolutely continuous with respect to some regular state so. Then, {mi I i E N) is uniformly absolutely continuous with respect to so. We conclude this section by exposing another contrast between L(H) and E(S) when S is incomplete. The Vitali-Hahn-Saks Theorem can fail in the E(S) setup even when we restrict to states. The inner product space that works out to be a counter-example for this is the same one considered in Theorem 36, i.e. E(S) = P(S). Let {siI i E N) be the sequence of vector states corresponding to the orthonormal basis {ei I i E N) of S . Put s = $si. Observe that each si is absolutely continuous with respect to so and {si ( i E N) converges pointwise to the state assigning 0 to finite dimensional subspaces and 1 to the co-finite dimensional ones. However, it is not difficult to check that {siI i E N) is not uniformly absolutely continuous with respect to so.

xiEN

David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

10 INFINITEVALUED MEASURES Here we present an overview of some of the results concerning measures on the projection lattice of a Hilbert space that can take also infinite values. An example of such a measure would be the dimension function. A mapping p : L(H) + [0,m] is said to be a measure if p(M @ N) = p(M) p(N) for all M, N E L(H) such that M I N . As we did with charges in Section 5, we can distinguish between the notions of (1) completely-additive, (2) a-additive and (3) regular measures. Observe that here by a regular measure we mean that given E > 0 and M E L(H), there is a finite dimensional subspace N of M such that p ( ~ Nn ~ H<) E . A measure p is afinite if there is a countable system of mutually orthogonal subspaces {Mi I i E N), Mi and p(Mi) < m for every i E N. such that H = VdEN For measures we can find a special formula, resembling the standard Gleason formula such that, when we restrict this formula to finite a-additive measures, we obtain the classical Gleason formula. A crucial notion, which substitutes trace class operators, is the notion of densely defined bilinear forms. In contrast to aadditive charges, there is no oneteone correspondence between the a-finite and a-additive measures and bilinear forms - not every bilinear form induces a a-finite and a-additive measure. A mapping f : B(H) + [0,m ] is a generalized frame function o n H if: (1) f (Ax) = f (x) for all x E B(H) and X E D; (2) there exists w E [0,m ] such that CiEr f (xi) = w for every orthonormal basis {xi I i E I ) of H; (3) for any closed subspace K of H , the restriction of f to B(K) is again a generalized frame function on K . It is straightforward to verify that the set of completely-additive measures on L(H) is in one-bone correspondence with the set of generalized frame functions on H. (We can view this as an extension of the one-teone correspondence that exists between the set of charges on L(H) and the set of frame functions on H.) Given a measure p on L(H) we define S, to be the subset of H consisting of vectors x such that p([x]) < m . In [Lugovaja and Sherstnev, 19801 G. D. Lugovaja and A. N. Sherstnev proved that if p is a a-finite measure on L(H), then S, is a dense subspace of H, see also [~vure~enskij, 1992, Section 3.41. The following lemma for infinite measures is of similar importance as that for finite states in the j a Sherstnev, 19801 (for an case of It3, and it was proved originally in [ ~ u ~ o v aand alternative prove see also [Chevalier et al., 20001).

+

>

LEMMA 85. Let Hn be an n-dimensional Hilbert space, where n 3. Let p be a measure o n L(Hn) such that p(Hn) = m and suppose that P i s a n ( n - 1)dimensional subspace such that p(P) < m. If q is a unit vector such that p([q]) < m , then q E P.

Proof. First assume that Hn = It3. Suppose that q is a unit vector making an acute angle 0 < a with the plane P such that p([q]) < m . We can choose a coordinate system as follows. Let XI be the unit vector bisecting the angle a. Let x2 be a unit vector orthogonal to X I and q. Observe that x2 E P . Let x3 be a

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115

unit vector orthogonal to xl and x2. Since 2 2 is orthogonal to q, it follows that the plane containing x2 and q has a finite measure. Denote this plane by Q. Put

F = {x E t5(iR3) 1 3

q') E P x Q such that pllq' and x E span{pl, q')).

Using the geometry of the sphere, one can show that F contains all the unit vectors with the absolute value of the latitude (with respect to the xlxz-plane) less than or equal to a. Hence we can find a unit vector ql and a plane PI such that p([ql]) < co, PI) < oo and the angle between ql and PI is 2a. One can repeat the same argument and deduce the p([x]) < oo for all x in B(W3) having absolute value of latitude equal to 2a. Therefore, we can repeat the above inductively and conclude that 4W3) < co, which is a contradiction. Now, we consider the case when H, = (C3. Let yo be a unit vector perpendicular to P and yl, y2 be two orthogonal vectors in P such that (q, yi) is real for i = 0,1,2. Let M be the threedimensional completely-real subspace generated by {yo,y1, y2}. Then q E M. The measure p induces a measure fi on L ( M ) by letting P([x]) = p([x]) for every x E B(M). Observe that fi([yo]) fi([y~]) fi([y2]) = oo and therefore, from the above arguments, it follows that q = Xlyl X2y2, where Xi E W, i.e. q E P . To prove it for every n > 3, one can apply induction. Suppose result is true for n < k and let P be a (k - 1)-dimensional subspace of Hk such that p(P) < oo. Let q be a unit vector with p([q]) < oo, and u a unit vector in P orthogonal to q. Put Hk-1 = [uILHkand P' = [u]IHkn P. Observe that q E Hk-1 and dim P' = k - 2. By induction hypothesis it follows that q E P'.

+

+

+

If t is a symmetric (not necessarily bounded) bilinear form and P is a projection on H such that P H D(t) then, we can define the symmetric bilinear form t o P on H x D(t) as follows: t 0 P(x, y) = t(Px, y). We write t o P E Tr(H) if t o P is bounded, and the associated bounded self-adjoint operator is a trace class operator. Moreover, we denote by tr t o P the trace of this operator, i.e.

where {xi I i E I}is any orthonormal basis of H . Using Lemma 85, it is possible to prove the following theorem. THEOREM 86 (Lugovaja and Sherstnev, 1980). Let H be a Hilbert space of dimension greater than 2. Let p be a n-finite, n-additive measure. The following assertions are equivalent: (1) There exists a (unique) positive, symmetric bilinear form t with the domain D(t) dense in H such that trtoPM iftoPMETr(H), otherwise.

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvurehnskij

(2) p has a support, i.e. a closed subspace M, of H such that p(N) = 0 if, and only if, N IM,. (3) p is a completely-additive measure. EXAMPLE 87. Any positive self-adjoint operator T defines by equation (1)a afinite and completely-additivemeasure, where t is defined by t(x, y) = (T'/'x, ~ ' 1 ' ~ ) with D(t) = D(T'/~). We now exhibit a difficulty that arises when we use bilinear forms that are not determined by positive self-adjoint operators. EXAMPLE 88. Let H be an infinite dimensional separable Hilbert space. Let {en I n E N) be an orthonormal basis, which is a part of a Hamel basis {ei I i E I)of H . Fix an element ei, which does not belong to the orthonormal basis {en I n E M) and define a linear operator T : H 4 H and a linear functional f : H 4 cC via

and

where cri, is the scalar corresponding to eb in the decomposition x = CiEI ai ei, x E H, with respect to the given Hamel basis. If we set t(x, y) = (Tx, Ty), then t defines via equation (1) a a-finite measure that is not a-additive.

G.D. Lugovaja [1983] showed that on a separable Hilbert space, a singular positive symmetric bilinear form does not define a a-finite, a-additive measure by equation (1). For that reason the following important result was proved by B. Simon [1978]: THEOREM 89. Let t be a densely defined positive symmetric bilinear form on a Hilbert space H. Then there exist two positive symmetric bilinear forms t, and t, with D(t,) = D(t) = D(t,) such that

where t, is the largest closable bilinear form less than t. THEOREM 90 (Lugovaja, 1983). Let H be an infinite dimensional Hilbert space and let t be a singular, positive symmetric bilinear form with dense domain D(t). Then, the mapping p : L(H) 4 [0,ca]defined by equation (1) is not a a-additive measure on L(H). The affirmative answer is also due to G. D. Lugovaja:

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THEOREM 91 (Lugovaja, 1983). Let H be an infinite dimensional Hilbert space. A positive symmetric bilinear form t with a dense domain D ( t ) defines through equation (1) a a-additive measure o n L(H) if, and only if,

(m), E Tr(H)

implies t o PM€ Tr(H),

for any M E L(H), where ( t o P M ) i~s the regular part of the closure t o PM. THEOREM 92. Let H be a separable Hilbert space of dimension greater than 2. Let p be a a-finite, regular measure o n L(H). Then, there exists a unique positive bilinear form t with a dense domain such that equation ( 1 ) holds. Conversely, for any positive bilinear form t o n H such that D ( t ) is dense in H, the mapping pt : L(H) + [0, m ] given by trtoP~

iftoPM€Tr(H), otherwise

i s a a-finite, regular measure. We end the survey by quoting another problem. PROBLEM 93. Characterize the a-finite, a-additive signed-measures taking infinite values on L(H) for a separable infinite dimensional Hilbert space H, where by a signed measure we mean a mapping p : L(H) [-m, m] preserving additivity on orthogonal subspaces. (It is possible to show that every signed measure attains at most one of the values k m . ) ACKNOWLEDGEMENTS The third author was supported by the Center of Excellence SAS - Physics of Information - 1/2/2005, the grant VEGA No. 216088126 SAV and by Science and Technology Assistance Agency under the contract No. APVT-51-032002. Bratislava, Slovakia. BIBLIOGRAPHY [Aarnes, 19701 J. Aarnes, Quasi-states on C*-algebras, Tl-ans. Amer. Math. Soc. 1 4 9 (1970), 601-625. [Amemip and Araki, 1966471 I. Amemiya and H. Araki, A remark on Piron's paper, Publ. Res. Inst. Math. Sci., Kyoto Univ. Ser. A 2 (1966-67), 423-427. [Bennett and Foulis, 19971 M. K. Bennett and D. J. Foulis, Interval and scale effect algebras, Adv. Appl. Math. 9 1 (1997), 823-834. [Birkhoff and von Neumann, 19361 G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. Math. 37 (1936), 823-843. [Bunce and Wright, 19941 L. J. Bunce and J. D. M. Wright, The Mackey-Gleason problem for vector measures on projections in von Neumann algebras, J . London Math. Soc. (2) 4 9 (1994), 133-149. [Buhagiar and Chetcuti, 20041 D. Buhagiar and E. Chetcuti, On isomorphism of inner product spaces, Math. Slovaca 5 4 (2004), 109-117.

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[Buhagiar and Chetcuti, t o appear] D. Buhagiar and E. Chetcuti, Quasi-splitting subspaces i n a prehilbert space, Math. Nach. ( t o appear). [Buhagiar and Chetcuti, submitted] D. Buhagiar and E. Chetcuti, Only 'freeJmeasures are admissible o n F ( S ) when S is incomplete, submitted. [Cattaneoand Marino, 19861 G. Cattaneo and G. Marino, Completeness ofinner product spaces with respect to splitting subspaces, Letters Math. Phys. 11 (1986), 1Ft20. [Chetcuit, 20021 E. Chetcuti, Completeness Criteria for Inner Product Spaces, MSc thesis, Univ. o f Malta, 2002. [Chetcuti and DvureEenskij, 2003al E. Chetcuti and A . Dvureknskij, Range of charges on orthogonally closed subspaces of an inner product space, Inter. J . Theor. Phys. 42 (2003), 19271942. [Chetcuti and DvureEenskij, 2003bl E. Chetcuti and A. DvureEenskij, A finitely additive state criterion for the completeness of inner product spaces, Letters Math. Phys. 64 (2003), 221227. [Chetcuti and DvureEenskij, 2004al E. Chetcuti and A . DvureEenskij, The aistence of finitely additive states o n orthogonally closed subspaces of incomplete inner product spaces, Letters Math. Phys. 67 (2004), 75-80. [Chetcuti and DvureEenskij, 2004bl E. Chetcuti and A. DvureEenskij, Measures on the splitting subspaces of an inner product space, Inter. J. Theor. Phys. 43 (2004), 364-384. [Chetcuti and DvureEenskij, 2005al E. Chetcuti and A. DvureEenskij, The state-space of the lattice of orthogonally closed subspaces, Glasgow Math. J . 47 (2005), 213-220. [Chetcuti and DvureEenskij, 2005bl E. Chetcuti and A. Dvureknskij, Recent progress o n preHilbert-space logics and their nzeasure spaces, Inter. J . Theor. Phys. 44 (2005), 2139-2151. [Chetcuti et al., 20061 E. Chetcuti, P. de Lucia and A. Dvureknskij, Sequential convergence of regular measures o n pre-Hilbert space logics, J . Math. Anal. Appl. 318 (2006), 194-210. [Chetcuti and Hamhalter, t o appear] E. Chetcuti and J . Hamhalter, Vitali-Hahn-Saks Theorem for vector nzeasures o n operator algebras, Oxford Quarterly J . Math. ( t o appear). [Chevalier et al., 20001 G. Chevalier, A. DvureEenskij and K . Svcrtil, Piron's and Bell's geometric lemmas and Gleason's theorem, Found. Phys. 30 (2000), 1737-1755. [Cooke, 19781 T . A. Cooke, The Nikodym-Hahn-Vitali-Sakstheorem for states o n a quantum logic, "Mathematical Foundations o f Quantum Theory" A. R . Marlow ed., Acad. Press, London (1978), 275-285. [Christensen, 19821 E. Christensen, Measures o n projections and physical states, Comm. Math. Phys. 86 (1982), 529-538. [Cooke et al., 19851 R . Cooke, M. Keane and W. Moran, A n elementary proof of Gleason's theorem, Proc. Cambr. Phil. Soc. 98 (1985), 117-128. [de Lucia and Pap, 20021 P. d e Lucia and E. Pap, Convergence theorems for set functions. In: Pap, E. (ed.), Handbook o f Measure Theory. Vol. I Amsterdam, North-Holland, 2002 pp. 125-178 . [Dorofeevand Sherstnev, 19901 S.V. Dorofeev and A.N. Sherstnev, Frame-type functions and their applications, Izv. vuzov matem. no. 4 (1990), 23-29 ( i n Russian). [Drish, 19791 T . Drisch, Generalization of Gleason's theorem, Inter. J . Theor. Phys. 18 (1979), 239-243. [Dunford and Schwartz, 19581 N. Dunford and J. T . Schwartz, Linear Operators, Vol. 1, Interscience, New York, 1958. [ A . DvureEenskij, 19781 A. Dvureknskij, Signed states on a logic, Math. Slovaca 28 (1978), 33-40. [Dvureknskij, 19861 A. Dvureknskij, Generalizations of Maeda's thwrem, Inter. J . Theor. Phys. 25 (1986), 1117-1124. [ ~ v u r e k n s k i j19881 , A. DvureEenskij, Completeness of inner product spaces and quantum logic of splitting subspaces, Letters Math. Phys. 15 (1988), 231-235. [Dvureknskij, 19921 A. DvureEenskij, Gleason's Theorem and Its Applications, Kluwer Acad. Publ., Dordrecht, Ister Science Press, Bratislava, 1992. [Dvureknskij, 19931 A. DvureEenskij, Gleason's t h w r e m and completeness criteria, Inter. J . Theor. Phys. 32 (1993), 2377-2388. [Dvureknskij, 19951 A. DvureEenskij, Frame functions and conzpleteness of inner product spaces, Ann. Inst. Henri PoincarB, 62 (1995), 429-438. [ ~ v u r e k n s k i j20011 , A. Dvureknskij, A new algebraic criterion for completeness of inner product space, Letters Math. Phys. 58 (2001), 205-208.

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[Dvureknskij, 20031 A. Dvureknskij, States on subspaces of inner product spaces with the Gleason property, Inter. J. Theor. Phys. 42 (2003), 1393-1401. [ D v u r e n s k i j and Pulmannovi, 19891 A. DvureEenskij and S. Pulmannovi, A signed measure completeness criterion, Letters Math. Phys. 17 (1989),253-261. [Dvureknskij and Pulmannovi, 20001 A. DvureEenskij and S. Pulmannovi, New 'Ii-ends i n Quantum Structures, Kluwer Acad. Publ., Dordrecht, 2000. [Dvureknskij et al., 19901 A. DvureEenskij, T. Neubrunn and S. Pulmannovi, Finitely additive states and completeness of inner product spaces, Found. Phys. 20 (1990), 1091-1102. [Dvureknskij and P t i k , 20021 A. DvureEenskij and P. P t i k , O n states on orthogonally closed subspaces of an inner product space, Lett. Math. Phys. 62 (2002),63-70. [Eilers and Horst, 19751 M. Eiers and E. Horst, The theorem of Gleason for nonseparable Hilbert space, Inter. J. Theor. Phys. 13 (1975),419-424. [Giuntini and Greuling, 19891 R. Giuntini and H. Greuling, Toward a formal language for unsharp properties, Found. Phys. 19 (1989), 931-945. [Gleason, 19571 A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. [Gudder, 19751 S. P. Gudder, Inner product spaces, Amer. Math. Monthly 82 (1975), 251-252. [Gudderbook, 19881 S. P. Gudder, Quantuna Probability, Academic Press Inc., Boston, San Diego, New York, Berkely, Tokyo, Toronto, 1988. [Gross and Keller, 19771 H. Gross and H. A. Keller, O n the definition of Hilbert space, Manuscr. Math. 23 (1977), 67-90. [Halmos, 19881 P. R. Halmos, Measure Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1988. [Hamel, 19051 G. Hamel, Eine Basis allen Zahlen und die unstetigen Losungen der Funktionalgleichung: f (x Y ) = f (x) f(Y), Math. Ann. 60 (1905),459-462. [Hamhalter, 20031 J. Hamhalter, Quantum Measure Theory, Kluwer Acad. Publ., Dordrecht, 2003. [Hamhalter and Pt&, 19871 J. Hamhalter and P. P t i k , A completeness criterion for inner product spaces, Bull. London Math. Soc. 19 (1987), 259-263. [Harding, 19911 J. Harding, Orthonaodular lattices whose MacNeille conapletions are not orthomodular, Order 8 (1991), 93-103. [Harding, 19961 J. Harding, Decompositions i n quantum logic, mans. Amer. Math. Soc. 348 (1996), 1839-1862. [Holland, 19691 S. Holland, Jr., Partial solution to Mackey's problem about modular pairs and completeness, Canad. J. Math. 21 (1969), 1518-1525. [Holland, 19951 S. Holland, Jr., Orthomodularity in infinite dimensions; a theorem of M. SolCr, Bull. Amer. Math. Soc. 32 (1995),20S234. [Husimi, 19371 K. Husimi, Studies on the foundations of quantum mechanics 1, Proc. Phys.math. Soc. Japan 19 (1937), 766-789. [Jajte, 19721 R. Jajte, O n convergence of Gleason measures, Bull. Acad. Polon. Sci., ser. Math. Astr. Phys. 20 (1972), 211-214. [Kadison and Ringrose, 19831 R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vols I and 11, Academic Press, London, 1983 and 1986. [Kalmbach, 19831 G. Kalmbach, Orthonaodular Lattices, Acad. Press, London, New York, 1983. [Kalmbach, 1986) G. Kalmbach, Measures and Hilbert Lattices, World Sci. Publ. Co., Singapore, 1986. [Keller, 19801 H. A. Keller, Ein nicht-klassischer Hilbertischen Raum, Math. 2. 172 (1980), 41-49. [ ~ o c h e and n Specker, 19671 S. Kochen and E. R. Specker, The problem of hidden variables i n quantum mechanics, J . Math. Mech. 17 (1967), 59-67. [Kolmogorov, 19331 A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung,Berlin, 1933. [KBpka and Chovanec, 19941 F. KBpka and F. Chovanec, D-posets, Math. Slovaca 4 4 (1994), 2134. [Lugovaja, 19831 G. D. Lugovaja, Bilinear forms defining measures on projectors, Im. vuwv rnatem. No. 2 (1983),88-88 (in Russian). [Lugovaja and Sherstnev, 19801 G. D. Lugovaja and A. N. Sherstnev, O n the Gleason theorem for unbounded measures, h v . vuzov matem. No. 2 (1980), 3&32 (in Russian).

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[Mackey, 19631 G. W . Mackey, Mathematical Foundations of Quantuna Mechanics, Benjamin, New Y o r k , 1963. [Maeda, 19801 S. Maeda, Lattice Theory and Quantuna Logic, Maki-Shoten, Tokyo, 1980 (in Japanese). [Maeda and Maeda, 19701 F. Maeda and S. Maeda, Theory of Symmetric Lattices, SpringerVerlag, Berlin, 1970. [Mayet, 19981 R. Mayet, Some characterizations of the underlying division ring of a Hilbert lattice by automorphisms, Inter. J . Theor. Phys. 37 (1998), 109-114. [Murray and von Neumann, 19361 F. J . Murray and J. von Neumann, O n rings of operators, Ann. Math. 37 (1936), 116-229. [Murray and von Neumann, 19371 F. J . Murray and J . v o n Neumann, O n rings of opemtors ZZ, 'Ikans. Amer. Math. Soc. 4 1 (1937), 208-248. [Murray and v o n Neumann, 19431 F. J . Murray and J . v o n Neumann, O n rings of opemtors ZV, Ann. Math. 44 (1943), 7164308. iron, 19641 C. Piron, Asionaatique quantique, Helv. Phys. Acta 37 (1964), 439468. [Piron, 19761 PirIPir C . Piron, Foundations of Quantum Physics, Addison-Wesley, Reading, Mass., 1976. [Piziak, 19901 R. Piziak, Lattice theory, quadratic spaces, and quantuna proposition systems, Found. Phys. 20 (1990), 651-665. [PtAk, 19881 P. P t i k , FAT - C A T ( i n the state space of quantum logics), Proceedings o f "Winter School o f Measure Theory", Liptovski J i n 1988, (Czechoslovakia) pp. 113-118. [ ~ t and & Pulmannovi, 199l] P. PtAk and S. Pulmannovi, Orthonaodular Structures as Quant u m Logics, Kluwer Acad. Publ., Dordrecht, 1991. [ P t i k and Weber, 19771 P. Ptdk and H. Weber, Two remarks on the subspace poset of an inner product space, Contrib. Gen. Alg. 10, Proceedings o f t h e Klagenfurt Conference, May 19-June 1, 1997, Verlag Johannes Heyn, Klagenfurt, 1998, pp. 263-267. [PtAk and Weber, 20011 P. P t i k and H. Weber, Lattice properties of subspace fanailies of inner product spaces, Proc. Amer. Math. Soc. 129 (2001), 2111-2117. [ ~ t and & Weber, 20041 P. PtAk and H. Weber, Order properties of splitting subspaces i n an inner product space, Math. Slovaca 54 (2004), No. 2, pp. 119-126. [PulmannovB, t o appear] S. Pulmannovi, A note on honaomorphisms of inner product spaces, Demonstratio Math., t o appear. [Sherstnev, 19741 A. N. Sherstnev, O n a notion of a charge i n nonconanautative scheme of measure theory, Veroj. metod i kibern. Kazan, no. 10-11 (1974), 68-72. (in Russian). [ ~ i m o n19781 , B. Simon, A canonical decomposition for quandmtic fornls with application t o monotone convergence theorems, J . Funct. Anal. 28 (1978) 377485. [Solkr, 19951 M. P. Sol&r, Characterization of Hilbert spaces by orhtomodular spaces, Comm. Algebra 23 (1995), 219-243. [Solbr, 19791 M. P. Solkr. Theory of Operator Algebras I, Springer, Berlin-Heidelberg-New Y o r k , 1979. [Varadarajan, 19851 V . S. Varadarajan, Geometry of Quantuna Theory, Springer-Verlag, New York Inc., 1985. [von Neumann, 19291 J . v o n Neumann, Zur Algebra dur Funktionaloperationen und T h w r i e der norn~alenOperatoren, Math. Ann. 102 (1929), 370-427. [von Neumann, 19401 J . von Neumann, O n rings of operators ZZZ, Ann. Math. 4 1 (1940), 94161. [von Neumann, 19491 J. von Neumann, O n rings of operators. Reduction theory, Ann. Math. 50 (1949), 401485. [Wigner, 19591 E. P. Wigner, Group Theory and its Applications t o Quantum Mechanics of Atomic Spectra, Acad. Press. Inc. New Y o r k , 1959.

HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

121

QUANTUM PROBABILITY Stan Gudder

1 INTRODUCTION

This chapter begins with the more concrete and familiar and then gradually progresses to the more general and abstract. To save space and spare the reader from too many technical details, we have decided to omit some of the longer proofs. However, we provide references for the proofs of all stated results and encourage the reader to further pursue topics of interest. Some proofs are provided for recent results to give the reader a flavor of the techniques that are now employed. It is widely believed that quantum theory is intrinsically statistical. That is, the study of elementary particles and their interactions is irreducibly probabilistic and cannot be derived from an underlying deterministic theory. Even those who advocate that such underlying hidden variable theories may be possible admit that they do not seem to be practical and do not appear to offer anything new or useful. In any case, just making the assumption that quantum theory is probabilistic goes a long way toward establishing the structure of quantum mechanics and understanding its interpretations. An important component of any physical theory is the concept of measurement. When a measurement is performed, the physical system interacts with a measuring apparatus and in general the environment cannot be neglected. The environment produces noise, any real apparatus has limited resolution and efficiency and the experimental control of initial conditions is never exact. It follows that no real measurement is absolutely precise and results of measurements must be described statistically. Hence we have a second reason for the importance of probability in quantum theory. Quantum probability generalizes classical probability theory in two ways. First, a traditional random variable f describes a measurement that is absolutely precise. After all, the values f (w) are sharply defined with no fuzziness involved. The uncertainty of classical probability comes from our lack of knowledge of the situation. We can only make statistical predictions about which sample point w occurs. However, as we have previously mentioned, real measurements are never completely precise. There are always errors involved and these should be taken into account in a realistic theory. Second, in a quantum mechanical world, two different measurements A and B frequently interfere with each other. One way this is manifested is that they

Stan Gudder

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cannot be performed simultaneously. We must specify which measurement is performed first and which is performed next. This results in a sequential or temporal product A o B of the measurements. Due to interference, in general, we have A o B # B o A so the theory becomes noncommutative. For this reason, quantum probability is sometimes called noncommutative probability theory. Classical probability theory contains no mechanism for describing such situations. Until recently the main framework for such descriptions has been operators on a Hilbert space which provides an obvious noncommutativity. In this case, fuzzy quantum events are represented by operators satisfying 0 5 A 5 I and observables or measurements are represented by positive operator-valued measures. These will be considered in Sections 4-8. In the last few years, researchers in the foundations of quantum mechanics have developed a more general formalism called a sequential effect algebra. In this case, fuzzy quantum events are represented by abstract objects with certain properties called effects and observables are represented by a-morphisms from the Borel subsets of the real line into the set of effects (effectvalued measures). These will be discussed in Section 9.

2 MOTIVATIONAL CALCULATIONS This section presents some standard concrete calculations that begin to show how the basic structure of quantum mechanics follows from probabilistic considerations and furthermore how quantum probability differs from classical probability theory. Suppose we measure the coordinate s of a quantum particle. Assuming that quantum theory is probabilistic, this measurement should be described by a random variable Q, on the probability space (B3,p) representing the position of the particle in three-space W3 where p(r, t) is a probability density for r E B3 depending on the time parameter t. The probability that Q, has a value in the Borel set A E B(B3) at time t is given by

where dV = dxdydz and the expectation of Q, at time t is r

So far, this is just classical probability theory. However, one of the first basic axioms of quantum mechanics is the Born rule which states that p(r, t) = I$(r, t)12 where $(r, t ) is a complex-valued function called a probability amplitude. We ~ , for conclude that $(r, t) is a unit vector in the complex Hilbert space L ~ ( RdV) each fixed t. We can write the probability amplitude as $ = f i e i s l where sl(r,t) is a measurable real-valued function. We can also define the measurable real-valued function s2(r,t) by s 2 = -(1/2)lnp so that fl = e-'=. Defining the complexvalued function s = sl is2 we obtain the eikonal form = eis. The function s 2

+

+

Quantum Probability

123

is directly related to the probability density p and is easily computed. However, the function 31, which is peculiar to quantum mechanics, is completely arbitrary unless the theory is developed further. We now show that the Heisenberg uncertainty principle essentially follows from the Born rule. We assume that p, s1,s2 are differentiable functions. Since p is a probability density, integrating by parts gives

lim px = 0 we obtain

Sx

dV = -1. Applying Schwarz's inequality and assuming that the integrals exist, we have

Under the reasonable assumption that

x+&m

To simplify the second integral on the right side of (1) we have

Since --

dx

dx

we obtain

Applying (2) and (3) we have

Substituting (4) into (1) gives

Now

(2) ( 2 )(2)

=i

so we conclude from (5) that

s2

l Za$l 2

2

= e2s2

1Ek1

Stan Gudder

124

Equation (6) is equivalent to the uncertainty principle which states that

where ti is a constant determined by experiment (Planck's constant). Assuming that p is symmetric so that Et(Q,) = 0 we conclude that the variance of Q, is

l2

Interpreting J 1-ih d V as the variance of a momentum measurement Px in the x-direction, (7) gives

The uncertainty principle (8) is a statistical inequality and in the process of o b taining (8) we have derived the momentum operator -ih& using probabilistic methods. The fact that p is a function of t is a concept that is usually not present in classical probability theory and is due to the motion of the particle in space. Information about the motion of the system can be described by the function sl. Using the notation XI = x, xz = y, xs = z, we define the probability velocity vector

where m is the mass of the particle. The change of probability in time is determined by the probability density current jk = puk, k = 1,2,3. The equation for jk is analogous to the expression for the density current in fluid dynamics. We then obtain

Since

+=fi

eisl

and

+*

=

we conclude that

Since jk and p are real-valued functions, taking the real part gives

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125

The existence of a nonzero jk depends on the fact that the probability amplitude I) is complex-valued. This is a feature of quantum mechanics that is not present in classical probability theory. Notice that jk can be written in terms of the momentum operators 1 j k = g ($*PXkI) - +PXk$*) which further indicates a relation between jk and the motion of the particle. The previous calculations suggest that the state of a quantum system is r e p resented by a unit vector $ in a Hilbert space and that position and momentum observables in the x-coordinate are represented by the self-adjoint operators

Motivated by these concrete calculations, the next section establishes the notation and definitions of a mathematical framework for quantum probability. 3 NOTATION AND DEFINITIONS

Let H be a complex Hilbert space that represents the state space of a quantum system S. The set of effects E(H) for S is the set of operators on H satisfying 0 5 A 5 I. More precisely, A E E(H) if and only if 0 5 (Ax, x) 5 (x, x) for all x E H. Effects represent yes-no measurements that may be unsharp (imprecise, fuzzy). F'rom a probabilistic viewpoint, effects can be thought of as fuzzy quantum events. It is interesting that many of the important classes of quantum operators are given by subsets of E(H). For example, sharp yes-no measurements are represented by the set of projection operators P(H) &(H). In traditional quantum mechanics projections correspond to (sharp) quantum events. A state for S is represented by an operator W E E(H) satisfying tr(W) = 1. We call W a density operator and denote the set of density operators by D(H). The density operators correspond to the probability measures of classical probability theory. The pure states of S are given by D(H) n P(H). A more general measurement (not just twevalued) is represented by a positive operator-valued (POV) or effect-valued measure X : B(R) -+ E(H) [Busch et al., 1995; Busch et al., 1991; Davies, 19761. That is X(B) = I and X(UAi) = C X ( A i ) for Ai f~ A, = 0, i # j, and convergence of the summation is in the strong operator topology. In this case, X(A) represents the effect observed when the measurement X has a value in the Bore1 set A. A projection-valued (PV) measure X : B(R) + P ( H ) corresponds to a sharp measurement and by the spectral theorem gives a single self-adjoint operator called an observable. In this work we shall mainly consider discrete measurements which are given by a countable set {Ai} G E(H) such that C Ai = I. Then Ai corresponds to the effect observed when the measurement results in outcome i. For A E E(H), the negation of A is defined by A' = I - A. Of course, A' E &(H) and if P E P ( H ) then P' E P(H). For A, B E E(H), if A B 5 I we

+

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+

define A @ B = A B. Then A @ B is defined if and only if A 5 B'. Roughly speaking, A@B corresponds to a parallel measurement of A and B. The algebraic system (E(H), @, 0, I) forms an effect algebra. Effect algebras are important in i j PulmannovB, 2000; foundational studies of quantum mechanics [ ~ v u r e ~ e n s kand Foulis and Bennett, 1994; Giuntini and Greuling, 1989; Gudder and Greechie, 20021. An orthoposet is an algebraic system (P, 0 , l ) where (P, <,0,1) is a bounded partially ordered set and ': P + P is an orthocomplementation which satisfies a" = a, a 5 b implies b' 5 a' and a A a' = 0. An orthoposet P is orthomodular if a 5 b' implies that aV b exists and a 5 b' implies that b = aV (bA a'). It is well known that (P(H), I,',0, I) forms an orthomodular lattice [Beltrametti and Cassinelli, 19811. It is instructive to notice the analogy between our previous concepts and those of a fuzzy set system [0, 1IY.An effect corresponds to a fuzzy set in [0, 1IYand a projection corresponds to a subset of Y. Thus, £(H) corresponds to a noncommutative fuzzy set theory and P(H) corresponds to a noncommutative set theory. For W E D(H) we can think of (£(H), W) as a noncommutative fuzzy probability space. A measurement in our framework is a noncommutative analog of a fuzzy random variable.

s,',

4 PROBABILITY AND CONDITIONAL PROBABILITY One of the main axioms of quantum mechanics says that the probability that the quantum event P E P ( H ) occurs when the system S is in state W E D(H) is

We generalize (1) by postulating that the probability that A E E(H) is observed (has value yes) when the system is in state W E D(H) is

In particular, for a discrete measurement X = {Ai), the probability that X results in outcome i is

Another axiom of quantum mechanics says that if the initial state of S is W E D(H) and event P E P(H) occurs then the final state is

assuming that tr(WP) # 0. This axiom is called the projection postulate. Now the projection postulate is somewhat controversial and we shall take the viewpoint that it is valid for certain ideal measurements which, in the sequel, we assume are operative. We now give a heuristic motivation for (4) that will be useful when we generalize it to the fuzzy case.

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Let us visualize P E P(H) as a filter for particle beams and suppose the incoming beam is in the pure state x E H , llxll = 1. After the beam impinges the filter P the outgoing beam is in the unnormalized state Px. Using Dirac notation, the incoming state is given by the density operator Win = Ix)(xl and the unnormalized outgoing state becomes

Moreover, the probability that a particle is transmitted becomes, by (1)

Since mixed states are convex combiiations of pure states, we can assume that (5) and (6) hold for any Win E D(H). Moreover, normalizing (5) gives

which is (4). Now suppose that A E &(H) is an imprecise filter for particle beams and as before the incoming beam is in the pure state x. Since A is unsharp we cannot assume that after the beam impinges A the outgoing beam is in the unnormalized state Ax. However, we do assume that the outgoing beam is in the unnormalized state Cx where C is a positive operator depending on A. As before the unnormalized outgoing state becomes

and the probability that a particle is transmitted becomes (8) PW,(A) = tr(CWinC) Applying (2) and (8) gives

Since (9) holds for all Win E D ( H ) we conclude that C2 = A SO C = All2 the unique positive square root of A. Applying (7),the normalized outgoing state is ~ 1 / 2 ~ ~ ~ ~ 1 / 2 Wout =

tr (WinA)

Motivated by the previous heuristic argument we postulate that if the initial state is W and effect A is observed then the final state is

For readers who object to this argument we may regard (10) as a postulate that only applies to certain ideal measuring apparatuses. Now suppose a discrete measurement X = {Ai), Ai E &(H), is performed. If outcome i occurs then by (10) the post-measurement state is

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If X is performed but the outcome is not observed, the post-measurement state becomes

Notice that (12) is a consequence of (11) because Waft,,should be a convex combination of the states in (11) with coefficients pw (Ai). Observe that Waft,,is indeed a state because

For A, B E &(H) the conditional probability that B is observed given that A is observed in the state W is by (10)

We then have that (13) P W ( A ' / ~ B A ~ /= ~ )PW(A)PW(B I A)

Equation (13) reminds us of the classical formula

so A 1 / 2 ~ A 1 / 2is analogous to an intersection A n B . We use the notation A o B = All2BA'/~ and unlike A n B we see that A o B # B O A in general. In fact, Ao B is the sequential product of A and B in which A is measured first and B is second. That A o B # B o A in general is due to quantum interference. For example, let A and B be polarization filters in planes perpendicular to a photon beam, where A polarizes vertically and B at a 45' angle. If the incoming beam is prepared in a state of horizontal polarization, then A o B transmits no particles, while B o A will transmit particles so A o B # B o A. Notice that A o B E &(H) because

for all x E H. The operation A @ B describes a parallel combination of A and B while A o B = A ' / ~ B A ' / ~ describes a series combination in which A is measured first and B second. We have seen that the probability law

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holds for all A, B E E(H), W E D(H). Now let X = {Ai), Ai E E(H), be a discrete measurement. The law of total probability would say that for any B E E(H)

If this law holds for all W E D(H), then it follows that

We call (14) the quantum law of total probability. In general, this law (14) does not hold. If AB = BA we say that A and B are compatible. Compatible effects have classical properties relative to each other. For example, if BAi = Ai B for all i, then (14) holds. It can be shown that (14) does not imply that B is compatible with all Ai [Arias, 19921. Compatible effects are sometimes called simultaneously measurable. The next result gives sufficient conditions under which h Singh, 1998; the quantum law of total probability implies compatibility [ ~ u s c and Gudder and Nagy, 20011. THEOREM 1. Equation (14) implies BAi = AiB for all i if any of the following conditions hold: (a) i = 1,2. (b) dim H < oo. (c) B has pure point spectrum that can be totally ordered in decreasing order. Equation (14) that B = C A i o B can also be interpreted as saying that B is not disturbed by the measurement X = {Ai). This suggests an application to axiomatic quantum field theory. Suppose the measurement X = {Ai) is performed in a bounded spacetime region Al and B E E(H) is performed in another spacetime region A2 that is spacelike separated from A1. According to Einstein causality, the measurement in Al should not disturb B so B = C A i o B. But as we said earlier this does not imply that BAi = AiB for all i. Thus, the axiom of local commutativity does not follow from Einstein causality. This indicates that the axiom of local commutativity may be too strong and should be replaced by a weaker axiom.

5 INDEPENDENCE Independence is one of the most important concepts in classical probability theory. However, it does not seem to be well developed in quantum probability theory. We say that A, B E E(H) are independent in the state W E D(H) if pw(A o B) = pw (A)pw (B). By (13) this is equivalent to pw ( B I A) = pw (B) whenever Pw(A) # 0. We now give a simple example in which

Recall that a pure state is a one-dimensional projection P, onto the subspace spanned by a unit vector z. For the pure state W = P, we have that pw(A)

= tr(WA) =

(Ax,x)

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Notice that A, B E E(H) are independent in the pure state W = P, if (A o Bx, x) = (Ax, x) (Bx, x) For our example, let A, B E P ( C 2 )be the projections

and let x = (1,O) E C2. Then

and

Since (Ax, x) = 1 and (Bx, a) = 112 we have that

1 (A o Bx, x) = -(Ax, x) = (Ax, x) (Bx, x) 2 but

1 # (Bx, x) (Ax, x) 2 This example shows that unlike the classical case, independence is not a symmetric relation. However, at the intuitive level this is not completely surprising. For example, my heart is independent of my little finger but my little finger is not independent of my heart. Recall that two bounded random variables f and g are uncorrelated relative t o a probability measure p if S f g d p = S f d p S g dp. Of course, two independent random variables are uncorrelated. The next theorem is a classical result of probability theory.

( B o Ax, x)

= - (Bx, x)

THEOREM 2. Two bounded random variables f , g are uncorrelated relative to every probability measure on their sample space if and only if f or g is constant. We now generalize Theorem 2 to fuzzy quantum probability theory [Gudder and Nagy, 200i] THEOREM 3. For A, B E &(H) the following statements are equivalent. (a) A and B are independent in every state W E Z)(H). (b) A and B are independent in every pure state in D(H). (c) A = cI or B = c I for some 0 5 c 5 1. We now give some examples of independent effects. Any two effects of the form A €31I,I @3 B E I ( H @ H ) are independent in a pure tensor state x @ y. If x is a unit eigenvector of A E &(H) then A and B are independent in the pure state P,

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for every B E & ( H ) . The next result generalizes this situation [Gudder and Nagy, 20011. THEOREM 4. The following statements are equivalent. (a) pw(A) = 1. (b) WA = AW = W. (c) pw (A o B) = pw(B) for every B E &(H). Notice that Theorem 4 says that if pw(A) = 1, then A and B are independent in W for every B E &(H). 6 PROPERTIES OF THE SEQUENTIAL PRODUCT

This section summarizes some of the properties of the sequential product. For proofs we refer the reader to u udder and Nagy, 20011. THEOREM 5. Let A, B E &(H). (a) If A o B = B o A then AB = BA. (b) If A o B E P ( H ) then AB = BA. Theorem 5(a) says that if the measurement order of A and B is irrelevant, then A and B are compatible. Of course, the converse is trivial. Theorem 5(b) says that even if A and B are both sharp, their sequential product is not sharp unless A and B are compatible. To give the reader a flavor of the subject we now present the simple proof of Theorem 5(a). Suppose that A o B = B o A. We then have that ( ~ 1 / 2 ~ 1 /(2~)1 / 2 ~ 1 / = 2 )( ~ 1 / 2 ~ 1 /(2~)1 / 2 ~ 1 / 2 ) This shows that the operators M = and N = are normal. Letting T = A'/' we conclude that MT = T N . It follows from Fuglede's theorem [Rudin, 19911 that M*T = T N * . Hence, B'/'A = AB'/' and we conclude that BA = AB. It is clear that the distributive law

always holds. However, as we have seen in Section 5 in general we have that

The next result shows that the associative law need not hold. THEOREM 6 . For A, B E &(H) we have that A o ( B o C) = (A o B) o C for every C E &(H) if and only if AB = BA. The fact that Ao ( B o C) # (A o B) o C in general is puzzling because physically one would expect the two sides to be equal. For example, suppose A, B and C represent beam filters. This would say that if we first apply A and then B o C considered as a single filter then this would be different than first applying A o B considered as a single filter and then applying C. Mathematically this can be written as

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For a further discussion on this point we refer the reader to u udder and Nagy, 20011. The next theorem gives even stronger results for other types of associativity. THEOREM 7. For A, B E E(H) the following statements are equivalent. (a) A o ( C O B ) = ( A o C ) o B for everyC E E(H). (b) C o ( A o B ) = ( C o A ) o B for every C EE(H). (c) A = c I or B = c I forsome0 < c < 1. The system (E(H), @, o, 0, I ) is a special case of a mathematical structure called a sequential effect algebra. This structure is important for foundational studies in quantum mechanics and will be discussed in Section 9 [Gudder and Greechie, 2002; Gudder and Nagy, 20011. Other examples of sequential effect algebras are boolean algebras and fuzzy set systems with their usual operations.

7 ALMOST SHARP EFFECTS We say that A E E(H) is almost sharp if A = P Q P = P o Q for P , Q E P ( H ) . This terminology stems from the fact that A may be obtained by measuring two sharp effects. In a sense, almost sharp effects are LLclose"to being sharp and we shall present a simple characterization of such effects. It follows from Theorem 2(b) that the set of almost sharp effects is strictly larger than P(H). Our characterization holds for the case dim(H) < m. Generalizations to dim(H) = m are possible but are a little more complicated t o state. We refer the reader to [Arias and Gudder, 20041 for generalizations and omitted proofs. We say that A E E(H) is nearly sharp if A and A' are almost sharp. For A E E(H) with dim(H) < m define the following nonnegative integers: no(A) = multiplicity of the eigenvalue 0 for A nl(A) = multiplicity of the eigenvalue 1 for A n(A) = dim(H) - no (A) - nl(A) Notice that n(A) is the sum of the multiplicities of the eigenvalues X for A satisfying 0 < X < 1. We call such eigenvalues fuzzy eigenvalues. The next theorem shows that A is almost (nearly) sharp if and only if the number of fuzzy eigenvalues is smaller than the number of sharp eigenvalues in a certain sense. For A E E(H) we denote the projection onto the range of A by PA. LEMMA 8. If A E E(H) with dim(H) A)

< m, then n(A) = dim(PAAf)= rank(PA -

Proof. By diagonalizing A we can assume without loss of generality that A has the form

Quantum Probability

where 0

< Xi < 1for i = 1 , . . .,m.

We then have that

AA' = diag (Xl(1- XI), .. . ,Xm(l - Am),0, . . . ,0) PA-A=diag(l-X1 ,..., l - X m , O ,...,0) It follows that

THEOREM 9. Let A E E(H) with dim(H) < co. (a) A i s almost sharp if and only zf n(A) 5 no(A). (b) A is nearly sharp zf and only zf n(A) 5 min {no( A ) ,nl ( A ) ) . Proof. (a) Suppose that A is almost sharp so that A = P Q P for P, Q E P(H). Since A 5 P it follows that PA5 P. Since APA = PAA = A we conclude that A = PAQPA. Letting NB be the projection onto the nullspace of B E E ( H ) , by Lemma 8 we have

n(A) = r a n k ( P ~- A) = rank (PA(I- Q)PA) 5 rank(] - Q ) = dim(NQ) 5 dii(NpAQpA) = d i m ( N ~= ) no(A) Conversely, suppose that m = n(A) 5 no(A). Without loss of generality we can represent A as a diagonal matrix (1). Let

and represent A by the block matrix

Since M 5 m(A) we can split the last block into an m x m block and an l x l block where l = %(A) - m. Let Q be the block matrix

It is easy to check that PAQPA= A and that Q E P ( H ) . The proof of (b) follows directly from (a).

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We now consider effects that are longer sequential products of sharp effects. The fuzziness index f (A) of A E &(H) is the smallest integer n such that A has the form A = PI ...PnQPn...Pl for Q, Pi E P(H), i = 1,.. .,n. Thus, A is sharp if and only if f (A) = 0 and A is almost sharp and not sharp if and only if f (A) = 1. If no such n exists, we write f (A) = 03. THEOREM 10. For A E &(H) with dim(H) < co we have that

We now give some examples in the Hilbert space H = (C3. In &((C3)define A = diag(X1, X2, As), B = diag(X1, X2, 0), C = diag(X1, 1, O), D = diag(X1, 0,O) where 0 < X i < 1. i = 1 , 2 , 3 . Wehavethat f(A) = c o , f ( B ) = 2 , f(C) = f(D) = 1. Notice that C is nearly sharp and D is almost sharp but not nearly sharp. We now write C = P Q P , P, Q E P(H).

The sets of almost sharp effects and nearly sharp effects have very weak structures. For example, C = diag(X1, 1,O) is nearly sharp, but XC is not even almost sharp for 0 < X < 1. Also, if A and B are nearly sharp and A B E &(H), then A + B may not be almost sharp. However, if A is almost sharp, then An is almost sharp for all n E N. The next result gives the structure of the set &(H),, of nearly sharp elements in &(H) when dim(H) < XI. This result also holds in the von Neumann algebra setting [Arias and Gudder, 20041.

+

THEOREM 11. If dim(H)

< co,

then (&(H),,,

<, ', 0, I ) is an orthoposet.

Proof. All the properties of an orthoposet are clear except for the condition AAA' = 0 for every A E E(H),, . To verlfy this condition suppose that B E &(H),, with B 5 A and B 5 A'. Then

Since (112)' = 112, it follows that 112 5 B' so that N B ~ = 0. Hence, no(Bf) = 0 and by Lemma 8 and Theorem 9 we conclude that

Hence, BB' = 0 so that B E P(H). Since B that A A A' = 0.

< 112, we have that B = 0. It follows

rn

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8 NON-DISTURBANCE FOR FUZZY QUANTUM MEASUREMENTS K. A. Kirkpatrick [Kirkpatrick, 20041 has recently discussed three ways of describ ing non-disturbance between quantum measurements. The first two are due to , and KirkLiiders [ ~ u s c hand Singh, 19981 and the third to Davies [ ~ a v i e s19761 patrick [Kirkpatrick, 20031. Letting X and Y be two quantum measurements, the three non-disturbance criteria are given roughly as follows. (1) The probability of an established value of Y is unchanged by the later occurrence of a value of X. (2) The probability of occurrence of a Y value is unchanged by a preceding execution of X. (3) If p and q are X and Y values, respectively, then the probability of p followed by q coincides with the probability of q followed by p. We said that the criteria are given roughly because the English language can be imprecise and ambiguous. We shall later translate these criteria into precise mathematical language and prove results about them. It is appropriate that these criteria are phased in terms of probabilities because statistical results are what are observed in the laboratory. Kirkpatrick showed that (I), (2) and (3) are equivalent to the compatibility (commutativity) of X and Y and hence are equivalent to each other. However, Kirkpatrick only considered sharp quantum measurements which are described by projection-valued measures. In the present work we shall generalize his results t o measurements that may be unsharp (or fuzzy) and these are described by positive operator-valued measures. Unsharp measurements are quite important in quantum measurement theory [Busch et al., 1995; Busch et al., 1991; Davies, 19761 and in applications such as quantum computation and quantum information [Nielsen and Chuang, 20001. Unlike the sharp case, we shall show that (I), (2) and (3) are not equivalent for general measurements. Only Criterion (3) is equivalent to the compatibility of X and Y. Although Criterion (1) implies compatibility it is not appropriate for the general case because it also implies that Y is sharp. Compatibility implies (2) but the converse does not hold. Although they are not explicitly symmetric relative to X and Y, the three criteria are implicitly symmetric in the sharp case because they are equivalent to the symmetric relation of compatibility. In the general fuzzy case, only (3) is symmetric in X and Y. This further justifies Kirkpatrick's choice of (3) for his discussion of noncompatibility in classical probability theory [Kirkpatrick, 20041. As in Kirkpatrick [Kirkpatrick, 20041 and Liiders [Busch and Singh, 19981, we shall only consider discrete quantum measurements. In the sequel we shall identify a (discrete) measurement with its set of measurement operators {Ai} without specifying its values. Two measurements {Ai} and {Bj} are compatible if their measurement operators are compatible, AiBj = BjAi for all i and j . We now translate the three non-disturbance criteria into

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mathematical forms and then draw conclusions concerning compatibility of the measurements. If X = { A i } and Y = { B j } are measurements, then the mathematical form of Criterion (1) for non-disturbance becomes

for all j, lc and W . Equation (1) says that if a Y-measurement results in outcome j and a later X-measurement results in outcome k, then a subsequent Ymeasurement will again result in outcome j with certainty in state W . Our first result shows that (1) holds if and only if X and Y are compatible and Y is sharp.

THEOREM 12. pw(Bj I Bj o Ak) = 1 for every j, k and W if and only if AiBj BjAi and Bi E P ( H ) for every i and j.

=

Proof. Since conditional probability is countably additive in its first argument, we have that (1) implies

(2) PW (Bi I Bj 0 Ak) = 0 for i # j . Applying the definition of conditional probability (2) becomes

' ' )can write (3) as for all i # j whenever t r ( ~ k ~ i ~ ' ~#~0.i We

Now (4) holds even if ~ ~ ( A ~ B ~ ' ' w B = ~ " )0 because in this case

A ~ ~ ' B ~ ~ ' W B ~ '= ' A0.~Since ~ ' (4) holds for every W we conclude that

for all i # j . We then obtain

Hence, B : ~ ' A ~ " B ~= ~ '0 so that B i ~ i l ' B j= 0 for all i # j . Summing over i # j and applying C Bi = I we have

Thus,

A;/'B~= B ~ A ; / ' B= ~ ( ~ ~ ~ : l= ~B g~ k112 A ~ ) * We conclude that AkBj = BjAk for all j and k. Now (5) becomes

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for all i # j . Summing (6) over k gives B;I2BiB;I2 = 0 for all i summing over i # j we have that

Hence, B j

= B;

# j and now

so Bj E P ( H ) for all j. The converse is straightforward.

It is easy to see that Criterion (1) does not imply that Ai E P ( H ) so X need not be sharp. This shows that Criterion (1) is not symmetric in X and Y. The mathematical form of Criterion (2) becomes

for every j and W. Equation (7) says the probability that a Y-measurement results in outcome j coincides with the sum of the probabilities that an X-measurement results in its possible outcomes and a later Y-measurement results in outcome j . THEOREM 13. If AiBj W.

= BjAi

for every i and j then (7) holds for every j and

Proof. In terms of traces, (7) becomes

If AiBj = BjAi, then the right side of (8) becomes

The converse of Theorem 13 does not hold except for some special cases. For example, the converse holds when dim H < ca or when X has just two measurement operators. In general, suppose that (7) holds for every j and W. Then by ( 8 ) we have that i

for every j . Now it is shown in [Arias, 19921 that for any infinite dimensional Hilbert space H there exists a measurement {Ai E E ( H ) : 1 5 i 5 5) and a B E E ( H ) such that B = C A:/~BA:/~ and yet BAi # AiB for all i. Letting X = {Ai) and Y = {B, I - B ) we see that X and Y satisfy Criterion (2) (Equation ( 7 ) )but are not compatible. This example also shows that Criterion (2) is not symmetric in X and Y. Indeed, writing Y = {B1, B2) where B1 = B, B2 = I - B, it follows from the work in [Busch and Singh, 1998; Gudder and Nagy, 20011 that Ai # Zj B:/~A~B:/~ for some 1 5 i 5 5. Finally, the mathematical form of Criterion (3) becomes

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for all i,j and W. By Theorem 5 Criterion (3) for non-disturbance of measurements is equivalent to their compatibility. This gives further justification for Kirkpatrick's choice of this criterion for his study of noncompatibility in classical prob ability theory [ ~ i r k ~ a t r i c20041. k, We have seen that Criterion (3) gives the most appropriate definition of nondisturbance for measurements. Following Kirkpatrick [Kirkpatrick, 2003; Kirkpatrick, 20041, we now briefly show that there are even disturbing measurements in classical probability theory. Suppose we have a deck of ordinary playing cards and we consider two measurements, FACE and SUIT. For the FACE measurement we draw a card at random and observe its face value, ace, two, . . ., king and for the SUIT measurement we draw a card at random and observe its suit, club, diamond, heart or spade. Let K be the value "king" for FACE and H the value "heart" for SUIT. Most of the usual probability problems for cards are one of two kinds, we either always replace the chosen card or always discard the chosen card. In either of these two cases, two card draws are independent. We then have for any standard probability distribution p that P(K 0 H ) = P(K)P(H) = P(H0 K ) Of course, this holds for any values of FACE and SUIT so these two measurements do not disturb each other in these two cases for the usual probability distributions. We now consider the situation in which we draw a card at random, replace it if its suit is spades and discard it otherwise. For simplicity, suppose we have a deck with just four cards, the kings and queens of spades and hearts K H , KS, QH, QS. Letting p be the uniform distribution, we obtain

and

In accordance with Criterion (3), FACE and SUIT disturb each other. A strange property of this example is that the measurement SUIT disturbs itself and this does not happen for sharp quantum measurements. To verify this disturbance, we have 1 2 1 p(HoS)=p(H)p(SI H ) = ? - ? = 3 and 1 1 1 p(S0H) =p(S)p(HIS)=- - - = 2 2 4

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Another problem with the previous example is that it does not have the important property of repeatability possessed by sharp quantum measurements [Kirkpatrick, 2003; Kirkpatrick, 20041. That is, if a sharp measurement is immediately repeated then the same outcome should be obtained. We now give an example that satisfies repeatability. We again have the two measurements FACE and SUIT. But now if a FACE measurement is performed and value F is obtained then all cards with other face values are discarded and we next draw from a deck all of whose cards have face value F. A similar procedure is followed for a SUIT measurement. For simplicity we assume that the deck contains only six cards, K S , QS, K H , QH, K D , QD and that p is the uniform probability distribution. We shall consider three preparation states depending on the cards available in the initial drawing. These states can be prepared by performing a sequence of measurements. The first two states may be considered to be "mixed states" and the third a "pure state." We first start with all six cards. We then obtain

and

We next start with KS, QS, K H , QH. We now obtain

and

Finally, we start with KS, QS. We now obtain

and p(HoK)=O We again conclude that FACE and SUIT disturb each other.

Stan Gudder

9 SEQUENTIAL EFFECT ALGEBRAS This section discusses a framework that unifies and generalizes our previous prob abilistic structures. The framework is obtained by abstracting the properties of the orthogonal sum @ and the sequential product o. If @ is a partial binary operation, we write a I b if a @ b is defined. An effect algebra [Dvure~enskijand Pulmannovh, 2000; Foulis and Bennett, 19941 is a system (E, O , 1 , @) where O , 1 are distinct elements of E and @ is a partial binary operation on E that satisfies the following conditions. (El) I f a I b , t h e n b I a a n d b @ a = a @ b . (E2) If a I b and c I(a@b),then b I c, a I (b@c)and a@(b@c)= (a@b)@c.

(E3) For every a E E there exists a unique a' E E such that a I a' and a@af=l. (E4) If a 1 1 , then a = 0. We call the elements of E effects and whenever we write a @b we are implicitly assuming that a I b. We define a b if there exists a c E E such that a @c = b. If such a c E E exists, then it is unique and we write c = b e a. It can be shown that (E, 5,') is a partially ordered set with 0 5 a 5 1for all a E E, a" = a, and a 5 b implies b' a'. Moreover, we have a I b if and only if a b'. If a I a we call a an isotropic element and when 0 is the only isotropic element of E, then we call E an orthoalgebra. An element a E E is s h a r p if a A a' = 0 and we denote the set of sharp elements by Es. It is easy to show that E is an orthoalgebra if and only if E = Es. Also, O , 1 E Es and a E Es if and only if a' E Es. If E and F are effect algebras, we say that 4: E -,F is additive if a Ib implies $(a) I +(b) and 4(a@b) = 4(a)@qj(b). If 4: E + F is additive and 4(1) = 1,then 4 is a morphism. If 4 : E + F is a morphism and 4(a) I 4(b) implies that a Ib, then 4 is a monomorphism. A surjective monomorphism is an isomorphism. It is easy to see that a morphism 4 is an isomorphism if and only if 4 is bijective and 4-I is a morphism. A s t a t e on E is a morphism s : E --t [0, 11. We interpret s(a) as the probability that the effect a is observed (has answer yes) when the system is in state s. We denoted the set of states on E by R(E). An effect algebra E is a a-effect algebra if for any increasing sequence a1 5 a2 5 -.. we have Vai exists in E. A morphism 4 : E -t F between a-effect algebras is a a-morphism if whenever a1 a1 we have q5(Vai) = V+(Q). Strictly speaking, in the sequel we should consider a-effect algebras and a-morphisms because our probabilistic structures satisfy the axioms of a a-effect algebra. However, to avoid technicalities that the interested reader can easily fill in, we shall only consider effect algebras and their morphisms. We now endow an effect algebra with additional structure by providing it with a sequential product. For a binary operation o, if a o b = b o a we write a I b. A sequential effect algebra (SEA) is a system (E, O , 1 , @, o) where (E, O,1, @)

<

<

<

< <

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141

is an effect algebra and o: E x E + E is a binary operation that satisfies the following conditions. (Sl) b H a o b is additive for all a E E. (S2) 1o a

=a

for all a E E.

(S3) If a o b = 0, then a I b. (S4) If a I b, then a I b' and a o (b o c) = (a o b) o c for all c E E.

We call an operation that satisfies (S1)-(S5) a sequential product on E. If a I b for all a, b E E, we call E a commutative SEA. We now present some examples of SEA'S including our previously studied probabilistic structures.

Example 1. For a Boolean algebra L3, define a Ib if a A b = 0 and in this case a @ b = a V b. Then (B,0,1, @) is an effect algebra that happens to be an orthoalgebra. In particular, if X # 0, then (2X,0, X, @) is an effect algebra. Defining a o b = a A b, (B,0,1, @, o) becomes a SEA. We shall show later that o is unique.

+

Example 2. For [O,1] R, define a I b if a b 5 1 and in this case a @ b = a b. Then ([0, 1],0,1,@) is an effect algebra. The only sharp elements are 0 and 1. Defining a o b = ab, ([O,1],0,1,@, o) becomes a SEA. We shall show later that o is unique.

+

Example 3. Let X # 0 and let 3C [0, 1IX.We call 3 a fuzzy set system on X if the following conditions hold. (i) The functions O , 1 E 3. (ii) If f E 3,then 1- f E 3.

(ii) If f , g ~ T w i t hf + g 5 1, then f + g € 3 . (iv) If f ,g E 3,then f g E 3. In this case, ( 3 , 0,1, @) is an effect algebra with f @ g = f + g whenever f - g 5 1. If 3= [0, 1IXwe call 3 a full fuzzy set system. The sharp elements of a fuzzy set system 3 are the characteristic functions in 3 which can be identified with the (sharp) subsets of X in 3. Indeed, if f E 3 is sharp, then f (1 - f ) 5 f , 1 - f implies that f (1 - f ) = 0. Hence, i f f (x) # 0,then f (x) = 1. A fuzzy set system 3 is a SEA under the operation f o g = fg. We shall show later that if .F is full, then o is unique.

Example 4. Let E(7-l) be the set of quantum effects on the Hilbert space 'FI. For A, B E E(7-l) we define A I B if A + B E &('FI) and in this case 0, I,@) is an effect algebra. The sharp elements of A €9 B = A B. Then (&(a), I(%) are the set of projection operators P(7-l)on H . Our previous work shows

+

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Stan Gudder

that E(7-l) is a SEA under the operation A o B = A~/'BA'/'. Unlike the previous three examples, E(7-l) is a noncommutative SEA. We do not know whether o is unique. However, o is unique if it satisfies some additional conditions. There are many other examples of SEA's but these four are the important ones for our purposes. We now observe that a SEA gives a general framework for noncommutative probability. If s E Q(E) is a state of E then s gives a (finitely additive) probability measure on E. A morphism h: B(W) t E corresponds to an observable and the morphism s o h : B(W) t [O,1] gives the distribution of h in the state s. We can now define the expectation and variance of h in the usual way. If a E E with s(a) # 0 we can define the conditional probability of b E E given a as s(b I a) = s ( a o b)/s(a). Notice that b I-+ s(. I a) is a state on E so we can develop a theory of conditional probabilities on E. We now state some basic results about a SEA E. The proofs are not difficult and may be found in [Gudder and Greechie, 20021. LEMMA14. (i) a o O = O o a = O a n d a o l = l o a = a f o r a E E . (ii) a o b s a for all a, b E E. (iii) If a 5 b, then c o a 5 c o b for all c E E. (iv) The following statements are equivalent. (a) a 6 Es. (b) a o a' = 0. (c) a o a = a. LEMMA 15. (i) If a o b = 0 then a Ib. (ii) For a E El b E Es, a o b = 0 if and o n l y i f a l b . (iii) F o r a , b ~ Es w i t h a I b we h a v e a @ b Es. ~ It follows from Lemma 15(iii) that Es is a sub-effect algebra of E that is an orthoalgebra. We have seen previously that in general, if a, b E Es then a o b $! Es so Es is not a sub-SEA of E . THEOREM16. L e t a E E a n d b E E s . (i) a I b i f a n d o n l y i f a o b = b o a = a and b 5. a if and only if a o b = b o a = b. (ii) If a I b, then a A b = a o b. (iii) If a Ib, then a @ b = a V b = (a' o b')'. We say that a, b E E coexist if there exist c, d, e E E such that c @ d @ e is defined and a = c@d, b = c@e [Busch et al., 1995; Busch et al., 19911. We say that a, b E Es are compatible if there exist mutually orthogonal elements c, d, e E Es such that a = c V d and b = c V e . THEOREM 17. (i) If a I b then a and b coexist. (ii) For a E E, b E Es, a I b if and only if a and b coexist. (iii) For a, b E Es the following statements are equivalent. (a) a I b. (b) a and b are compatible. (c) a and b coexist. It is well known that the converse of Theorem 17(i) does not hold [Gudder and Greechie, 20021. We now show that for certain important examples of SEA's, the sequential product is unique. THEOREM 18. For an eflect algebra E that is a Boolean algebra, there is a unique sequential product a o b = a A b.

Proof. Since E is a Boolean algebra, we have E = Es. For a, b E E, define c = a A b , d = a G a A b , e = b e a A b . It iseasy tocheckthat c,d,earemutually

Quantum Probability

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orthogonal and that a = c V d and b = c V e. Hence, any two elements of E are compatible. By Theorem l7(iii) we have that a I b for every a, b E E and any rn sequential product o. It follows from Theorem 16(ii) that a o b = a A b. THEOREM 19. There i s a unique sequential product o n the effect algebra [O,1] C

R. Proof. Let o be a sequential product on [0, 11. Then for any integer n 2 1 and a E [0, 11 we have

so that a 0 ( l l n ) = (1ln)a. Also, for any integer 1 5 m 5 n we have

Hence, for any rational number r E Q n [O, 11 we have a o r = ar. Now let b E [O,1] be irrational. If r E Q n [O, 11 and b < r then by Lemma 14(iii) we obtain

Similarly, if r E Q n [O,1] and b > r, then

Since Q n [O, 11 is dense in [0, 11, we obtain a o b = ab.

1

A SEA isomorphism is an effect algebra isomorphism that preserves the sequential product. The proof of the next result is a straightforward verification. THEOREM 20. Let E, F be effect algebras and let 4: E + F be an effect algebra isomorphism. If o i s a sequential product o n E, then a * b = 9 [+-'(a) o +-'(b)] i s a sequential product on F . Moreover, ( E ,o) and (F,*) are SEA isomo7phic. COROLLARY 21. If E and F are isomorphic effect algebras and E admits a unique sequential product o, then F admits a unique sequential product. Proof. By Theorem 20, F admits a sequential product *. Let 4 : E -+ F be an effect algebra isomorphism and define -:E x E + E by a . b = 4-' [+(a)* +(b)]. By Theorem 20, . is a sequential product on E so we have a . b = a o b. Hence,

Thus, for every c, d E F we have

It follows that

* is unique.

Stan Gudder

144

Let (Ei, Oi, li,@) be a collection of effect algebras. We obtain a new effect algebra by taking the cartesian product Ej and defining ai I bi if ai Ibi for each i in which case ai @ bj = (ai CBbi). Then ( n Ei, Oi, li,a ) is an effect algebra. If each of the Ei's is a SEA, then Ei becomes a SEA with the componentwise sequential product.

n

n

n n

n

n

n n n n

THEOREM 22. Let (Ei, Oi, li,ai,oi) be SEA'S i E I. Then E = Ei admits a unique sequential product zf and only if each Ei, i E I, admits a unique sequential product.

n

Proof. If Ej admits two sequential products for some j E I, then clearly Ei admits at least two sequential products. Conversely, suppose Ei, i E I, admits a unique sequential product oi and let * be a sequential product on Ei. For j E I, let f j E Ei be defined by

n

n

Clearly, f j E Es. Let o be the sequential product on E given by (f o g)(i) = f (i) oi g(i), i = I, for any f , g E E. Since f o f j = f j o f , by Theorem 17(ii), f and f j coexist. But coexistence is independent of the sequential product so again by Theorem 17(ii), f * f j = f j * f for any f E E . It follows from Theorem 16(ii) that f * f j = f A fj. Hence,

For any f , g E E we have

Now [0, fj] given by

E is an effect algebra with greatest element f j and 4: Ej + [0,fj]

is an effect algebra isomorphism. Since Ej admits a unique sequential product, by Corollary 21, [O, fj] admits a unique sequential product. Hence, (f

* g>(j)= [(f * 9) * fjl (j) = [(f * fj) * (9 * fj)l (j) = [(f * f j ) 0 (g * fj)I (j) = (f * f d j ) O j (9 * fj)(j) = f (j)

oj ~ ( j = ) (f og)(j)

Hence, o is the unique sequential product on E.

145

Quantum Probability

COROLLARY 23. A full fuzzy set system E product.

=

[0, 1IXadmits a unique sequential

Proof. This follows from Theorem 22, because by Theorem 19, [O, 11 admits a rn unique sequential product. Whether a general fuzzy set system admits a unique sequential product is an unsolved problem. One can give examples of effect algebras that admit no sequential product and effect algebras that admit many sequential products u udder and Greechie, 20021. THEOREM 24. Let E be a SEA. For p, q E Es we have p o q E Es if and only if P l q-

Proof. If p I q it is clear that p o q E Es. Conversely, suppose that r = p o q E Es. We first show that r 5 q. Since r o q 5 r and r 5 p, we have

By Lemma 15(ii), p I(r 8 r o q). Hence,

Since r 8 r o q 5 r E Es we have that r 8 r o q = 0. Thus, r = r o q so that r o q' = 0. Again by Lemma 15(ii) we have that r 5 (q')' = q. We now show that p o q ' s q ' . S i n c e p o ( q 8 p o q ) = O we have that q e p o q < p ' s o that

Hence, p o q' 5 q'. Applying Theorem 16(i) we have that p o q I q and p o q' Since p = p o q CBp o q' we conclude that p I q.

I q.

rn

For arbitrary elements p, q E E, it is an unsolved problem whether p o q E Es implies that p I q. This result does hold for p, q E E(7f). LEMMA 25. If a o b = a, then a I b and a 5 b.

Proof. Since a o b = a we have

Hence, a o b' = 0. Thus, a I b' so that a I b. Therefore, a = a o b = b o a

5 b.

H

It is unknown whether b o a = a implies that a I b THEOREM 26. Let E be a S E A and suppose a , b E Es. (i) a o b 5 b if and only if a 1 b. (ii) a o b b if and only if a o b = b o a = b.

>

Stan Gudder

146

Proof. (i) If a I b, then a o b = b o a 5 b. Conversely, assume that a o b 5 b. We now show that a o b E Es. Since a o b 5 a, b, applying Theorem 16(i) we have a o b I a and a o b I b. Hence,

The result follows from Theorem 24. (ii) Suppose that a o b 2 b. Then since a o b I b, a o b I a as before we have ( a o b ) o ( a o b ) = [(aob) o a ] o b = ( a o b ) o b = b Taking the sequential product on the left with a gives

Since a o b E Es by Theorem 24, a I b. Hence,

The converse is trivial.

BIBLIOGRAPHY [Arias, 19921 A. Arias, A. Gheondea and S. Gudder, Fixed points of quantum operations J. Math. Phys. 43 (2002 , 5872-5881. [Arias and Gudder, 2004 A. Arias and S. Gudder, Almost sharp quantum effects, J. Math. Phys. 45 (2004), 4196-4206. [Beltrametti and Cassinelli, 19811 E. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics, Addison-Wesley, Reading, Mass., 1981. [Busch et al., 19951 P. Busch, M. Grabowski and P. J. Lahti, Operational Quantum Physics, Springer, Berlin, 1995. [Busch et al., 19911 P. Busch, P. J. Lahti and P. Middlestaedt The Quantuna Theory of Measurements, Springer, Berlin, 1991. [Busch and Singh, 19981 P. Busch and J. Singh, Liiders theorem for unsharp measurements, Phys. Lett. A 249 (1998), 10-24. [Davies, 19761 E. B. Davies, Quantum Theory of Open Systems, Academic Press, New York, 1976. [Dvureknskij and Pulmannavli, 20001 A. DvureEenskij and S. Pulmannovfi, New !&ends in Quantum Structures, Kluwer, Dordrecht, 2000. [Foulis and Bennett, 19941 D. Foulis and M. K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325-1346. [Giuntini and Greuling, 19891 R. Giuntini and H. Greuling, Toward a formal language for unsharp properties, Found. Phys. 19 (1989), 931-945. [Gudder and Greechie, 20021 S. Gudder and R. Greechie, Sequential products on effect algebras, Rep. Math. Phys. 49 (2002), 872-111. [Gudder and Nagy, 20011 S. Gudder and G. Nagy, Sequential quantum measurements, J. Math. Phys. 42 (2001), 5212-5222. [Kirkpatrick, 20031 K. A. Kirkpatrick, "Quantal" behavior in classical probability, Found. Phys. Lett. 16 (2003), 199-224. [Kirkpatrick, 20041 K. A. Kirkpatrick, Compatibility and probability, amiv: quant-ph/0403021, 2004. [Nielsen and Chuang, 20001 M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Canabridge University Press, Cambridge, 2000. [Rudin, 19911 W. Rudin, Functional Analysis McGraw-Hill, New York, 1991.

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HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

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QUANTUM LOGICS AS UNDERLYING STRUCTURES OF GENERALIZED PROBABILITY THEORY

Pave1 Pt &k, Sylvia Pulmannovb

1 INTRODUCTION

As convincingly argued in [ ~ a c k e1963; ~ , Beltrametti and Cassinelli, 1981; Gudder, 1979; Varadarajan, 1985; Sarymsakov et al., 1983; Svozil, 19981, etc., one of the structures especially suited for considering a general (quantum) probability theory is that of a countably complete orthomodular poset (a "quantum logic", a term having its origin in the seminal paper by Birkhoff and von Neumann [~irkhoffand von Neumann, 19361). From the point of view of theoretical physics, this structure seems to be well suited for dealing with the phenomenon of noncompatibility. From the purely mathematical point of view, it is challenging too because it permits a uniform generalization of probability theory on both projection lattices and Boolean algebras. The conceptual generality inherent in the quantum logic approach opens a new field for interesting combinatorial and measure-theoretic considerations which, in turn, may enrich the study of quantum theory by which they were motivated. (In the short appendices we consider the uncertainty relations and the Bell inequalities as matters related to quantum stochastics.) This chapter adds some novelties - both in the sense of updating and in style - to the stochastic results published in the book [ P ~and B PulmannovL, 19911. Conceptually, the treatment presented here to a certain extent overlaps with the book cited, of course. Our intention was to make this presentation as self contained as possible. In this, however, we had to make a compromise between size and readability. The reader will see that there are some important technical results which we use in our considerations and for which we refer him to other sources. Certain important and interesting topics related to quantum probability theory such as e.g. quantum measurements [ ~ u s c het al., 19911 (see also [Pulrnannovb, 1994bl) or unsharp quantum logics [Dalla Chiara et al., 20041 are not included, and concerning the probability theory on projection lattices the reader is referred to [Hamhalter, 20031.

Pave1 Ptik, Sylvia Pulmannovi

2 BASIC DEFINITIONS AND FACTS In this section, we introduce basic definitions and results which we will need in the sequel. The emphasis is on the intrinsic properties of quantum logics and on the compatibility relation. More details can be found in [Beran, 1985; Beltrametti and Cassinelli, 1981; Kalmbach, 1983; Piron, 1976; PtLk and PulmannovL, 1991; RCdei, 1998; Varadarajan, 19851. DEFINITION 1. A partially ordered set (L, 5) endowed with a unary operation ' : L + L is called a quantum logic, or simply a logic, if (i) there exists a greatest element, 1, in L; (ii) a 5 b

=+

a' 2 b' (a, b E L);

(iii) a" = a (a E L);

(iv) a V a' = 1 (a E L); (v) for every sequence (ai)i of elements of L such that ai the supremum Vi ai exists in L;

< a$ whenever i # j

<

b (a, b E L) then there is a c E L with the properties c 5 a' and (vi) if a b=aVc.

In the above definition, the symbols V and A denote supremum and infimum in L respectively, if they exist. Property (vi) is called the orthomodular law. A quantum logic is alternatively called an orthomodular a-poset. If property (v) holds only for finite sequences, L is called an orthomodular poset. Property (ii) implies the existence of a smallest element of L, which we denote by 0. Obviously, 0 = 1'. It may be noted that (iv) follows from (vi) and, also, (v) for finite sequences follows from (vi), but economy in our axiomatic setup is not our aim. Property (ii) implies that the de Morgan laws hold in L : if Vai exists in L, then ~ a exists : in L and the equality (Vai)' = Aai holds true. DEFINITION 2. Two elements a, b E L are said to be orthogonal (written a Ib) if a b'.

<

DEFINITION 3. Two elements a, b E L are said to be compatible (written a * b) if there are pairwise orthogonal elements a l l bl, c in L such that a = a1 V c and b = bl V C. LEMMA 4. If a I b (a, b E L), then (a V b) A a' = b. Proof. If a I b, then (a V b) A a' exists, in view of a I (a V b)' and (a V (a V b)')' = a' A (a V b). Clearly, b (a V b) A a'. The orthomodularity implies that (a V b) A a' = b V d, where d I b. From d 5 (a V b) A a' a' we infer that d a' A b' = (a V b)', but this together with d a V b yields d = 0. rn

<

<

LEMMA 5. For any a, b E L,

<

<

Quantum Logics as Underlying Structures of Generalized Probability Theory

(i) i f a Ib, then a

o b;

(ii) if a 5 b, then a

o b;

149

(iii) i f a i+ b, then { a , at, b, b') is a set of pairwise compatible elements.

.

Proof. I t suffices t o prove that a o b implies at * b. Let a * b, then a = a1 V c, i b = bl V c , where a l , bl, c are pairwise orthogonal. T h e n b y Lemma 4, ( ab l~) ~ b = a, hence a' = (a' A b:) V bl, and therefore the elements c, bl, ( a V bl)' are pairwise orthogonal.

L E M M A 6 . If a o b, then a V b and a A b exist in L and a A b = c, a V b = a1 V bl V c, where a l , bl, c are the mutually orthogonal elements in the definition of compatibility. Moreover, a1 = a A b', bl = b A a'.

Proof. B y orthogonality, a1 V bl V c exists, and this element clearly dominates both a and b. O n the other hand, if e E L dominates a and b, then a1 V c 5 e, bl V c 5 e and this implies that a1 V bl V c 5 e. W e see that a1 V bl V c = a V b. T h e existence o f a A b follows from the fact that a A b = ( a fV b')' and from Lemma 5 (iii). B y Lemma 4, a A b = (al V c) A b 5 (b' V c) A b = c, which entails that a A b = c. Finally, a1 5 a a1 5 b', hence a1 5 a A bt. O n the other hand, a A b ' = ( a l V c ) A b f 5 ( a l V c ) A d = a l . Theproofofbl=bAa'isanalogous.

L E M M A 7. For a,b E L , a anda~c'sb'.

.

o

b i f and only i f there is a c E L with c 5 a , c 5 b

Proof. Suppose a * b. T h e n a = a1 V c, b = bl V c, where c 5 a, c 5 b and a A c' = a1 5 b'. Conversely, let c 5 a , c 5 b and let a A c' 5 b'. From the orthomodularity, there are a l , bl such that a = a1 V c, b = bl V c, where a1 Ic, bl Ic. Moreover, ¤ a1 = a A c' 5 bt 5 b:. I t follows that a b.

-

L E M M A 8. If b tt a$, i = 1 , 2 , . . . and the elements V z l ai, L, then b * V z l ai and b A ( V z l ai) = V iM = l ai A b-

V z l ai A b exist in

Proof. Obviously, V z l ai A b is not greater than V z l ai and b. B y Lemma 7, b o ai implies that b A (b A ai)' 5 a: for all i. T h i s entails that b A ( V z l b A ai)' 5 (b A (b A ai)' 5 a: for all i and hence b A ( V z l b A ad)' 5 (VE1ai)'. T h i s implies again by Lemma 7 that b o V z l ai. T h e rest o f the proof can b e obtained b y reasoning as in t h e second part o f the proof o f Lemma 7 with c = VEl ai A b and an application o f Lemma 6 .

In what follows, let us denote b y [a,b] the set { c E L : a 5 c 5 b), where a , b L~, a 5 b. L E M M A 9. Let a E L , a # 0 . Then the set [O,a]with the ordering inherited from L and with the orthocomplementation defined by setting bla := b' A a, b E [0,a ] , is a logic.

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Pave1 PtAk, Sylvia Pulmannovd

Proof. Let b 5 a. Then b' A a exists by orthomodularity. The inequalities bl j b2 j a ilpp?y b> j b k , and thus b V bta = b V ( M A a) = a by orthomodularity, too. Also, (b ") a = (b' A a)' A a = (b V a') A a = b by Lemma 4. This proves that b H bfa is an orthocomplementation in [0, a]. Orthomodularity follows from ort homodularity in L. rn The logic of Lemma 9 will be denoted by L[O, a]. LEMMA 10. If for every a, b E L the supremum a V b exists in L then the logic L is a a-lattice.

Proof. Let (ai)i be a sequence of elements of L. Put bl = a l , . . . ,b, = a1 V a2 V . . V a, (n E U). Then bl ,b2 A b', , . . . is a sequence of orthogonal elements, hence rn ai. its supremum, b, exists. It is easily seen that b =

-

VE1

Note that a logic L becomes a Boolean a-algebra if it is a distributive a-lattice. The next theorem provides a characterization of Boolean a-algebras among logics. THEOREM 11. A logic L is a Boolean a-algebra if and only if a a, b E L.

tt

b for all

Proof. Necessity is clear. For sufficiency, if a * b for all a, b E L, then by Lemma 6, a V b and a A b exist in L for all a, b E L. By Lemma 10, L is a a-lattice. Distributivity follows from Lemma 8. A logic L which is also a lattice is called a lattice logic. Obviously, a lattice logic is an orthomodular a-lattice. Let us say that (al, a2, as) C L is a distributive triple if ai A (aj V ak) = (ai A aj) V (ai A ak) for every permutation {i, j, k) of the set {1,2,3). THEOREM 12 (Foulis-Holland). Let a, b, c be elements in a lattice logic L such that a chosen one is compatible with the other two. Then (a, b, c) is a distributive triple.

Proof. Assume, e.g., that a * b, a * c. By Lemma 8 we have a tt b V c and a ~ ( b v c= ) ( a ~ b ) V ( a ~ cIt) .suffices toprove that (aVb)Ac= (aAc)V(bAc). By the De Morgan laws this is equivalent to (a A b) V c = (a V c) A (b V c). But ( a v c ) ~ ( b ~=c( )a ~ c A) [ ( ~ A ( ~ V C ) V ~ ' A = ( ~a~ VC ( b) V ] c ) ~ c =( a ~ bV)C , where we used Lemma 8 and compatibility of aV c with a A ( b c)~and a' A (b V c).

rn A subset M of a lattice logic L which has the property that among any three arbitrarily chosen elements of M there is one which is compatible with the other two, is called a Foulis-Holland set. Greechie [Greechie, 19791 (see also [~almbach, 19831 for a simpler proof) proved that the sublattice of L generated by a FoulisHolland set M is distributive. The following lemma is a strengthening of Lemma 8 for lattice logics.

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THEOREM 13. Let L be a lattice logic. Let M be a subset of L such that V M := V{m : m E M ) exists in L, and let b cr m for all m E M. Then b V M and V{bAm:rn€m)=(VM)~b.

-

Proof. Evidently, (V M ) A b 2 m A b for all m E M. Let u E L be such that u > m A b f o r a l l m E M . T h e n z : = u A ( V M ) A b ~ m ~ b f o r a l l rM n .~ m for all m E M it follows that We will show that z = (V M) A b. From b bA(btV(VM)') 5 bA(b'vmt) = b A r n t < bA(VM)'. By Lemma7, b - V M . From orthomodularity we have that (V M ) A b = z V d, where d I z. Thus d 5 ( V M ) A b a n d d 5 z' 5m'Vb'. Thisentailsthat d 5 (m'Vbf)Ab=m'Abfor all m E M . Hence d 5 (V M)', which together with d 5 V M entails that d = 0. We have shown that u 2 (V M) A b, so that (V M) A b is the supremum of the set { b ~ m : m ~ M ) . DEFINITION 14. Let L1 and L2 be logics. A mapping h : L1 + L2 is said to be a a-homomorphism if (i) h(1) = 1; (ii) h(at) = h(a)' (a E L1);

VEl h(ai)

(iii) h ( V z l ai) = elements of L1.

for every sequence (ai)i of mutually orthogonal

A bijective a-homomorphism h such that h-I is also a a-homomorphism is called an isomorphism. If L1 = La, then an isomorphism is called an automorphism. A a-homomorphism is a lattice a-homomorphism if h(a V b) = h(a) V h(b) whenever a V b exists in L1. If the condition (iii) is required to be satisfied only for finite orthogonal sequences, then h is called a homomorphism. Let us observe that if h : L1 t Lz is a a-homomorphism and (ai)i is an arbitrary sequence of elements in a lattice logic L1, then h(Vi ai) = Vi h(ai). Note also that if L1 is a lattice logic and L2 is an arbitrary logic, then every a-homomorphism h : L1 + L2 is a lattice a-homomorphism exactly when L1 is a Boolean a-algebra [~afiasovb,19811. DEFINITION 15. A subset L1 of a logic L is called a sublogic when L1 is a logic with the ordering and complementation inherited from L and when the identity mapping id : L1 -+ L is an injective a-homomorphism. A sublogic is called a lattice sublogic if the following implication holds: Whenever (ai)i is a sequence of elements in L1 such that Vi ai exists in L, then Via* belongs to L1. A sublogic L1 of a logic L is called a Boolean sublogic if (i) for every sequence (ai)i of L1 the supremum Vi ai exists in L and belongs to L1; (ii) every (a, b, c) c L1 is a distributive triple. DEFINITION 16. We say that a logic L is regular if for every a, b, c in L which are pairwise compatible we have a * b V c (equivalently, a * b A c).

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Let A be an arbitrary subset o f a logic L . Put C ( A ) := {a E L : a * b f o r every b E A } . W e easily see that, for every A , B

(i) A C B (ii) A

c L,

=+ C ( B ) c C ( A ) ;

c C(C(A));

(iii) C C ( C ( A ) )= C ( C C ( A ) )= C ( A ) . T h e set C ( L ) is called the center o f L. L E M M A 17. For every subset A L , the set C ( A ) is a sublogic of L . Proof. Evidently, 0 and 1 belong t o C ( A ) . If a E C ( A ) , then also a' E C ( A ) b y Lemma 5. I f (ai)iENis a sequence o f pairwise orthogonal elements in C ( A ) , then Viewai exists in L , and for every b E A we have b * ai, i = 1 , 2 . ... Hence b A ai exists. A s ( bA ai)iEn is an orthogonal sequence, ViEN bA ai exists in L . B y Lemma rn ai, and hence ViEN ai E C ( A ) . 8, b * ViEN L E M M A 18. If B is a subset of pairwise orthogonal elements in a regular logic L , then C ( C ( B ) ) is a Boolean sublogic of L . Proof. B y Lemma 17, C ( C ( B ) )is a sublogic o f L . Since B is a pairwise compatible set, we have B C C ( B ) , which entails that C ( C ( B ) ) = C ( C ( C ( B ) ) ) ,so that C ( C ( B ) )is a compatible set. Let a , b E C ( C ( B ) )and c E C ( B ) . T h e n a , b, c are pairwise compatible. T h e n regularity implies that c * a V b, so that a V b E C ( C ( B ) ) . It follows that C ( C ( B ) )is a lattice. Moreover, ( a ,b, c) C C ( C ( B ) )is a distributive triple for every a, b, c. PROPOSITION 19. The center C ( L ) of any logic L is a Boolean sublogic of L . Proof. B y Lemma 17, C ( L ) is a sublogic o f L . I f a , b E C ( L ) , then a * b, therefore a V b exists in L . Moreover, a A c and b A c exist in L for all c E L , and in addition, a A c tt b A c. B y Lemma 8, aV b E C ( L ) . Since a * b for all a, b E C ( L ) , C ( L ) is a Boolean sublogic. THEOREM 20. Let L be a regular logic. Let K := { a , : a E I } be a subset of pairwise compatible elements in L . Then there exists a Boolean sublogic B of L such that K B C_ L . Proof. W e have K C C ( C ( K ) ) ,and b y Lemma 18, C ( C ( K ) )is a Boolean sublogic of L. rn DEFINITION 21. A logic L is called separable if every set o f nonzero pairwise orthogonal elements in L is at most countable. PROPOSITION 22. Let L be a separable lattice logic. For every set A C L there exists a countable subset (ai)iErpisuch that V A = V z 1ai. Consequently, L is a complete lattice.

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Proof. Let A c L. Let B denote the set of all suprema of at most countable subsets of A. Let C := {b A c' : b E B, c 5 b}. By Zorn's lemma, there exists a maximal subset of pairwise orthogonal elements in C , let us say { b l A d , b2Ac&,. . .}, bi. Then clearly b E B. If there which is at most countable. Let b = is a E A such that a $ b, then 0 # (b V a) A b' E C. Then the inequalities b i A d 5 b 5 (bVa)'Vbfori=1,2, ...imp lythat ( b ~ a ) ~ Ibb i' A c ' , , i = 1 , 2 , ..., which contradicts the maximality of the set { b i < ~ ) E l . This proves that b = V A.

Vzl

rn

DEFINITION 23. An element a in a logic L is called an atom if a inequality b 5 a implies either b = 0 or b = a.

# 0 and the

DEFINITION 24. A logic L is called atomic if for every b E L, b # 0, there is an atom a E L with a 5 b. A logic L is called atomistic if every element in L can be written as a supremum of all atoms that are less than or equal to L. PROPOSITION 25. A lattice logic is atomic if and only if it is atomistic. Proof. Let L be an atomic lattice logic. Let b E L, b # 0 and let A be the set of all atoms lying under b. Then A # 0. Let c E L be such that a 5 c for all a E A. Then a 5 b A c, and by orthomodularity, b = (b A c) V (b A (b' V c')). If b A (b' V c') # 0, then, by the atomicity of L, there is an atom e 5 b A (b' V c'). But e 5 b implies e 5 b A c, in contradiction with e 5 b' V c'. Hence b = b A c, and rn therefore b = V A.

Let us note in concluding this paragraph that Prop.25 does not have to hold for general logics (see [Greechie, 19691). 3 COMPATIBLE SUBSETS OF A LOGIC In the quantum logic approach to quantum mechanics, a quantum logic L is interpreted as the set of all experimentally verifiable propositions about a physical system. The compatible subsets of L should correspond to those propositions that can be simultaneously verified, and hence can be dealt with as classical propositions. Generally, we may assume that the elements of a subset M of L are compatible if they can be embedded into a Boolean sublogic of L, in accord with the assumption that the logic of a classical system is a Boolean D-algebra. In this section, we will find intrinsic characterizations of compatible subsets of a logic. If a logic L is regular, then by Theorem 20, every pairwise compatible subset of elements of L is contained in a Boolean sublogic of L. The following example shows that this need not hold in an arbitrary logic. EXAMPLE 26. Let M = {1,2,3,4,5,6,7,8) and let L be the set of all subsets of M with an even number of elements. Then L ordered by inclusion and endowed with set-theoretical orthocomplementation is a logic. Let A = {{I,2,3,4}, {1,2,5,6}, {2,3,5,7}}. It is easy to check that elements of A are

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pairwise compatible. But they cannot be contained in a Boolean sublogic of L since {1,2,3,4,5,6)) A {2,3,5,7) does not exist in L. As a result, a stronger condition of compatibility is needed to characterize those subsets of a logic that can be embedded into Boolean sublogics of L. Such a condition, so-called strong compatibility, was found independently in [GUZ, 19711 and [Neubrunn, 19741. Let us introduce it in the next two definitions. DEFINITION 27. Let M be a subset of a logic L. We say that a, b E M are M compatible in M , written a * b, if there are pairwise orthogonal elements a l , bl, c in M such that a = a1 V c and b = bl V c. DEFINITION 28. A subset A of a logic L is said t o be strongly compatible if for every two elements a, b E A, a L(A) * b, where L(A) is the smallest sublogic of L that contains A. Obviously, the smallest sublogic of L that contains A, L(A), exists (L(A) is the intersection of all sublogics of L that contain A). Recall that the smallest sublogic of L that contains A is called the sublogic of L generated by A. THEOREM 29. Let A C L, and let L(A) be the sublogic of L generated by A. Then L(A) is a Boolean sublogic of L if and only if A is strongly compatible. Proof. If L(A) is a Boolean sublogic, then A C L(A) implies that for every a, b E A, a = a A (b V b') = (a A b) V (a A b') by distributivity in L(A) . Since L(A)

is a Boolean sublogic, a A b, a A b' E L(A), hence a L(A) ++ b. Conversely, assume that A is strongly compatible. For a E A, put B, := {b E L(A) : a L(A) * b). By the properties of compatibility, B, is a sublogic of L, and since A c B, by strong compatibility, we have L(A) c B, for every a E A. Now let b E L(A), and put Bb := {c E L(A) : b LLA) c). The first part of this proof implies that A c Bb, and hence L(A) c Bb for all b E L(A). This entails that every pair a, b E L(A) is compatible in L(A). As a consequence, L(A) is a Boolean sublogic of L. Obviously, strong compatibility implies pairwise compatibility. The next example shows that strong compatibility is not equivalent to pairwise compatibility, and also that strong compatibility is not necessary for a subset of L to be embeddable into a Boolean sublogic of L. EXAMPLE 30. Let M = {1,2,3,4) and let L be the Boolean algebra of all subsets of M. Let A = {{1,2), {2,3), {3,4)). The smallest sublogic L(A) containing A consists of all subsets with an even number of elements. The elements of A are pairwise compatible in L but not strongly compatible, (e.g., {1,2) n {2,3) = (2) which does not belong to L(A)). In what follows, we will use the following notation. Let D = (-1, I), and let A C L. Write A' := {a' : a E M) and let a' := a, a-I := a'. Put D := {-1,l).

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Let A = {al, az, . . . ,a,) be a finite subset of L such that for every d E Dn the infimum a? A a? A . . . A a$ exists in L. Then put

and set

com(al,a2,. . . , a n ) =

V{f : f E FA).

DEFINITION 31. Let A = {a1, az, . . . ,a,) be a finite subset of a logic L. We say that A is a compatible set if all elements in FAexist and com(al, az, . . . , a,) = 1. If A is an arbitrary subset of L then A is said to be compatible if all finite subsets of A are compatible. The element com(a1, az, . . . ,a,) (if it exists) is called the commutator of the set {al, az, . . . ,a,). If L is a lattice, then com(A) exists for any finite subset A c L. For any two elements we have com(a, b) = (a A b) V (afA b) V (aA b') V (afA bf ) . This commutator in , (see also [chevalier, 1984; orthomodular lattices was introduced in [ ~ a r s d e n19701 Chevalier, 1989; MatouSek, 19921, etc.). THEOREM 32. A subset A of L is contained in a Boolean sublogic of L if and only if A is a compatible set. Proof. If there is a Boolean sublogic Lo of L such that A C Lo, then A is a compatible set. Assume that A is a compatible subset of L. If M = {al, az, . . . ,a,) is a finite subset of A, then FM= {a:' A a$ A .. A a$ : d E D) is a set of pairwise orthogonal elements of L the supremum of which equals 1. Let B(M) denote the set of suprema of all subsets of FM.It is easily verified that B(M) is a Boolean sublogic of L. Put B = U{B(M) : M a finite subset of A). Then B is also a Boolean sublogic of L. Indeed, if al, az, a3 E B and if MI, M2, M3 are finite subsets of A with ai E B(Mi), i = 1,2,3, then MI U Mz U M3 is a finite subset of A, and B(Ml U M2U M3) contains all a l , a2, as. It follows that any three elements of B are contained in a Boolean sublogic of L. Therefore B is a Boolean subalgebra of L. We still need to verify that B is closed under the formation of suprema of countable subsets. Let Lo be the sublogic of L generated by B. We will show that Lo is a Boolean sublogic of L. Take a E B and put K, := {b E Lo : b 3 a). By Lemma 17, K , is a sublogic of Lo. Since B is Boolean, we have B C Ka. Thus Ka = Lo for all a E B. Further, take b E Lo and put Kb := {c E Lo : c 3 b). Then, again, Kb is a sublogic of Lo. By the first part of this proof, B C Kb. It b in Lo for all a, b E Lo. follows that Kb = Lo for all b E Lo. Hence we have a By Theorem 11, Lo is Boolean a-algebra, hence Lo is a Boolean sublogic of L.

-

The following condition was introduced in [Brabec, 19791.

.

DEFINITION 33. A subset A of a logic L is f-compatible if for every finite subset F = {al, a2,. . . ,a,) of A there exists a finite set G = {g1,g2,. . . ,gk) of pairwise orthogonal elements of L such that for every i = 1,2, . . . ,n there is a subset Hi C G such that ai = V{fj : f j E Hi). The set G is called an orthogonal cover of F.

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The following statement can be proved by a technique similar to the proof of , Theorem 32 ( [ ~ t b kand PulmannovB, 1991, Prop. 1.3.221, see also [ ~ r a b e c1979; Brabec and PtBk, 19821). PROPOSITION 34. Let A be a finite subset of L. Then A is compatible if and only if it is f-compatible. By Lemma 9, for any b E L, the interval [0, b] can be viewed as a logic with the smallest element, 0, the greatest element, b, and with the relative orthocomplementation xlb = x' A b. In the next lemma, we prove that compatibility in L and compatibility in [0,b] are equivalent. LEMMA 35. Let b E L. The elements al, a2,. . . ,a, in the interval [0,b] are compatible in L if and only if they are compatible in [0,b].

Proof. If com(al, a2,. . . , a,) exists and equals to 1, the it is easy to see that com(al, as,. . . , a,) A b is the commutator of al, aa, . . . , a, taken in [0, b], and since it is equal to b, the compatibility in [0,b] follows. Conversely, if elements are compatible in [0,b], then they have an orthogonal cover, which means that they are f-compatible and hence by Proposition 34 they are compatible in L. It is easily seen that for a tweelement set A = {al, a2) the notions of pairwise compatibility, f-compatibility and compatibility coincide. We have already shown that pairwise compatibility is strictly weaker than compatibility. The next theorem and Pulmannovb, 1991, Theorem 1.3.251) shows that also "triplewise", (see [ ~ t B k etc. compatibility is too weak to guarantee compatibility. THEOREM 36. Let n (n 2 2) be a natural number. Then there exists a (finite) logic L, with a subset A, such that the following statements hold: (i) the set A, is not compatible in L,; (ii) every subset of A, which contains less that n elements is compatible in L,. Let us recall ([Harding, 1998; Harding, 1839-1862; PtSk and Pulmannovb, 19911, etc.) that a logic L is said to be regular if for any set {a, b,c) C L of pairwise compatible elements we have a tt b V c. As seen before, there are logics that fail to be regular (Example 26). Interestingly, projection logics are usually regular e r KatrnoBka, 1979; Harding, 19981 and [ P ~and B Weber, (see e.g. [ ~ l a c h s m e ~and 20011). PROPOSITION 37. Every lattice logic is regular. Proof. The proof follows immediately from Lemma 8. PROPOSITION 38. A logic L is regular if and only if every pairwise compatible subset of L admits an enlargement to a Boolean sublogic of L.

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Proof. By Theorem 20, every pairwise compatible subset of a regular logic is contained in a Boolean sublogic. To prove the converse, assume that every pairwise compatible subset of a logic L is compatible. Then if a, b, c are pairwise compatible elements of L, then they are contained in a Boolean sublogic Lo of L, hence b V c rn exist in L and belongs to Lo, and a tt b V c in Lo implies a * b V c in L. PROPOSITION 39. (i) Let A be a compatible subset of L. Among the Boolean sublogics of L containing A there is a maxzmal and a minimal one. (ii) The union of all maximal Boolean sublogics of L is L. (iii) The intersection of all mosimal Boolean sublogics of L is C(L).

Proof. Let K A denote the collection of all Boolean sublogics of L that contain A. By Theorem 32, K A is nonempty. The intersection, Lmi,, of K A is the smallest Boolean sublogic that contains A. Let us consider the collection K A ordered by inclusion. By a standard application of Zorn's lemma the collection K A conThis completes the proof of the statement. The tains a maximal element, L,,. rn statements (ii) and (iii) are obvious consequences of the statement (i). DEFINITION 40. A maximal Boolean sublogic of L is called a block of L. In view of Proposition 39 every logic can be viewed as a union of its blocks. A more detailed analysis of the configuration of blocks in a logic can be found in [Greechie, 1971; Matouavsek, 20041 and [Navara and Rogalewicz, 19911. 4 STATES ON A LOGIC DEFINITION 41. A state on a logic L is a mapping s : L -+ [0, 11, where [O,1] denotes the unit interval of the real line W,which satisfies the following conditions: (i) s(1) = 1; (ii) if (ai)i is a sequence of pairwise orthogonal elements in L, then 00 s (Vi=l ai) = ~(ai).

czl

+

If sl and sz are states on L, then their convex combination s = am1 (1 -a)sz, where 0 5 a 5 1, are also states on L. A state s on L is called pure if s = as1 (1 - a)s2, 0 < a < 1 implies that s = sl = sz. A state which is not pure is called a mixture. If for every a E L, s(a) E (0, I), we say that s is a two-valued state. Let S(L) denote the set of all states on L and let Sz(L) denote the set of all twevalued states on L. Let us note that every s E S2(L) is pure. Indeed, if we can write s = as1+ (1 - a)s2 for a E (0,l) and for distinct sl, s 2 E S(L), then there is 0 < a < 1with sl(a) # s2(a), and the value asl (a) (1-a)s2(a) is neither 0 nor 1. Moreover, if L is a Boolean logic, then s E L2(L) 8 s is pure. Indeed, if s E S(L) is such that 0 < s(b) < 1 for some b E L, we can put s = as1 (1 - a)sz, where

+

+

+

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a = s(b) and sl, s 2 are states on L defined by the formulas sl(a) = s(b)-ls(a A b) and s2(a) = (1 - s(b))-ls(a A b'). Thus s is not pure. LEMMA 42. The set S(L) is a a-convex set, that is, if sl, s 2 , . . ., are states and q 2 0, Czl q = 1, then s = C z f l s i is also a state on L.

Proof. It follows from the fact that s(a) = CEl cisi(a) 5 CEO=, ci = 1 for every rn a E L. DEFINITION 43. We say that a state s on a logic L has the Jauch-Piron property, or is a Jauch-Piron state if the following implication holds: If s(a) = 1and s(b) = 1 for some a, b E L, then there is c E L, c 5 a, c 5 b such that s(c) = 1. Equivalently, a state s is Jauch-Piron if s(a) = 0 = s(b) implies that there is d E L, a, b 5 dl and s(d) = 0. If L is a lattice logic, then a state s is Jauch-Piron iff s(a) = 1 = s(b) implies s(a A b) = 1 (a, b E L). In the study of Jauch-Piron states we encounter new phenomena in (non-commutative) measure theory see e.g. [ ~ Lucia e and Ptik, 1992; Bunce et al., 1985; Muller, 1993; Ruttimann, 1977; Ptik, 19981. DEFINITION 44. We say that a set S of states on L is (i) order determining if s(a) 5 s(b) for all s E S implies a 5 b; (ii) unital if for every a E L, a

# 0 there is s E S such that

s(a) = 1;

(iii) rich if {s E S : s(a) = 1) C {s E S : s(b) = 1) implies a 5 b. (Equivalently, if a , l b implies 3s E S : s(a) = l,s(b) # 0). We say that a logic L is (j) unital if it has a unital set of states, (jj) rich if it has a rich set of states, (jjj) rich2 if it admits a rich set of (0,l)-states. LEMMA 45. (i) A rich set S is unital and order determining. (ii) If L is a lattice logic, then a unital set M of Jauch-Piron states is rich.

Proof. Part (i) is straightforward. For (ii) Let a $ b. Then a = (a A b) V c with c # 0, c Ia A b. Let s(a) = 1 s(b) = 1. Then the Jauch-Piron property implies that s(a) = 1 =+ s(a A b) = 1. By unitality there is s E M such that s(aAb) = 1, contradicting s(c) = 1. But then s(c) = 1 =+ s(a) = 1 cIaAb. rn s(a) 5 s(b). In the As a consequence of orthomodularity we have a 5 b definition of a state, orthomodularity is not explicitly used. Consequently, the n e tion of a state makes sense for any partially ordered orthocomplemented set which

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is closed under suprema of countable pairwise orthogonal sets. Orthomodularity 19931 and or am halter and Ptbk, 19871). is often forced (see also [~vure~enskij, PROPOSITION 46. Let (L, 5,') be a partially ordered orthocomplemented set with the properties (i)-(v) of Definition 1. Let S be an order determining set of states on L. Then L also has property (vi) of Definition I , that is, L is a logic. Proof. Let a 5 b, a, b E L. Then a V b' exists in L, and s(a V b') = s(a) + s(bf) = s(a) + 1- s(b). For any s E S we have s ( a V (a v b')')

=

s(a) = s(a)

+ s((a V b')') + 1- s(a v b')

= s(b),

and since S is order determining, this implies that a V (a V b')' = b. This is equivalent to (vi) of Definition 1. The following definition introduces an important class of set-representable logics. It should be noted that even finite logics are not necessarily set-representable, see [Katrno~ka,1982; Ovchinnikov, 1997; Tkadlec, 1993; Sherstnev, 19681, etc. DEFINITION 47. Let R be a nonempty set and let A be a collection of subsets of R. Then A is said to be a concrete logic if the following conditions are satisfied:

3. if {Ai : i E N) C A is a countable family of pairwise disjoint subsets of R, then U{Ai : i E N) E A It is interesting to note that the notion of a concrete logic has long been known in classical probability theory in connection with generating Bore1 sets (see e.g. [Olej~ek,19951 for more details). There, it is called a Dynkin system of sets. Note also that there are Boolean logics (Boolean cr-algebras) which are not concrete [~tbk and Pulmannovi, 19911. The following statement was proved in udder, 19791. THEOREM 48. A logic L is isomorphic (as a logic) to a concrete logic if and only if L is rich2. Proof. Suppose first that L is isomorphic to a concrete logic (R, A). Then we may identify L with A. Take A, B E A such that A $ B. Then A \ B # 0 and therefore we can choose a point p E A \ B. Consider a state sp concentrated at p (i.e., sp(C) = 1 iff p E C). Then sp(A) = 1 and sp(B) = 0. Thus, L is rich2. Conversely, suppose that L is rich2. Put R = S2(L), and A = {A E S2(L) : there is an a E L such that A = {s E S2(L) : s(a) = 1)). We show that (0, A) is a concrete logic. If A = {s E S2(L) : s(a) = I), then R\A = {s E S2(L) : s(af) = 1). Hence A E A implies R \ A E A. Let Ai = {s E S2(L) : s(ai) = I), i = 1 , 2 , .. .,

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.

and let Ai be mutually disjoint sets (i E N). As L is rich2, it follows that the elements a i , i E N are mutually orthogonal. We may then write Ui,,Ai = {s E S2(L) : s(ViENai) = 1). Hence (a,A) is a concrete logic.

The following proposition gives a characterization of compatibility in concrete logics. PROPOSITION 49. Let A be a concrete logic of subsets of $2. Then A, B E A are compatible if and only .if A n B E A.

Proof. Assume that A * B. Then there are mutually disjoint sets A1, B1, C in A such that A = A1 U C, B = B1 U C. Then A n B = C E A. Now assume that A n B E A. Then we have A = ( A n B ) u ( A \ ( A n B ) ) , B = ( A n B ) U ( B \ ( A n B ) , where the sets A n B, A \ ( A n B), B \ (A n B) are mutually disjoint sets. Since for disjoint elements of A the set-theoretical union U coincides with the supremum in A. It follows that A * B.

.

The state space S(L) of a logic can be rather poor (even for finite L we can have S(L) = 8 - see [Greechie, 1971; Navara, 19941 and [Weber, 19941) or rather bizarre ([Navara et al., 1988; Navara and Rogalewicz, 19881. To illustrate that, let us consider the following four classes of (unital) logics, denoted by C1, La, C3, C4:

- L E L1 e if a cft b, there is a state s E S(L) such that s(a) = 1and s(b) # 1; if a cft b, and if we are given an s E S(L) such that s(a) = 1 and s(b) 2 1- E ;

- L E C2

E

> 0,

then there is a state

- L E C3 H if a cft b, then there is a state s E S(L) such that s(a) = 1= s(b); - L E L4 e if a cft b, then there is a twevalued state s E S2(L) such that s(a) = 1= s(b) and, moreover, the set S2(L) is unital for L. THEOREM 50. [ ~ t i and k Pulmannovi, 1991, Th. 2.4.121 (i) L E C1 e L is a rich logic. (ii) L E L4 H L is a concrete logic. (iii) The inclusions C4 C C3 C L2 C L1 hold.

Proof. (i) We rephrase the definition of a rich logic: a logic L is rich if for any a, b E L with a 5 b there is a state s E S(L) such that s(a) = 1 and s(b) # 1. Thus every rich logic obviously belongs to L1. Now let L E L1 and let a, b E L, a 5 b. If a * b, we have a = a1 V c, b = bl V c, where a l , bl, c are mutually orthogonal. Obviously, a1 # 0. Take a state s E S(L) with s(al) = 1. Then we have s(a) 2 s(al) = 1. From b 5 a; we have s(a;) = 0 2 s(b). Hence s(a) = 1 and s(b) = 0. (ii) Follows by (i) and Theorem 48. (iii) The inclusions C4 C C3 C C2 follow immediately from the definitions. It remains t o verlfy that C2 c C1. Suppose that L E C2 and take elements a, b E L. bb', and therefore there is a state s E S(L) such that s(a) = 1 If a cft b, then a and s(bb')2 112. Since s(b) = 1- s(b1), we see that s(b) < 1/2, as required.

+

.

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It can be proved that all the inclusions in Theorem 48 are proper [ ~ t b k and Pulmannovb, 1991]. The counter examples are, usually, constructed with the help of so called Greechie past job [Greechie, 1971; Greechie, 19691 or its generalizations [Mayet et al., 20001. Let us present here a typical way of constructing interesting features of state spaces in this manner. Let us first introduce another class of logics ([~ogalewicz,1984b] and [ ~Btand Rogalewicz, 1983bl). We point out that various other such classes having their origin in mathematical analysis have been r Navara, 1987; D7Andreaand de Lucia, 1991; De Simone and studied [ ~ i n d e and Navara, 2001; De Simone et al., to appear], etc. DEFINITION 51. Let us denote by L2- the following class of logics: L E La- (j if a * b and if we are given an E > 0 then there is a state s E S(L) such that s(a) 2 1- E and s(b) 2 1- E. The class Lz- is interesting in its own right but it is also relevant to the study of observables as we will see later. PROPOSITION 52. C2- is a proper subclass of L2. Proof.We assume that the reader is to a certain extent familiar with the interpretation of the Greechie diagrams of a logic (see our next Figures 1,2). Let us just recall that the "points" of a Greechie diagram are interpreted as the atoms of the logic and the straight line segments (or possibly the curve segments) group together those atoms which belong to a maximal orthogonal set (Boolean block). There is a oneto-one correspondence between the states on the logic constructed and the so called weights on the Greechie diagram. (A weight is such a (nonnegative) evaluation of the points of the diagram that the sums over Boolean blocks equal to 1.) We shall not make any distinction between the weights on a Greechie diagram and the states on the corresponding logic. The precise description of Greechie diagrams may be found in [~reechie,19711 or [~chultz,19741. Let us return to the example disproving L2 c Lz-. We start with a preliminary construction (Figure 1). Claim. The logic L given by Figure 1 has the following properties: 1. L E L2-, 2. If s is a s t a t e o n L and if s(b) = 1 , then s(a) = O . Proof. Suppose that s(b) = 1 for a state s on L. Then s(fi) = 0 for any i = 1,2,3, . . . . Therefore s(d) C s(zi) 5 s(d) C s(cj). Since s(e) = 0, the left side of the last inequality equals to one. Hence s(d) C s(q) = 1 and therefore s(a) = 0. This was to prove. Observe that we have also proved that the identity s(b) = 1 for a state s implies s(a') = 1. On the other hand, if E > 0 and r E (0,l) are given, we may find a state s such that s(b) = 1- E and s(a) = r. It is easily seen that if we take two noncompatible elements p, q E L such that (p, q) # (a, b), there exists a state s on L with s(p) = 1= s(q). Hence L E Lz-.

+

+

+

Pavel Pt6k, Sylvia Pulmannov6

e

Figure 1. The desired example establishing La[Rogalewicz, 1984al.

\ L2 # 0 is now

constructed as follows

Claim. The (lattice) logic L given by Figure 2 belongs to La- but does not belong to L2. Proof. If we take noncompatible atoms (p, q) # (a, b), then one easily shows that there exists a state s on L such that sCp) = 1 = s(q). Let us consider the pair (a, b). If s(b) = 1, then s(fi) = 0 for i = 1 , 2 , 3 , . . . and therefore s(d) C s(ci) = 1. As

+

a result, s(a) 5 $ and L does not belong to Lz. On the other hand, L belongs t o La-. Indeed, if we are given a E > 0, we may simply construct a state s with s(b) = 1- E and s(fi) E ( 0 , ~ ) . An alternative proof of the previous result can be obtained by the following result, which uses the convexity of the state space. PROPOSITION 53. A logic L belongs to Lz- if and only if it satisfies the following condition: If a, b are two noncompatible elements of L and if we are given a real number T E (0,l) then there exists a state s on L such that s(a) = T = s(b). Proof. The condition is obviously sufficient. To prove necessity, let a real number

>4

< 1be given. We may suppose that T - otherwise we take up the equivalent assertion with a', b' and T' = 1- T . We shall construct two states sl,s 2 on L such

T

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Figure 2.

i)

that s l ( a ) = sl(b) = rl E (0, and s2(a) = s ~ ( b = ) 7-2 E ( r ,1). The required state s on L can then be constructed by setting

We shall first define s l . If there is a state s on L such that s(al) = 1 = s(bl), we put sl = s. If this is not the case, the assumption guarantees the existence of states s3,sq on L with s3(a1)= 1, s3(b1)= 7'3 < 1, sq(al) = 7-4 < 1, s4(bf)= 1. Let us put

Let us now consider the construction of s2. If there is a state s on L with s(a) = 1= s(b), we put s2 = s and the proof is complete. If there is no such state,

Pave1 Ptdk, Sylvia Pulmannovi

164

we consider states 95, s 6 and s 7 such that sg(a) = 1, s5(b) = r5 < 1, s6(a) = rs < 1, s6(b) = 1 and, further, if we set q = max(r,r5, r6), we require s7(a) = r7 > q, s7(b) = r8 > 2. The assumptions guarantee the existence of such states. If r 7 = T,, we put s 2 = 37. If r7 < T,, then

If r,

< r7, the construction

of

92

proceeds dually. Now the proof is complete.

DEFINITION 54. An element a E L is called a support of a state s on a logic L, if for any b E L, a Ib if and only if s(b) = 0. If a is a support of s, we will write a = supp s. PROPOSITION 55. Every Jauch-Piron state on a separable lattice logic has a support. Proof. Let L be a separable lattice logic and let s be a state on L. Put so := {a E L : s(a) = 0). By Proposition 22, the element a = VsO exists in L, and a = ViENail where ai E so,i E N. Put bl = a l l bz = b l V az, . . . , b, = bn-1 V a,, . . . . Clearly, (bi)i is a nondecreasing sequence with Vi bi = a. From the JauchPiron property we obtain s(bi) = 0, i = 1,2,. . .. By a-additivity of s, s(a) = s(Vi bi) = limn,,s(bi) = 0. We will show that supp s = a'. If b E L, ~ ( b = ) 0, then b E so, and hence b Ia'. Conversely, if b Ia', then b 5 a, hence s(b) = 0.

The notion of a Jauch-Piron state can be strengthened further by introducing , RieEanovA, 1998; subadditive states (or so called valuations, see e.g. [ ~ t f i k1998; Sarymsakov et al., 19831, etc.). DEFINITION 56. Let L be a logic. A state s on L is said to be subadditive if for any a, b E L there exists c E L such that a 5 c, b 5 c and s(c) 5 s(a) s(b).

+

It is an easy exercise to show that each subadditive state is Jauch-Piron. Indeed, if s(a) = 1, s(b) = 1 then s(af) = 0, s(b') = 0. Since s is subadditive, there is c E L such that a' 5 c, b' 5 c with s(c) 5 s(af) s(br) = 0. Thus, s(d) = 1 and c' 5 a and d 5 b. If a logic has "enough" states and if each state on it is subadditive, the logic is in a sense "almost Boolean". This situation is, however, fairly subtle as the following example shows (see also [Navara and Pt&, 19881 and [Ptiik, 19981).

+

EXAMPLE 57. There is a concrete non-Boolean logic such that each state on it is subadditive. Let us indicate the construction of such an example. Let P be a set of the first uncountable cardinality. Then each state on exp P lives on a countable set a well-known result of measure theory (by exp P we mean the Boolean a-algebra of all subsets of P). In other words, if s is a state on exp P, then there is a countable subset, M, of P such that s(M) = 1. Let L be the logic of all subsets

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of {1,2,3,4) whose cardinality is even. Take {1,2,3,4) x P and let us make the latter set an underlying set of the logic in question. Let us call this logic K , K = A, where A is the following family of subsets of {1,2,3,4) x P: A set T, T c {1,2,3,4) x P, belongs to A precisely when the cardinality of the set Q = P- {p E P I the set of all w E {1,2,3,4) such that ( w , p ) E T is of an even cardinality ) is at most countable. Then A is a concrete logic (of subsets of {1,2,3,4) x P ) . Indeed, {1,2,3,4) x P belongs to A. Furthermore, it is easily seen from the definition of A that if T E A, then ({1,2,3,4) x P ) \ T belongs to A as well. Finally, let T, (i E N) be a collection of pairwise disjoint elements of A. Then U Ti E A by a simple cardinality argument. We have to show that any state on A is subadditive. Let s be a state on A. Since exp P can be embedded in A, the restriction of s to e x p P is a state on exp P and as such it must live on a countable set, S. But then s is essentially a (Boolean) state on exp S and therefore it must be subadditive. This completes the construction. 5 OBSERVABLES ON A LOGIC DEFINITION 58. An observable on a logic L is a a-homomorphism from a aalgebra C of subsets of a set R to L, that is, a mapping x : C + L is an observable if (i) x(R) = 1;

(ii) x(EC)= x(E)', E E C (ECis the complement of E in R), w (iii) x ( U z l Ed)= Vi=l x(Ei) for every sequence of C.

of pairwise disjoint subsets

The set R is called the value space of the observable x. Let us notice that if (Ei)i is any sequence of elements of C, then x ( U z l Ei) = x(Ei). This is obtained by replacing (Ei)$ by the sequence of pairwise disjoint sets (Fj)j, where Fl = El, F n = E n \ (uT~; E ) , and Ui Ei = Uj Fj. This entails that the range R(x) := {x(E) : E E C) of an observable x is a Boolean sublogic of L. A family (x,), of observables on L is called compatible if the ranges of all x, belong to the same Boolean sublogic of L. If 52 = IR and C is a suba-algebra of the Bore1 a-algebra B(B) of subsets of W,then an observable x : C -t L is called a real observable: In this case, if L is a concrete logic, then an observable is nothing but a classical random variable [~aradarajan,19851, and this is the case in many other situations [ ~ t i k19811. , If f : R + IR is a measurable function and x : C + L is an observable, then f (x) : B(W) + L, f (x)(E) = x(fP1(E)) is also an observable, which is called the function f of the observable x.

Vzl

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Pave1 Ptik, Sylvia Pulmannovi

The theory of real observables on a logic was developed by Varadarajan [Varadarajan, 19851. The corner stone is the celebrated Loomis-Sikorski theorem. The statements in [Varadarajan, 19851 are formulated for a lattice logic but if we replace pairwise compatibility by the stronger notion of compatibility, the statements hold for arbitrary logics as well. THEOREM 59. (Loomis-Sikorski), [Varadarajan, 1985, Theorem 1.31 Let B be a Boolean a-algebra. Then there exists a measurable space ( R ,C ) , where C i s a a-algebra of subsets of a nonempty set R , and a a-homomorphism h from C onto B. THEOREM 60. [Varadarajan, 1985, Theorem 1.6, Lemma 3.161 Let x be a real observable o n a logic L. Then the range R ( x ) = { x ( E ) : E E B ( R ) ) of x i s a countably generated Boolean sublogic of L. Conversely, if B i s a countably generated Boolean sublogic of L , then there exists a real observable x such that R ( x ) = B. THEOREM 61. [Varadarajan, 1985, Theorem 1.41 Let ( R , C ) be a measurable space, and let h be a a-homomorphism from a onto a Boolean a-algebra B. Let u be a real observable with range in B. T h e n there exists a C-measurable function f : R + B such that u ( E )= h ( f -'(E)), E E B ( B ) . The function f is unique in the sense that if g i s any other function with the same properties, then { w E R : f ( w ) # g ( w ) ) belongs to the null space of h. THEOREM 62. [Varadarajan, 1985, Theorem 3.91 Let { x , : a E A ) be a n arbitrary family of compatible real observables on a logic L. T h e n there exists a set X , a a-algebra t3 of subsets of X , real-valued B-measurable functions g, on X ( a E A), and a a-homomorphism r of t3 into L such that

for all a E A and E E B ( B ) . Suppose further that either L is separable, or that A i s countable. T h e n there exists a n observable x and real valued Borel functions f, of a real variable such that V a E A, x , = fa o x . Notice that for at most two observables the condition of compatibility in Theorem 62 may be replaced by pairwise compatibility [Ramsay, 19661. Theorem 62 enables us to construct a calculus of functions of several compatible observables. THEOREM 63. [Varadarajan,1985, Theorem 3.171 Let L be a logic and x l , x 2 , . . . ,x, compatible observables on L. T h e n there exists one and only one a-homomorphism T of t3(Rn) into L such that for all E E B ( R ) and all i = 1,2,. . . ,n,

xi ( E ) = T (TF' ( E ) ) , where ~i i s the projection ( t l ,t 2 , .. .,t,) Borel function on Bn,

ti of Bn to B . If g i s any real-valued

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is an observable. If g l , gz, . .. ,gk are real valued Borel functions on Bn and yi = gio(xl,x2, . . . ,xn), then yl, y2, . . . ,yk are compatible, and for any real valued Borel function h on W k ,

where h(g1,gz, . . . ,g k ) is the function

The a-homomorphism T of Theorem 63 is called the joint observable of X I , 22,. . .,x,. In analogy to self-adjoint operators, we can also introduce the notions of an eigenvalue and the spectrum of a real observable. DEFINITION 64. A real number t is an eigenvalue of a real observable x if x ( { t ) )f 0. DEFINITION 65. Let x be a real observable on a logic L. Put

a ( x ) := n

{ : C~is a closed subset of W with x ( C ) = 1).

Then the closed set a ( x ) c W is called the spectrum of the observable x. PROPOSITION 66. The spectrum a ( x ) of a real observable x is the smallest closed subset of R such that x(C) = 1.

Proof. Since the topology of the real line W satisfies the second countability axiom, there exists a sequence of closed subsets C1,C2,.. . such that z(C,) = 1for emh n , and a ( x ) = r),"==, C,. Since x(W\C,) = 0 for all n , we have x(W \a(x)) = X(U,"=~(W \ C,) = V,"==l x(W \ C,) = 0, which entails that x ( a ( x ) )= 1. rn A number T E a ( x ) is a spectral value of x. An eigenvalue is a spectral value, the converse need not hold. LEMMA 67. A real number is a spectral value of an observable x if and only if x(U) # 0 for every open subset U c W which contains r .

Proof. Let U be an open set containing r and let x(U) = 0. Then W \ U is closed and x ( B \ U ) = 1. This yields W \ U > o ( x ) ,so that r $ a ( x ) . Conversely, assume that r $ a ( x ) . Since W is a metric space, there exist open subsets Ul,U2such that r E Ul, a ( x ) c U2 and Ul n U2= 0. Then x(U2)= 1 and rn this implies that x(Ul) = 0. DEFINITION 68. A real observable x is said to be (i) discrete if a ( x ) = { T I , r2, . . .); (ii) constant if a ( x ) = { r ) ;

Pave1 Ptdk, Sylvia Pulmannovd

168

(iii) simple if a(x) = {rl,r 2 ,

...,r,);

(iv) a proposition or 0 - 1observable if o(x)

c (0, 1).

For a E L, denote by qa the unique 0 - 1 observable for which qa({l)) = a. The observable corresponding to 0 E L, go, is the null observable. The observable corresponding to 1 E L, ql, is the unit observable. The null and unit observables are constants, and r.ql (defined by the function calculus) is a constant observable with the spectrum a(%)= {r). LEMMA 69. (i) For every a E L, q,, = f (9,) with f (t) = 1 - t (t E R). (ii) If x is an observable and XE is a characteristic function of a set E E B(R), then XE(X)= qz(E). PROPOSITION 70. The following conditions are equivalent: (i) x is a proposition observable; (ii) there exists an observable y and E E B(R) such that x = XE(Y); (iii) x2 = x. Proof. (i) & (ii): By Lemma 69, xil1(x) = x. (ii) & (iii): If x = x ~ ( y )then , (i): Put f(t) = t, g(t) = t2, t E W. We have x2 = Xs(y) = xE(y) = x. (iii) {t : f (t) # g(t)) c UrEqf -l(-oo, r)Ag-l(-oo, r), where Q is the set of rational numbers, and E A F = (Er7FC)u( E c nF) is the symmetric difference of sets. Then ~ ( f - ~ ( - o o , r ) A ~ - ~ ( - or)) o , = 0, which entails that x({t : t2 = t)) = x({O, 1)) = 1, hence x is a proposition observable.

6 PARTIAL COMPATIBILITY AND JOINT DISTRIBUTIONS OF OBSERVABLES In the sequel we will introduce the notion of partial compatibility and that of a commutator. We will then study properties of these concepts and relations between them. These results will be used in our study of joint distributions of observables. We mostly restrict ourselves to lattice logics. The notion of a commutator of two elements in an orthomodular lattice L was , The main result was that the orthomodular ideal introduced in [ ~ a r s d e n19701. J generated by the commutators of all pairs of elements of L is the smallest orthomodular ideal such that the quotient L / J is a Boolean algebra. The commutator , of a finite subset of an orthomodular lattice L has been used in [ ~ e r a n 1979; Beran, 1985; Bruns and Greechie, 1990; Bruns and Kalmbach, 1973; Greechie and Herman, 1985; Chevalier, 1984; Chevalier, 1989; Chevalier and Pulmannovi, 2000; e, Kalmbach, 19831, and [ ~ o ~ u n t k19831. The generalization of the concept of a commutator to arbitrary subsets of an orthomodular lattice L was studied in a series of papers [Dvure~enskijand Pulmannovi, 1982; DvureEenskij and Pulmannovb, 1984; LutterovB and Pulmannovi,

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1985; Pulmannovii, 1980; Pulmannovii, 1981b; Pulmannovi, 1985; DvureEenskij 1993; Ptik and Pulmannovi, and Pulmannovii, 19821, see also [~vure~enskij, 19911, the notion of partial compatibility was introduced in [~ulmannovii,1981bl. The notion of a joint probability distribution of Gudder type was introduced in [Gudder, 1968; Gudder, 19791. The subsequent developments can be followed 19821-[~vure~enskij, 1987131, [Lahti and Ylinen, 1987; Lutterovi in [~vure~enskij, and Pulmannovi, 1985;Pulmannovii, 1978;Pulmannovii, 1980; Pulmannovi, l98la; Pulmannovi, 1985; DvureEenskij and Pulmannovi, 1982; DvureEenskij and Pulmannovi, 1984; Pulmannovii and DvureEenskij, 19851, and [Pulmannovii and Stehlikovi, 19861. Heisenberg's uncertainty relations were expressed in the language of quantum logics in [Lahti, 1980; Bugajski and Lahti, 1980; Lahti and Mqczyliski, 19871. Results on the relation between uncertainty relations and total noncompatibility can be found in [~ulmannoviiand DvureEenskij, 19851 The notion of complementarity was introduced in [Lahti, 19801. There is a huge list of publications devoted to these problems, which are relevant to quantum theory see, e.g., [Gudder, 1979; Jammer, 1974; Jauch, 1968; Holevo, 1980; Mackey, 1963; von Neumann, 1932; Primas, 19811. The existence of sums of observables has already been emphasized by von Neumann [von Neumann, 19321. In some axiomatic approaches to quantum mechanics the existence of sums of bounded observables is postulated (e.g., in [Segal, 19471, and the C*-algebraic approach [Alfsen and Schulz, 1979; Emch, 1972; Haag and Kastler, 19641). The logics with the property that every two bounded observables are summable were investigated in [Gudder, 1966; Riittimann, 19851, and [DvureEenskij and Pulmannovii, 19801. Joint distributions of Urbanik type were introduced in [Urbanik, 19611 and in [Varadarajan, 19851 for observables on a Hilbert space logic. In [Gudder, 19661 a generalization of this notion to observables on a larger class of logics was studt, Urbanik, 1985; ied. For the further development see, e.g., [ ~ u ~ m g a a r1983; DvureEenskij and Pulmannovii, 1989; Pulmannovii and DvureEenskij, 19801.

6.1

Partial compatibility and the commutator

Let L be a logic, D = {-1,1), and for a E L, let a w l = a', a1 = a. DEFINITION 71. We say that a subset A of a logic L is partially compatible with respect to an element b E L (in short, A is p.c. b) if the following conditions are satisfied: (i) b tt A, i.e., b * a for all a E A; (ii) the set b A A := {b A a : a E A) is a compatible subset of L. Recall that for a subset A C L, C(A) = {b E L : b * A ) is a sublogic of L, where b * A means that b * a Va E A. It what follows, let us agree to write CC(M) instead of more correct C(C(M)).

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Pave1 Ptik, Sylvia Pulmannovi

THEOREM 72. If for a E L a finite subset M of L is p.c. a, then CC(M) is p.c. a.

Proof. Let M = {al, a2,. . . ,a,) and let {el, e2,. . . ,ek) be a minimal orthogonal cover of M A a in [0,a]. Since a 6 C(M), a cr b for all b E CC(M). From ei * a j A a and ei a for j 5 n, i 5 k, we easily derive that ei E C(M). Hence b tt ei, i k. Therefore {{b A ei, b' A ei)f=l, b A a A (Visk ei)') is an orthogonal ' p.c.a. cover of the set {al A a, a2 A a, . . . , a, A a, b A a), which shows that M U {b) is By induction, we see that every finite subset {bl, b2,. . . , b,) c CC(M) is p.c.a, hence CC(M) B ' p.c. a.

<

<

If A c L is p.c. a, we can prove, using Zorn's lemma, that there is a maximal p.c. a subset of L containing A. THEOREM 73. A subset Q of L which is a maximal p.c a is a sublogic of L. If L is a lattice then Q is also a lattice.

Proof. Let (ai)iEN be a sequence of pairwise orthogonal elements of Q. We need to show that Vi ai E Q. From ai cr a V i we derive that a * Vi ai and (Vi ai) A a = Vi a A ai. The set Q A a is compatible in [0, a], hence it is contained in a Boolean sublogic of [0,a], and this Boolean sublogic contains also Vi a A as. This entails that Q u {Vi q) is p.c. a, and maximality of Q implies that Vi ai E Q. If b E Q, then the Boolean subalgebra of [0, a] containing b A a contains also (b A a)' A a = b' A a, hence b' E Q. This proves that Q is a sublogic of L. If L is a lattice, then for b, c E Q, a v b exists in L, and the relations a * b, a * c imply that a * b V c (a lattice logic is regular), and a A (b V c) = (a A b) V (a A c). The Boolean sublogic of [0,a] that contains Q A a then contains also ( a b)~V (a AC), whence Q U {b V c) is p.c. Q. By the maximality of Q, a V b E Q. We conclude that Q is a lattice. THEOREM 74. Let F = {al, aa, . . . ,a,) be a finite subset of L. If com(F) exists in L, then (i) F is p.c. com(F); (ii) if a E L is such that F is p.c. a, then a 5 com(F).

Proof. (i) From the definition of the commutator it is clear that ai * com(F) for all i 5 n. Put c := com(F), and consider the interval [0,c]. From F cr c, and existence of com(F) in L, we derive that com(al A c, . . . ,a, A c) in [0,c] exists, and is equal to com(F) A c = c, hence the elements a1 A c, a2 A c, . . . ,a, A c are compatible. This proves that F is p.c. com(F). (ii) The statement is trivial for a = 0. Assume that F is p.c. a for a # 0. Then F A a is compatible in [0,a], hence VdED,, (al A a)d; A ... A (a, A = a, where = a: A a if di = -1. Hence = ai A a if di = 1 and (ai A we put (ai A VdEDna:' A . . .a$ A a = a, and as a ++ F, this implies that com(F) A a = a, hence a com(F). H

<

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In what follows we will tacitly assume that L is a lattice logic. Let us first extend the notion of a commutator to arbitrary subsets of L . DEFINITION 75. Let M be a subset of a lattice logic L . If the following element com(M) exists, (1) com(M) := l \ { m m ( ~ ): F is a finite subset of M ) ,

it is called the commutator of the set M . THEOREM 76. Let M be a subset of a lattice logic L. If com(M) exists, then M is p. c. com(M).

Proof. It is easy to see that if Fl C F2, where F l , F2 are finite subsets of M , then com(Fl) 2 com(F2). Let b E M be an arbitrary element. We have w m ( M )=

/\ w m ( F ) = /\ c o m ( U~ {b)), F

F

and since b * com(F U {b)) for all F C M, by Lemma 8 we have b ct w m ( M ) . Hence M t)com(M). Let bl, bz E M . An easy computation shows that com(bl A w m ( M ) ,bz A c o m ( M ) ) = com(b1,b2) A com(M) = com(M). This entails that bl A com(M) and b2 A com(M) are compatible in [0,com(M)].Since a lattice logic rn is regular, we obtain that M is p.c. com(M). COROLLARY 77. Let M C L. If com(M) exists, then for every a E L , M is p.c. a i f and only i f M * a and a 5 com(M).

<

Proof. If M is p.c. a, then every finite subset F of M is p.c. a, so that a com(F) by Theorem 74, and consequently a 5 com(M). Conversely, assume that M * a and a tt com(M). Let b, c E M . Then the relations b A com(M) et c A w m ( M ) , b A com(M) * a, c A com(M) * a imply that b A a A com(M) t)c A a A w m ( M ) . Now if a 5 com(M), then we obtain b A com(M) t)c A com(M), i.e., M is p.c. a.

rn COROLLARY 78. Let A be a subset of L such that com(A) exists. If L ( A ) is the lattice sublogic of L generated by A , then com(A) = w m ( L ( A ) ) .

Proof. Observe that L ( A ) is p.c. com(A). Indeed, by Theorem 73, the maximal subset Q of L which contains A and is p.c. com(A), is a lattice sublogic of L. The result then follows from L ( A ) Q. By Corollary 77, com(A) 5 com(F) for every finite set F c L ( A ) . Let c E L be an element such that c 5 com(F) for every finite F C L ( A ) . Then c 5 com(G) for every finite G C A. Hence c 5 com(A). This proves that w m ( A ) = / \ { c o m ( ~ ): F is a finite subset of L ( A ) ) = com(L(A)).

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In what follows, we will often make use of the following important theorem, which extends the Foulis-Holland theorem [~reechie,19791, see also [~almbach, 1983; Navara and HyEko, to appear] and [ ~ t and ~ k PulmannovB, 1991, Prop. 5.1.13.a]. THEOREM 79. Let L be a lattice logic and let S c L have the property that among any three elements of S there is one which is compatible with the other two. Then the sublattice [S] of L generated by the set S is distributive (i.e., for any three elements a, b, c E [S]we have a A (b V c) = (a A b) V (a A c)). In the next part we will use the following conventions. We will write com(Ml , M2, . . . , M,) instead of com(Ml U M2 U . . . U M,), and sometimes we will use the symbol com(al, a2, . . . , a,, Q) instead of the more correct notation of com({al, a2, . . . ,a,), Q). THEOREM 80. Let Ai (i E N) be finite subsets of mutually orthogonal elements in L. Then for any finite subset Q of L we have com(Al,A2, ..., An,Q)

=

A{com(al,az, . , . , a n , Q ) : ( a 1 7 a z i . . . , a n )

E A1 x A2 x

... x A,)).

Proof. We will proceed by induction. Assume that n will show that

(2) com(al, a2, b2,. Consider the set

. . ,b,)

= com(a1, b2,

=

1 and A1 = {al, az). We

.. .,bn) A com(a2, b2, . . . ,bn).

where bd = b2 A . . A b$. The set B contains four-element classes for different d E Dn-l. Any two elements from different classes are orthogonal. Any two elements in the same class are mutually compatible, with the possible exception of a; A bd and a; A bd. Consequently, among any three elements of B there is always one compatible with the other two. Therefore we may use distributivity to obtain (2). The proof then continues by induction, first with respect to the cardinality of A1, then with respect to the number of sets Ai, i 5 n. For details, see [Pt& and PulmannovB, 1991, Prop. 5.1.141. In the following results we will apply properties of commutators to observables. We will view observables as logic morphisms x : B(M) -+ L, where B(M) is the a-algebra of Bore1 subsets of a separable Banach space M and L is a (lattice) logic. We will write R(x) := {x(E) : E E B(M)) for the range of the observable x. let us recall that R(x) is a Boolean sublogic of L. Let {x, : a E I) be a system of observables. Let us write

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If com{x, : a: E I ) exists, it is called the commutatorof the system { x , : a: E I ) of observables. k Pulmannovi, In what follows, we will need the following result (see [ ~ t i and 1991, Theorem 5.1.171). THEOREM 81. Let M be a separable Banach space and let L be a logic. A Boolean sublogic B of L is the range of an observable x : B ( M ) + L i f and only i f B is countably generated. THEOREM 82. Let I be an at most countable set and let {xi : i E I ) be a system of observables on a lattice logic L. Then com{xi : i E I ) exists in L. Moreover, there is a countable set Q such that Q C UiEIR ( x i ) such that com{xi : i E I ) = com(Q).

Proof. Let Qi denote a countable generator of R(xi). Let Li be the lattice sublogic of L generated by Qi. It is easy to see that Li C R(xi). Therefore Li is compatible, hence a Boolean sublogic of L, and therefore Li = R(xi). Further set Q = UiEIQi. Evidently, Q is at m a t countable. Since the set of all finite subsets of a countable set is countable, and since L is a a-lattice, the element com(Q) = n { c o m ( F ) : F a finite subset of Q ) exists in L. Let L ( Q ) denote the lattice sublogic of L generated by Q. By Corollary 78 com(Q) = com(L(Q)). By the first part of the proof we have R(xi) C L(Q)(i E I ) . This implies that com(F) 2 corn(&) for every finite subset F C UiEIR(xi). Let c E L be any element such that c 5 com(F) for every finite subset F of UiEIR(xi). Then com(G) 2 c for every finite subset G C Q (i.e., com(Q) 2 c). Hence com(Q) = com{xi : i E I ) . COROLLARY 83. Let

X I , 52,.

. .,xn

be observables on L. Then

Moreover, there exists a countable set {Et,. . . ,Ek : i E I ) such that

Proof. Observe that if a l , a2,. . . ,a, are compatible elements of L and F is a finite subset of L, then com{al, a2, . . . ,a,, F ) = com({a$ A .A a$ : d E Dn), F ) . We see that if G is a finite subset of UiEIR(xi), then G = G I U ..-U G,, where Gi C R ( x i ) (i 5 n ) are orthogonal sets. This implies that for every such finite set F we have com(F) = Aj com{xl (E:, . . . ,x, ( E i ) ) ,where the index j runs over a finite set. By Theorem 82, there is a countable system {Fn)nEWsuch that com(x1, 2 2 , . . .I 5,) = AnENcOm(Fn)In what follows, we will use the expression "observables are partially compatible" if the union of ranges of the observables is partially compatible. In what follows we will often make use of the following proposition the proof of which is straightforward.

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PROPOSITION 84. Suppose that a # 0 and suppose that {x, : a E I) i s a system of observables which are partially compatible with respect t o a. T h e n the mappings x, A a : B(M) + L[O,a] defined via (x, A a)(E) = x,(E) A a are compatible observables o n the logic L[O,a].

6.2 Joint probability distribution of Gudder type In the following definition, we consider observables on the Borel a-algebra B(M) of a separable Banach space M. Notice that the functional calculus for real compatible observables (Theorem 63) can be extended also to the latter case ([Ptbk and Pulmannovb, 1991, Chap. 41) in view of the fact that all separable Banach spaces have isomorphic algebras of Borel sets. Throughout this paragraph, let us assume that L is a lattice logic. DEFINITION 85. We say that the observables xl,x2,. . . , xn on L have a joint probability distribution of Gudder type in a state s if there is a (probability) state P,81,...,,n on B(M) such that, for every rectangle El x ... x En E B(Mn), the following equality is satisfied:

If such a state P,Sl ,...,,n exists, we will call it the probability distribution of Gudder type of the observables X I , . . . ,x,. Since a state on B(Mn) is uniquely defined by its values on the rectangle sets, the joint probability distribution of Gudder type is uniquely defined whenever it exists. The notion of joint probability distribution can be generalized in the following natural way. We say that a system 0 of observables {x, : a € J) has a joint distribution of Gudder type in a state s if the joint probability distribution of Gudder type in the state s exists for any finite subsystem x,,,~,,, . . . ,xan (a1, a 2 , . . . , an E J) of (3. THEOREM 86. T h e joint distribution of Gudder type for compatible observables XI,.. . ,an exists in a n y state s o n L.

Proof. Let $1, x2, . . . ,xn be compatible observables. By the functional calculus (Theorem 63), there exist an observable z and measurable functions fi : M + M(i 5 n) such that xi = z.f
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It follows that s.w is the joint distribution of Gudder type of X I , . . ,xn in the rn state s. As a corollary we obtain that the joint distribution of Gudder type exists for an arbitrary system of compatible observables. The following theorem gives a general criterion of the existence of the joint distribution of Gudder type in a given state 3. THEOREM 87. Let J be an arbitrary set. The observables { x , : a E J } have a joint probability distribution in a state s zf and only if for every a l , . . . ,an E J and every E l , . . . ,En E B ( M ) the following equality holds:

(3) s(com(xa,( E l ) ,xa2( E z ) ,. - - ,xan (En))= 1. Proof. Assume that equation (3) holds for every a l , . . . ,an E J and every E l , . . .,En E B(M). Fix a l , a2, . . . ,an E J and put xai = xi,i = 1,2,. . . ,n. By Corollary 83, there is a countable set { E f , . . . ,NA : i E I } such that

By (3), s ( c o m { x 1 ( ~ f. ). ., , x n ( E i ) } ) = 1 for all i E I . The a-additivity of s implies that s ( c o m { x l , x ~. ., . ,xn}) = 1. Write c := com{xl,x2,.. . ,xn}. Then xi, i = 1 , 2 , .. . ,n , are partially compatible with respect to c and by Proposition 84, xi A c, i = 1,2, . . . ,n are compatible observables on the logic [0,c]. Since s(c) = 1, the restriction of s to [0,c] is a state on [0,c]. It follows that the observables X I A c, 2 2 A C , . . . ,xn A c have a joint distribution of Gudder type in the state s, and hence there is a measure p on B(Mn) such that for every rectangle El x . x En,

p(E1 x

.

x En) = S

( X ~A

c(E1)A . . . A xn A c(En)).

Let Lo be the lattice sublogic of L generated by Ui,, R(xi). By Corollary 78, com(Lo)= c, and Lo is p.c. c. The restriction of s t o Lo is clearly a state on Lo. Therefore for every a E Lo, a = a A c V a A c'. Further, s(c) = 1 implies that s(a) = s(a A c). This entails that

p(E1 x . . . x En) = s(xl(E1)A xz(E2) A . . A xn(En)A C ) = s(x1( E l )A xz (E2)A ' ' ' A ~ n ( E n ) ) , hence the joint distribution of X I , . . . ,xn in the state s exists. Since the choice of a l , . . . , an was arbitrary, the system { x , : a E J } has a joint distribution. Conversely, assume that the joint distribution of Gudder type for observables { x , : a E J } exists in state s. Let xl,x2,. . . ,xn be arbitrary finite subsystem of the system {x, : a E J}. Then there is a probability measure p on f ? ( M n )such that p(E1 x x En) = s(xl(E1)A . . . A xn(En))for every E l , . . . ,En E B(M). From the properties of a measure p it can be easily derived that

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Recall that by Theorem 82, the commutator com{xi : i E I) exists for every at most countable system of observables {xi : i E I). Using properties of the commutator, we can easily derive the following result. THEOREM 88. Let I be a n a t most countable set. A system of observables {xi : i E I) has a joint distribution in a state s if and only if s(com{xi : i E I ) ) = 1. Let us note that if the joint distribution of Gudder type for observables {xi : i E I) exists in the state s, then the statistical properties of these observables with respect to the state s are the same as the properties of the compatible observables {xi A c : i E I ) , where c = com{xi : i E I). We may say that the existence of a joint distribution enables one to "compatibly reduce" the observables. In the next theorem, we introduce a criterion which does not depend on the notion of a commutator. THEOREM 89. T h e observables XI,x2, . . . ,x, have a joint distribution of Gudder type in state s if and only if s ( x i ( ~ :U E:)

A

A

xj(Ej))

=

s(x,(E:)

A

A

jSn,j#i

xj(Ej))

(4)

jSn,j#i

+

/\

s(xi(E3A

xj(Ej))

jSn,j#i for all i 5 n and Et, E:, Ej E B(R) such that E; n E: = 8.

Proof. We sketch the proof for n = 2, for n > 2 the proof is analogous. Let D be the class of all measurable rectangles E x F E B(R2) and let A be the algebra of all finite disjoint unions of elements in D. For E = El x E2 in D, define a set function POby putting Po(E) = s(xl(E1) A xz(E2)). Using condition (4) it can be proved that ,YO is finitely additive on 2). Let E E A, E = U i l n E i , E i E D , E i n E j = f l , i , j € { 1 , 2,..., n). Put

Then the set function p l : A 4 [O,11 is well defined and finitely additive. By a well known theorems of classical measure theory (e.g, [Halrnos, 1962; Pt& and Pulmannov6, 1991]), p l is a-additive, and hence it has a unique extension to a a-additive measure p on B(R2). Thus, p is the joint distribution of Gudder type of XI, 22,. . . ,x, in the state s. In the following definition, we introduce the concept of "degree" of noncompatibility for a countable system of observables. DEFINITION 90. Let I be an at most countable set. We say that the observables {xi : i E I) are:

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(i) compatible if com{xi : i E I))= 1; (ii) partially compatible if 0 < com{xi : i E I) < 1; (iii) totally noncompatible if com{xi : i E I}= 0.

6.3 Uncertainty relations and complementarity In this section we introduce a generalized definition of the famous Heisenberg uncertainty relations and a definition of complementarity for real observables on a unital lattice logic. We establish connections between concepts studied previously and commutators of observables. Recall that a set S of states on a logic L is called unital if for every a E L, a # 0, there is a state s E S with s ( a ) = 1. Note that the previous results allow us to formulate the following result, which is a sort of "classi6cation" of observables. PROPOSITION 91. Let I be a n at most countable set and let S be a unital set of states o n L. The observables { x i : i E I) on L are (i) compatible if and only if the joint distribution of Gudder type exists for all states s E S . (ii) partially compatible i f and only if there are states s l , s2 E S such that the joint distribution of Gudder type exists in s1 but does not exist in s2. (iii) totally noncompatible if and only if the joint distribution of Gudder type does not exist in each state s E S. The proof is obvious from Theorem 88. In the sequel, let us consider real observables o n a lattice logic L with a unital set of states S. The expectation of a real observable x in a state s is defined by r

provided the integral on the right exists. Let us first mention a problem which was popular in the eighties and nineties: the problem of uniqueness of observables (see [Gudder, 1979; Gudder, 19661, etc.). This problem is still not completely solved. It reads as follows: Suppose that for two bounded observables x , y we have s ( x ) = s ( y ) for any state s E S(L), does it follow that x = y? This question, which is very important for quantum stochastics, has the answer yes for projection logics as well as for Boolean logics, but in general the answer is no, (see [Navara, 19951 for a counterexample and review). Let us here present one of the positive results in order to gain insight into the problem. We will show that the question has a positive answer for the logics of the class C2- considered in section 4 (we will recall the definition in the formulation of the result). THEOREM 92. Let L be a logic such that

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(i) for any a E L there is a state s E S such that s(a) = 1 (i-e., L is unital), (ii) for any E E ( 0 , l ) and any a, b E L with a that s(a) 2 1 - E , s(b) 2 1 - E .

cft

b there is a state s E S such

Let x , y be two bounded observables on L such that s ( x ) = s ( y ) for any s E S. Then x = y. Let us first prove a lemma. LEMMA 93. Let L be a logic such that for any a E L there is a state s on L with s(a) = 1. Let x , y be two observables on L. If s ( x ) = s ( y ) for any s E S , then for any real r the relation x(r, +m)tt y(r, +m) implies x ( r ,+m)= y(r, +m).

Proof. Let x , y be two observables and let r be a real number. Put a = x(r, +m), b = y(r, +m) and suppose that a tt b. Then there are elements al,bl,c E L such that al 5 bi, a1 5 c', bl 5 c' and a = a1 + c , b = bl +c. Let a1 # 0. Then there is a state s on L with s ( a l ) = 1. It follows that s(bl) = 0 = s(c) and thus s(a) = 1, s(b) = 0, s(bl) = 1. Hence s ( x ) r, s ( y ) < r - a contradiction. Analogously, bl # 0 is not possible and we get a = c = b. This completes the proof. H

>

Let us come to the proof of Th. 92. Suppose that x # y. By the preceding lemma, there exists an r E R such that x(r, +m)cft y(r, +m).Hence x ( - m , r ) cft y(r, +m).Since Vr-, x ( - m , r - $ ) = x ( - m , T ) , there exists a number n E N such that x ( - m , r cft y(r, +a). Choose real numbers K , E such that y(-K, K ) = 1 and ~ ( 2 n K 1) < 1. Take a state s E S that satisfies

+

i)

Then

1 ( 1 - E ) . ( T - - ) + K E = ( ~ - E ) T - -( l n n s(y) 2 ( 1 - e)r - K E

s(x)

<

+KE,

and

Thus s(y) > s ( x ) , which is a contradiction. This completes the proof. The variance of x in the state s is defined by

provided the integral on the right exists. For observables X I , x2, . . . ,x, we put

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(7) V(xl, . . . ,x,) = {s E S : var,(xi) < oo, i 4 n). The following definition can be considered an expression of Heisenberg's uncertainty relations in the logicealgebraic approach.

DEFINITION 94. Let xl, xz, . . . ,x, be real observables on L. Then the following two cases can occur: E v ( x l , . . . ,x,)) : uar,(x~)var,(xz).. . var,(x,)

2 6;

2. (YE> 0)(3s E V(xl,. . . , x,)) : vars(xl)vars(xz). . . var,(x,)

< E.

1. (36 > O)(VS

In the case (1) we say that the observables XI,.. . ,x, satisfy the uncertainty relation. In the case (2) we say that the observables X I , . . . ,x, do not satisfy the uncertainty relation. PROPOSITION 95. If the observables XI, . . . ,x, are compatible, they do not satisfy the uncertainty relation. Proof. Let w : B(M) + L be the joint observable for X I , .. . ,x,. By Lemma 67 (generalized in an obvious way to observables on B(Rn)), a point (TI,rz, . . . ,r,) belongs to the spectrum w(w) of the observable w iff for any 7 > 0 we have

Since S is unital, there is a state s E S such that

The properties of the joint observable imply that s ( ~ i ( ( ri 7, Ti

+ 7))) = 1 (i 4 n).

This yields, for all i (i 4 n),

The last inequality implies that the observables XI,.. . ,x, do not satisfy the unrn certainty relation. THEOREM 96. If the observables XI,.. . ,x, satisfy the uncertainty relation, then com(xl, . . . ,xn) = 0. Proof. Put c = com(xl, . . . ,x,) and suppose that c # 0. Define Sc := {s E S : s(c) = 1). Since S is unital, S, # 8. If we restrict S, to the logic [0,c], we easily verify that Sc is a unital set of states for [O, c]. The observables xi A c (i 5 n) are compatible and have the same probability distributions in every state s E Sc as the observables xi (i 4 n). By Proposition 95, the observables xl A c, . . . ,x, A c on [0,c] do not satisfy the uncertainty relation. Since the variances of the observables xi A c (i 4 n) are equal to the variances of the observables xi (i 4 n) for all states in S,, the latter observables do not satisfy the uncertainty relation. This concludes rn the proof.

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The following definition is an attempt to formulate the notion of complementarity in the frame of quantum logics. It is motivated by the notion of complementarity of observables on the Hilbert space logic L(H). On L(H) the observables P and Q corresponding to momentum and position, respectively, are complementary. DEFINITION 97. We say that two observables x and y are complementary if for any bounded sets El F in B(W) such that w(x) (f El w(y) (f F we have x(E) A y(F) = 0. In the following propositions we list elementary facts concerning complementarity. The simple proofs are left to the reader. PROPOSITION 98. Two compatible observables are complementary if and only if at least one of them is a constant. PROPOSITION 99. Let a, b (a, b # 0 , l ) be two elements of L and let q,, qb be the cowesponding proposition observables such that q,({l)) = a, qb({l)) = b. Then q, and qb are complementary if and only if they are totally noncompatible. PROPOSITION 100. If the observables x and y are complementary and noncompatible, then they are totally noncompatible. The following example shows that the converse of Proposition 100 does not hold. Consider the Hilbert space W3. Notice that there exists no pair of nontrivial complementary observables. Let B1 = {el, e ~es) , and B2 = { f 1, f2, f3) be two bases of W3 with B1 n B2 = 0. Let [el denote the one-dimensional subspace generated by e E W3. Define two observables x and y on L(W3) as follows. Choose three real numbers TI, r z , T-3and put x({ri)) = [ei], y({ri)) = [fi] (i = 1,2,3). Then x and y are totally noncompatible but not complementary.

6.4

Sum logics and joint distributions of Urbanik type

The notion of a sum logic has its motivation in the quantum logic L(H) of closed subspaces of a (complex, separable) Hilbert space H. By the spectral theorem, there is a one-to-one correspondence between the real observables on L(H) and the self-adjoint operators on H . In particular, for every two bounded observables, their sum is defined as the observable corresponding to the operator sum of the corre sponding bounded self-adjoint operators. In certain cases the sums of unbounded operators exist, too. We will try to find a generalization of this phenomenon in the study of "sum logics". We note that in the definition of a sum logic L we need not require L to be a lattice logic. Let us recall the following notions (compare with Definition 44). A set S of states on L is said to be rich if the inclusion {s E S : s(a) = 1) C {s E S : s(b) = 1) implies a 5 b. A set of states S on L is said to be order determining if s(a) 5 s(b) for all s E S implies a 5 b. For a set S of states, we denote by conv(S) the a-convex envelope of S. Let L be a logic and let S be a rich a-convex set of states on L. For a real observable x on L, set V(x) := {s E S : s(x2) < 0 0 ) . In view of Schwarz

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inequality, s ( x ) ~5 s(x2), and since var, (x) = s(x2) - s ( x ) ~we , have V(x) = {s E S : var, (x) < m}. PROPOSITION 101. [Gudder, 19791 Let L be a logic and let S be a a-convex set of states on L which is rich. Let x be a real observable on L. Then x is bounded if and only if V(x) = S.

Proof. If x is bounded, then x2 is also bounded, hence V(x) = S. Conversely, let x be unbounded. We will show that there is a state s E S such that s(x2) < m does not hold. Let rn E w(x) be numbers such that 2n+2 - 1 > lrnl > 2,+l (n E N). Let U, be disjoint open balls in W with diameter less that 1 and with centers at r,. Put a, = x(U,). Since any Un is open and contains an element from the spectrum of x, we have a, # 0. Let s, E S be such that s,(a,) = 1. Since the elements ai(i E N) are orthogonal, we obtain sj(ak) = 0 for j # k. The state s = CjEN 2-jsj belongs to S. Assume that s(x2) < m . Then

which is absurd.

H

DEFINITION 102. We say that the real observables X I , (or that their sum exists) if

22,

.. . ,x, are summable

(i) the set V(xl,x2, . . . ,x,) := Uisn V(xi) is order determining; (ii) there is a unique real observable z such that V(z) > V(xl ,x2, . . . ,x,) and s(z) = ~ ( $ 1 ) s ( x ~ ) . . . s ( z ~for ) all s E V(xl, 5 2 , . . . ,5,).

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The observable z is called the sum of the observables xl, x2, . . . ,x,, and we write z = x 1 + x 2 + ...+a,. DEFINITION 103. A pair (L, S ) , where L is a logic and S is a a-convex rich set of states, is called a sum logic if the following two conditions are satisfied: (i) every two bounded observables are summable; (ii) every two compatible observables are summable. The above definition implies that summability and the sum of observables xl,x2,. . . ,x, are invariant with respect to permutations of the set {1,2,. . . ,n} and with respect t o the mappings xi + rxi (i 5 n) for every r E W. Moreover, if X I , x2,. . . ,x, are summable and if the sums yl = XI x2 . . . xk: and y2 = xk+l x, exist, then XI + x 2 + - - - + x n = yl +y2. Notice also that if xi,i 5 n are compatible and xi = u. fsrl(i 5 n), then 2 1 . xn = u.(fl . . . fn)-l. We will see that the logic L(H) with S = S(L(H)) is a sum logic, and so are the logics of projections in a von Neumann algebra. To our knowledge, a characterization of sum logics is not known.

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Proposition 99 and Proposition 101 imply that every two bounded observables have a bounded sum. This allows us to prove the following statements. PROPOSITION 104. Let (L,S) be a sum logic. Then (i) if x, y are bounded observables, then x = y if and only if s(x) = s(y) (that is, L has the Uniqueness property); (ii) L is a lattice; (iii) every state s E S has the Jauch-Piron property (that is, for every s E S , if a, b E L and s(a) = 1 = s(b), then s(a A b) = 1). Proof. (i) Suppose that s(x) = s(y) for all s E S. Let qo be the proposition observable qo({O)) = 1. Then the observables x qo and y qo are sums of the observables x and qo. Since the sums are unique, we have x qo = y qo. Hence x = y. (ii) Let a, b be elements of L and let q,, qb be the corresponding proposition observables. We will show that (q, qb)({2)) = a A b. From s((q, qb)({2))) = 1 it immediately follows that s(q, qb) = 1, hence s(a) = 1 = s(b). Since S is rich, it follows that (q, qb)({2)) is a lower bound of a, b. Let c E L be such that c 5 a, b. Then the equality s(c) = 1implies s(a) = 1 and s(b) = 1. Hence, s(q,+qb) = s(a) +s(b) = 2. Since 2 is the maximal point of the spectrum w(qa+qb), we see that s(q, qb)({2)) = 1. This implies that c 5 (q, qb)({2)). We have shown that the infimum a A b exists in L for every a, b E L, i.e., L is a lattice. (iii) Suppose that s(a) = 1 and s(b) = 1 for s E S. By the proof of (ii), we conclude that s(a A b) = 1. Thus s has the Jauch-Piron property.

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The notion of another type of joint distribution is based on the CramCr-Wold theorem in classical probability theory. Let p be a probability measure on B(Wk). For a E Wk, define ha : Wk + W by putting h,(r) = a.r = CiCkairi. If a is a real number, then the set {r : a.r 5 a) = P,(a) is a closed half-space in lRk. Further, we have p(P,(a)) = p.h;l(-oo,cr]. We recall that the CramCr-Wold theorem states that if pl and pz are probability measures on B(Wk) such that pl (Pa(a)) = pz(Pa (a) for every a E Wk and every a E W, then pl = p2. DEFINITION 105. We say that the observables X I , xz, . . . ,xn on a sum logic have a joint distribution of Urbanik type in a state s if the following two conditions are satisfied: aixi exists for any a (i) the sum Xi<, -

=

(al, az, . . .,h)E Wn,

(ii) there exists a probability measure p on B(Wn) such that, for any a E Wn and any a E R, p(Pa(a)) = s ( ( C aixi)(-m, 4 ) . i
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If such a measure p exists, we call it the joint probability distribution of Urbanik type for the observables xl,x2,. . . ,xn in the state s. We will write p = s, ,,...,., By using results from classical measure theory the following statement was , see also [ ~ t & and Pulmannovb, 1991, Th. 5.5.31). Let proved ( [ ~ r b a n i k19611, in the us denote by QS, the characteristic function of the observable xi state s, i.e. let us write Q: (u) =

((c

eiuts

aixi) (dt)).

ij n

THEOREM 106. The observables X I , x2, . . . ,xn have a joint distribution of Urbanzk type in a state s if and only if the function @(a) = @;(I) : Bn + C is the characteristic function of some n-dimensional probability distribution. PROPOSITION 107. Let X I , 5 2 , . . . ,xn be compatible observables on a sum logic (L, S). Then the joint probability distribution of Urbanik type exists in every state s E S. Proof. Let w be the joint observable of XI, x2, . . . ,xn. Then we have xi = w.?r1, where ri : Wn -4 W is the corresponding projection (i 5 n). By the functional calculus for compatible observables we obtain

for any a E Wn. Hence, for any E E B(W) we have

where s.w is the probability measure on L3(Wn) which is identical with the joint distribution of Urbanik type for the observables X I , xz, . . . ,xn in the state s. Indeed, we have airi)-l(-m, a] = P: (a).

(x isn

It should be noted that the joint distribution of Urbanik type for compatible observable~is identical with the joint distribution of Gudder type. As a consequence, we obtain the following result. PROPOSITION 108. Let (L, S ) be a sum logic and let XI, $2,. . . xn be bounded observables on L. If xl, x2, . . . ,xn have the joint distribution of Gudder type in a state s E S then they also have the joint distribution of Urbanik type in the state s, and both joint distributions are identical.

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184

Proof. The proof follows by compatibility of the reduced observables xi Ac (i 5 n), rn where c denotes the commutator of XI,XZ, . . .,x,. The converse of Proposition 107 has been proved for discrete observables only

udder, 1966; Urbanik, 19851, [ ~ t &and k PulmannovB, 1991, Prop. 5.5.51. The more general case needs special properties of sum logics, which are satisfied, e.g., by the Hilbert space logic. 7 THE LOGIC OF CLOSED SUBSPACES OF A HILBERT SPACE In this section we consider the Hilbert space logic in more detail. This logic plays an important role in investigations into the mathematical foundations of quantum mechanics. We restrict ourselves to a complex separable Hilbert space H, although many results remain valid in more general situations. (It should be noted that there is a way of extending several techniques of Hilbert space logics to Prehiibert spaces. Interesting results along this line can be found in [~hetcuti, 2005; Chetcuti and DvureEenskij, 2003; DvureEenskij, 1993; Hamhalter and Pt&, 1987; Pt& and Weber, 2001; Buhagiar and Chetcuti, 2005; Cattaneo and Marino, 1986; Mushtari and Matvejchuk, 19851, etc.) By a subspace we here always mean a linear subspace of H and by a closed subspace we mean a subspace of H closed in the norm induced by the scalar product. Let M, N be subspaces of H. Write M N = {x y : x E M, y E N), denote by the norm closure of M in H , and set M' = {x E H : (x, y) = 0 Vy E M). Obviously, MI is a closed subspace of H . Consider the partial ordering of L ( H ) defined by set inclusion, and denote the least element by 0 (i.e., 0 = {O), and the greatest element by 1 (thus 1 = H). It is easily seen that with the inclusion ordering, L(H) becomes a lattice with the operations A, V defined by M A N = ~ n ~ a n d ~ ~ ~ = m . The following result is the so called projection theorem.

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THEOREM 109. If M E L(H), then every vector x E H can be uniquely written in the form x = y z, where y E M and z E MI.

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Proof. (see, e.g., [ ~ a l m o s 19511). , We may assume that x E H \ M . Put 6 = inf{llx - yll : y E M). By the definition of 6, there exists a sequence {y,) in M such that 11yn - XI 1 -+ 6. We then obtain

Thus {yn) is a Cauchy sequence. Since M is closed and H complete, there exists y E M such that yn -+ y. Since the norm on H is continuous, we obtain Ily-XI[ = limn,, llyn - 211 = 6. We will show that z = x - y E MI. Suppose that for some

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v E M w have cr = @(z,v ) , where /? is a real number. Using the equality llzll = 6, we obtain

This implies that for certain small negative values of /? we find (z, v ) = 0. We conclude that x = y z, where y E M , z E MI. This expression is unique: if x = yl 21, then y - yl = z - zl and since y - yl E M , z - zl E M', we obtain rn y = y1, z = z1.

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THEOREM 110. The set L(H) endowed with the inclusion ordering and with the orthocomplementation operation defined above i s a lattice logic.

Proof. The infimum A of two elements becomes set intersection. It follows that the infimum of an arbitrary family of elements of L(H) exists and becomes set intersection. Assume that Mn E L(H) (n E N). Then V M, = A{M E L(H) : M > Un M,). Thus, L(H) is a a-complete lattice. The mapping M -+ M' is an orthocomplementation. Indeed, the inclusion M C N obviously implies N' c MI. Further, M A M' = 0, because x E M and x E M' yields (x, x) = 0, which means x = 0. The inclusion M C M u is obvious. Assume that x E M u \ M. By Theorem 109, x = y z with y E M and z E M'. We have z = x - y E M u , and therefore z E M u A MI. Thus z = 0, and hence M = Mu. Finally, assume that M c K ~ But and M ' A K =O. I f x K~ , thenwe h a v e x = y + z for y~ M a n d z MI. z = x - y E K , and, therefore, z = 0. Thus, M = K , and we have verified the orthomodular law.

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By an operator on H we mean a linear mapping A : H -+ H. The n o r m of an operator A is defined by IlAll = sup{IIAxII : x E H, IIxII = 1). An operator A is bounded if the norm of A is finite. If A is bounded, the adjoint operator A* of A is defined by the equality (Ax, y) = (x, A*y) for any x, y E H . A bounded operator A is self adjoint if A = A*. A projection operator (also called a projection or a projector) is a bounded operator P such that P = P2= P*. THEOREM 111. To every closed subspace of H we can associate a unique projection, and every projection is associated with a unique closed subspace of H .

Proof. Assume that N E L(H). The projection associated with N , denoted by pN,is defined as follows: p N x = y, where y is the unique vector of N given by the decomposition x = y + z in the sense of Theorem 109. On the other hand, if P is a projection, the set Np = {x E H : P x = x) is the subspace of H associated with P. One can simply verify that pNis a projection, Np is a closed subspace, rn and that Nplv = N a n d pNp= P. The set of all projections on H can be endowed with a partial ordering by writing P 5 Q if P Q = Q P = P. The following statements are well known and easy to prove (e.g., [Halmos, 19511).

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PROPOSITION 112.

pN be the corresponding (i) Let M, N be closed subspaces of H and let pM, = pNpM, i.e., projections. Then M c N if and only if pM= pMpN pM5 pN.

+

(ii) M IN if and only if pMpN = pNpM = 0 if and only if pM pN is a projection. (iii) M, N are compatible in L(H) if and only if pMpN = PNPM (iv) For two projections P, Q, the operator Q - P is a projection if and only if p I Q(v) P Q is a projection if and only if P Q = Q P . (vi) If P is a projection, then 1- P is also a projection, and N 1 - ~= Nb. The states on the logic L(H) are characterized by the famous Gleason theorem [Gleason, 19571 stated below. THEOREM 113. (Gleason theorem) Let H be a separable Hilbert space with dim H 2 3. If s E S(L(H)) is a pure state on L(H), then there exits a unit vector v, E H such that s ( P ) = (v,, Pv,) (P E L(H)). Moreover, every state on L(H) can be written as a a-convex combination of pure states such that the unit vectors corresponding to them are mutually orthogonal. Gleason's theorem implies that the pure states s on L(H) are in a one-to-one correspondence with unit vectors v, in H (i.e., v, E H, Ilv,l\ = 1) given by s(P) = ( P v , , ~ , ) . Therefore pure states are also called vector states. Moreover all states on L(H) are in a one-bone correspondence with the positive self-adjoint traceclass operators with unit trace (so-called density operators). That is, to every state s there corresponds a positive operator T,= CiEN c ~ P [ ~ ~ I ,where CiEN cj = 1, and {vj : i E N} is a complete orthonormal set of vectors in H. We then have that s(P) = tr(T,P) = CiEN ~ ( P v jvj). , Let us note that Gleason's theorem also holds for separable real and quaternionic Hilbert spaces with dimension at least three and it can be extended to certain nonseparable Hilbert spaces (see [~vure~enskij, 1993; Hamhalter, 20031). Let ( 0 , C) be a measurable space. A spectral measure is a mapping E : C + L(H) such that the following conditions are satisfied: (i) E ( 0 )

=

1;

(ii) if S1,S2E C and S1n S2= 0, then E(Sl)

IE(S2);

(iii) if Sj, i E M is a collection of pairwise disjoint sets of C, then E(UjENSi) = CiENE(Si), where the sum on the right converges in the strong operator topology.

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In other words, a spectral measure is an observable E : C 4 L(H). If C = B(W), then E is a real observable. The real observables on L(H) are characterized by the spectral theorem. Before stating it, let us first generalize the notion of bounded operator. Let D be a dense subspace of H and let T : D + H be a linear mapping. Call T again an operator on H. Then one can define the operator T* adjoint to T such that the following conditions are satisfied: The domain of T* is determined by the requirement D(T*) = {y E H : there is a unique y* E H such that (Tx, y) = (x, y*) for any x E D(T)), and we define T*y = y* (y E D(T*)). An operator T is called self-adjoint if T = T*.

THEOREM 114 (Spectral theorem). For any self-adjoint operator A on a Hilbert space H there exists a unique observable pA: B(B) + L(H) such that (i) the domain D(A) of A consists of all vectors x E H for which the integral J X2 ( p A(dX)x, x) converges, (ii) for any x E D(A) and any y E H we have (Ax, y) = 5 x(P*(~x)x,y). Conversely, for any real observable E there exists a unique self- adjoint operator

A such that E = pA. For the proof of the spectral theorem see, e.g., [Halmos, 19511. The spectral measure corresponding to a self-adjoint operator A is also sometimes called the spectral resolution of A. The spectrum of an operator A is defined as the set of all complex numbers c such that the operator A- cl has no bounded inverse mapping. It can be proved that the spectrum of a self-adjoint operator A coincides with the spectrum of the corresponding observable pA. Let s be a state on L(H) and let T be the corresponding density operator. Let A be a self-adjoint operator and let pAbe the corresponding observable. Then for the expectation of A (equivalently, of p A ) in the state s we have

if the integral converges. Let us conclude this section by showing that the Hilbert space logic is a sum logic in the sense of Definitions 102, 103. This fact is important in quantum axiomatics. By the spectral theorem, if A is a self-adjoint operator, we can identify it with the corresponding observable, and hence view A as a mapping from B(W) to L(H). If f : W + W is a measurable function, then f (A) = A.f-I is an observable too (for example,A2 = A.f -', where f (t) = t2). Denote by D(A) the domain of a self-adjoint operator A. Recall that, in case A is unbounded, D(A) is a dense subset of H . Moreover, we can write D(A)

J t z ( ~ ( d t )u)~ ,< m )

=

{u E H :

=

{ U E H : s,(A2)

< m).

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PROPOSITION 115. Let Sv be the set of all vector states on L ( H ) . Then S C Sv is rich for L ( H ) i f and only i f S = Sv. Proof. Richness of Sv is obvious. If S # Sv, then there is a vector v E H such that sv 4 S . Let [v]be the subspace generated by the vector v . Then the equality s,([v]) = 1 holds for no s, E S , contradicting the hypothesis that S is rich.

COROLLARY 116. Let A be a self-adjoint operator. Then the following statements are equivalent: (1) the set S ( A ) of all vector states corresponding to the vectors of D ( A ) is rich for L ( H ) ; (2) D ( A ) = H ; (3) A is bounded. PROPOSITION 117. For a linear subspace K of H , write S ( K ) = {s, : u E K , llull = 1). Then K is dense i n H i f and only i f S ( K ) is order determining for L(H)Proof. Let K be dense. Then if s,(P) 5 s,(Q) for all unit vectors in K , we infer that ( P v ,v ) 5 (Qv,v ) for all v E H , and hence P 5 Q. This shows that S ( K ) is order determining. Conversely, let KO be a subset of H be such that the set S ( K 0 ) = {s, : u E K ) is an order determining set of states. Let K l be the linear subspace generated by KO. Obviously, S ( K l ) is order determining for L ( H ) . Let K be the closure of K 1 in H . Then for every u E KO we have s,(K) = 1. Since S ( K o ) is order determining, we have K = H .

Observe that if s = CiEN aisi and s(A2) < CO, then si(A2) < ca for all i E W. Moreover, if P, Q E L ( H ) and if si(P) 5 si(Q) for all i, then s ( P ) 5 s(Q). This implies that the set {s E S ( L ( H ) ) : s ( A 2 ) < ca) is order determining for L ( H ) if and only if the set of vector states {s, : s,(A2) < ca) is order determining for L(H). If A is a self-adjoint operator, its domain D ( A ) is dense in H . If A , B are selfadjoint operators, then their sum A B is defined on D ( A ) n D ( B ) . It is known that A B need not be self-adjoint. But if at least one of A, B is bounded, then A B is self-adjoint (see e.g. [ ~ u n f o r dand Schwarz, 19571, [~ru~ovecki, 19711). If A , B commute, i.e. if they correspond to compatible observables, then we have by functional calculus of compatible observables that A B is self-adjoint. Let us finally describe an algebraic construction in connection with sum logics. Let x , y be two bounded real observables on a sum logic. The so called Segal product is defined by 1 x o y = z [ ( x + ~ ) 2 (- X - Y ) ~ ] .

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In [ ~ u d s o nand Pulmannovb, 19931, the following statement was proved. THEOREM 118. Let L be a sum logic such that on the set of Ob(L)of all bounded observables on L , the Segal product is distributive (with respect to the sum of observables). Then Ob(L)admits the structure of a Jordan algebra.

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8 JOINT DISTRIBUTIONS ON THE HILBERT SPACE LOGIC The main result of this section is the statement that observables having a joint probability distribution of Gudder type in a state s also have a joint probability distribution of Urbanik type in the state s, provided that all required sums exist. Let L(H) be the logic of a separable Hilbert space H (real or complex) with dim H 2 3. Let B(M) be a Bore1 a-algebra of subsets of a separable Banach space M (later on we will restrict ourselves to the real observables). Recall briefly our notations. If C is a closed subspace of H , then pCdenotes the corresponding projection. For a vector v, the symbol [v] denotes the one-dimensional subspace generated by v. If llvll = 1, then s, : C + (v, pCv) denotes the corresponding state on L(H). By Gleason's theorem, for every state on L(H) there is a density ~ ) that T is positive, self-adjoint and operator T with s(C) = s ( p C ) = ~ T ( T P such has a unit trace. In agreement with the preceding sections, we write, for all d = (dl, dz, . . . ,d,) E Dn, D = (-1, I), and all F E L(H), Fdi = F provided di = 1, and ~~i = F' provided di = -1. THEOREM 119. Let v be a unit vector in H. The observables X I , xz, . . . ,xn on L(H) have a joint distribution of Gudder type in the state s, zf and only if for any El,Ez,.. .,En E B(M) and any permutation p of the set {1,2,. . . , n ) the following equality holds: (8) p x l ( E 1 ) . . . p ~ n ( E n=) P~X P ( ~ ) ( E P. .( .~p)~ p ( n ) ( E ~ ( n ) ~ . For the proof of this theorem we need some simple lemmas the proofs of which are left to the reader. LEMMA 120. If the equality (8) holds for a vector v E H , then ~ X ~ ( E I ) A . . . A X ~=( E pzi(E1). ~), .. p ~ n ( E n ) ~ . LEMMA 121. Let {vi : i E N) be an orthonormal set of vectors. If llv112 = CienI(", vi)I2 for a vector v E H , then v = CiEn(v,vi)vi. LEMMA 122. Let VI, Vz, for a d € Dn. Then

.. .,Vn E L(H) and let v E H, v # 0.

pVp(1)

*

. .pVp(n),

Let v E Ai<, -

~ i ~ '

= pV1. . . pVnv

for every permutation p of the set {1,2,. . . ,n). Let us now prove Theorem 119. Proof. Assume that the equality (8) holds for v E H , llvll = 1. Then by Lemma 120 we have

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Using Theorem 87, we get that the observables X I , x z , . . . ,x, have a joint distribution of Gudder type in the state s,. Suppose that a joint distributions of Gudder type for the observables X I , xz, . . . , x, in the state s, exists. Let El, E2,.. .En E B(M). By Theorem 87, we have

This equality yields

where we put M(d) = Ai<, x ~ ( E ~ )( d~ E ' Dn). Obviously, the vectors { P ~ ( ~ :) v d E Dn) are mutually orfiogonal. Applying Lemma 121, we see that the vector v is a linear combination of the vectors P ~ ( ~ )Byv Lemma . 122, condition (8) is satisfied. This completes the proof of Theorem 119. The following theorem gives a necessary and sufficient condition for the existence of a joint distribution of Gudder type. THEOREM 123. Let X I , x z , . . . ,x, be observables and let s be a state with the density operator T = CiEN Ci~[v"I. (i) T h e joint distribution of Gudder type for X I , x z , . . . ,x, in the state s exists if and only if the condition 8 of Theorem 119 is satisfied for each vector vi ( i E M ) . (ii) If X I , 2 2 , . . . ,x, are real bounded observables and if A1,A2,. . . ,A, are the corresponding bounded self-adjoint operators o n H , then the following conditions are equivalent:

(a) the observales X I , xz, . . . ,xn have a joint distribution of Gudder type in the state s;

(b) the condition (8) of Theorem 119 holds for each vector v i ( i E N ) ;

( c ) the equality Al . . . Anvi = Ap(l).. .Ap(,)vi (i E M ) holds for a n y permutation p of the set { 1 , 2 , . . . ,n ) ;

(d) the equality A1 . . .A,T the set { 1 , 2 , . . . ,n ) .

=

Ap(l).. .Ap(,)T holds for every permutation p of

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Proof. By Theorem 87 we see that a joint distribution of Gudder type in the state s = CiENsi exists iff it exists in state si for all i E M. Combining this with Theorem 119, we obtain that (i) holds. From the properties of spectral measures and Theorem 119 we deduce that statement (ii) holds. THEOREM 124. Let { x , ) , ~ ~be a system of observables on L(H). Denote by Ho the set of all vectors in H that satisfy the condition (8) for any finite choice xal, x a 2 , . .. ,x,,, all02,. . . a, E I. Then Ho is a closed linear subspace of H and the equality Ho = com{x, : a E I) holds. Moreover, if s is a state with the C i ~ [ v 'then ] , the joint distribution of Gudder type for density operator T = CiEN the observables {x, : a E I) in the state s exists if and only if vi E Ho for all i (i E N).

Proof. It is easy to check that Ho is a closed linear subspace of H . Since L(H) is a complete lattice, the commutator C := com{x, : a E I) exists. Moreover, since L ( H ) is separable (in the sense that every orthogonal set of projections is at most countable), we can find a countable subset J of I such that C = com{xj : j E J) (see Proposition 22). Let v be a unit vector and s, the corresponding state. By Theorem 119, v E Ho if and only if the joint distribution of Gudder type for {x, : a E I} exists in the state s,. Since C = com{xj : j E J), we deduce (Theorem 87) that the joint distribution for the observables {x, : a E I) in the state s, exists if and only if s,(G) = 1, i.e., if and only if v E C . It follows that Ho = C . The remaining part of the proof follows by Theorem 123. The next theorem is a converse of Proposition 107 for bounded observables on L(H). We will follow the proof in [Varadarajan, 1985, Theorem 4.251. Let us first introduce some notations. Let x j ( j 5 k) be bounded observables on L(H), and let Aj be the bounded self-adjoint operator corresponding to xj. If cl, c2,. . . ,c, are real numbers, put c = (cl,. . . ,ck), A(c) = clAl + c2A2 + . - -+ cnAn Let E H P ~ ( ~ ) ( E=: ) Q(c; E ) be the spectral measure of A(c). If q5 E H is a unit vector, we have s + ( p A ( " ) ( ~ )= ) s+(c;E) = IIQ(c;E)II, so that s+(c;E ) is a probability measure on B(R). Let us recall that observables X I , . . . ,xk have a joint distribution of Urbanik type in a state s+ if there is a probability measure p+ on B(W) such that

where E(c) = {(tl, t2,. . . ,tk) : cltl

+ + cktk E E). ''

For the proof of the theorem announced we need the following lemma. LEMMA 125. Let B be a nonnegative self-adjoint operator and S,T two projections such that B 5 S and B 5 T. Then B 5 S A T.

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Proof. Let Ns, NT be the ranges of S,T, respectively. Then l l ~ ~ / ~ 5 v 1I I1S~V ~ ~ ~ , and hence B vanishes on Nk. Similarly, B vanishes on N+ and hence B = 0 on (NsflN~)'.Since B is self-adjoint, it leaves NsnNT invariant, and for v E N,nNT, (Bv,v) 5 ll~11~. Thus B 5 pNsnNT =S A T . THEOREM 126. T h e bounded observables xl, x2, . . . ,xk o n L(H) have a joint distribution of Urbanik type in all states o n L(H) if and only if the operators Al, A2,. . . , Ak corresponding t o X I , x2,. . . ,xk commute with one another.

Proof. Sufficiency follows by Proposition 107. To prove necessity, assume that the observables in question have a joint d i s tribution of Urbanik type in all states on L(H). Let us restrict ourdelves to the case k = 2 and prove that AlA2 = A2A1, the in the general case the argument is similar. For any nonzero 4 E H, we write 11, = (11411)-14 and define q+ := 11~$11~~+. Then q+ is a nonnegative measure on B(W2) and q+(W2) = 11q5112. Set qo = 0. For any pair for vectors 4,4' E H, let us define a C-valued measure q+,y by the polarization formula

We claim that for each M E B(R2), the map (q5,4') sesquilinear on H x H . Indeed, let us write

I+

q+,y (M) is bounded and

If M = E(c), E E B(W), then q+(M) = (Q(c;E)@,4) and hence

which entails v(M) = 0. Since this equality holds for all M = E(c), it follows v = 0. Similar arguments complete the proof that (q5,qh1) I+ q+,y is sesquilinear. Fkom 0 5 q+,y(M) 5 11q5112 we conclude that (#,+I) H q+,v is bounded. Using well-known techniques, we can construct a unique bounded self-adjoint operator PMsuch that q+,v(M) = (PM414)Clearly, 0 5 PM5 1 for all M , P@ = 0, P R z = 1. Moreover, as M I+ Q+,+I(M)is a measure, PMis countably additive with respect to the disjoint sets MI, M2,. . .. We also have q+,+(M)2 0. Moreover, MI C M2 implies PMl5 PMz . If M = E(c), then PM= Q(c; E). We have to show that for any two sets El, E2 E B(W), the operators Q1 := Q((1,O);E l ) and Q2 := Q((0,l); E2) commute. Since El x E2 C El x R, we have

Similarly,

P E ~~ ~P WE X~E=Q2. ~

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From Lemma 125 we conclude that

Replacing E2 by W \ E2,we obtain

The last two inequalities give us

where the last inequality holds because Q1 A Q2 and Q1 A (1 - 92) are orthogonal projections. Hence we obtain

from which we conclude that Q1 and Q2 commute.

rn

We now establish an important connection between the joint probability distributions of Gudder and Urbanik type on L(H) (L(H) can be regarded as a prototype of a sum logic). This extends the result of Proposition 108 to certain unbounded real observables on L(H). We do not go here into details concerning sums of unbounded operators, see e.g. [Ptiik and PulmannovS, 1991, Theorem 5.6.81. THEOREM 127. Let x and y be real observables on L(H), and let for all a,P E W the observables a x + Py exist. Then if the observables x and y have a joint distribution of Gudder type i n a state s , they also have a joint distribution of Urbanik type in the state s, and these joint distributions coincide. Proof. We sketch the proof of Theorem 127. In view of Theorem 123, it is sufficient to consider only vector states. Let v be a unit vector in H and let the joint distribution of Gudder type exist in the state s,. Applying Theorem 124, we find that v E Ho, where Ho = com(x, y). Let A, and A, denote the self-adjoint operators on H corresponding to x and y, respectively. Then PH0 commutes with A, and A,. That is, Ho reduces A, and Ay, the reduced operators A,/Ho and A,/Ho are self-adjoint and the corresponding observables xo and yo on L(Ho) are compatible. The observables a x and Py are summable if and only if aA, + PAy is essentially self-adjoint (i.e., aA, PAy has a unique self-adjoint extension aA, + PA,, corresponding to the observable ax+Py). Then Ho reduces also the self-adjoint operator corresponding to ax+Py, and (ax+Py)o = axo+Pyo. Since xo and yo are compatible, they have a joint distribution of Urbanik type in the state s, (restricted to Ho). This joint distribution is identical with the joint distribution of Gudder type (Proposition 107). This means that there is a measure p on 13(W2) such that

+

Pave1 Ptik, Sylvia Pulmannovi

194

for all a,p E R and all E E B(W). We can further write 3,

(axo + pya) ( E ) )

=

(v, p ( a " ~ + P ~ o ) ( E )=) ~(v, ) P(""+~Y)~(~))V)

-

(v, P ( ~ " + ~ ~ ) (=~ S), V (ax )

+ py) (E)).

This implies that the joint distribution of Urbanik type for x and y in the state s, exists. If E, F E B(W), then p(E x F )

s,(xo(E) A yo(F)) = (v, = (v, P ~ ( ~ ) P ~=( s,(x(E) ~ ) ) A y(F)), =

which implies that the joint distribution of Gudder and Urbanik type for the observables x and y are equal (since they are equal for the observables xo and yo). The proof is complete The following example shows that the converse of Theorem 127 does not hold. Let us consider operators P and Q of momentum and position on L2(R). Thus the operator Q has the domain D(Q) = {f E L2(B) : 5 lrf(r)12dr < m), and (Qf)(r) = r f (r) for f E D(Q). The operator P has the domain D(P) = { f E L2(R) : ~ p j (12dp ~ ) < m ) , where f" is the Fourier-Plancherel transform of f (r), and (Pf)(r) = - i h q . It is well-known that P and Q satisfy the Heisenberg uncertainty relation, and therefore they do not have a joint distribution of Gudder type in any state. In fact, they are even complementary. But it was proved in [Urbanik, 19611 that they have a joint distribution of Urbanik type in the state

5

and do not have one in the state

9 APPENDIX 1: UNCERTAINTY RELATIONS Let L be a quantum logic and S(L) the state space of L. Let x be a real observable on L. Let var,(x) denote the variance of x in the state s. Put V(x)

= {s E S(L) : var,(x)

< m).

For any two real observables, the following alternative possibilities can occur:

(A) (VE > 0)(3s E V(x) n V(y))(var,(x).var,(y) < 6). (B) (36 > O)(Vs E V(x) n V(y))(var,(x).var,(y) 2 E).

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If (B) occurs, we say that the uncertainty relation holds for x and y. For t E R, 6 > 0, put U(t, 6) = {r E W : It - rl < 6). Notice that if t belongs to the spectrum of x, then x(U(t, 6)) # 0. Recall that L belongs to the class L3 if S(L) is unital (i.e., to every a E L there is a state s E S(L) with s(a) = 1) and for any a, b E L such that a cft b there is s E S(L) such that s(a) = s(b) = 1. Let a, y be real observables on L with the spectra a(x), a(y), respectively. The following two cases can occur:

(b) (3(u7V,6) : u E u ( x ) , ~E o ( Y ) , ~> 0) (Vqo rla)(x(U(u, 6)) cft Y ( U ( ~rl))). ,

>

0) (3q,0

<

q

<

In [PulmannovA, 1988], the following results were obtained. THEOREM 128. Let L be a logic with the unital state space. If for the observables x and y on L the condition (a) holds, then the uncertainty relation does not hold. In other words, (a) =+ (A). Moreover, if L is a lattice, then (a) implies that x and y are compatible. THEOREM 129. Let L E L3. Then the uncertainty relation does not hold for any pair of observables x, y. Proof. Let x, y be observables on L. By Theorem 128, (a)+-(A).Suppose that (b) holds for x and y. Then there are u E a(x), 6 > 0, v E a(y) such that to any qo > 0 there is q < qo with x(U(u,6)) cft y(U(v,q)). AS L belongs to L3, there is a state s E S(L) such that s(x(U(u, 6))) = 1= s(y(U(v, 71))). Choosing q sufficiently small, we obtain var,(x).var,(y) < 6 for any given E > 0.

10 APPENDIX 2: BELL INEQUALITIES ON QUANTUM LOGICS J.S. Bell ell, 19601 introduced examples of inequalities valid in classical prob ability theory. However, these inequalities were found to be violated in certain quantum mechanical experiments. The Bell inequalities are sometimes considered as "tests" of "hidden variables", i.e., tests of the nearness of the probability model used to the classic Kolmogorov model. Many results can be found in the literature that suggest the non-existence of hidden variables in quantum mechanics (see e.g., [Kochen and Specker, 1967; Beltrametti and Cassinelli, 1981; Giuntini, 1991; Bell, 19601). In the logicealgebraic approach to quantum mechanics, the standard model of the set of experimentally verifiable propositions of a physical system, which is based on a Boolean a-algebra, is replaced by a more general model based on a "quantum logic". The quantum logic is commonly supposed to be an o r t h e modular a-poset. In the standard Hilbert space approach, the quantum logic is the projection lattice of a Hilbert space.

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There are several formulations of the problem of hidden variables in the logice algebraic approach. Typically, the existence of hidden variables is related to the existence of a rich supply of dispersion-free states on the set of propositions. A physical system is considered to be L'classical" if its set of propositions forms a Boolean a-algebra. The problem that arises now is that there are Boolean aalgebras which posses many states but no dispersion-free states. As a result, even certain classical systems would on this view not admit hidden variables. So the existence of hidden variables as a test of whether or not a system is "classical", resp. "quantum" may not be entirely convincing. An attempt to overcome this difficulty has been made in [Deliyannis, 19711, where in the criterion of the existence of so-called quasi hidden variables the dispersion-free states have been replaced by dispersion-free quasistates, i.e., finitely additive states. Recently q isi is, 20001, a suitable definition of so called approximate hidden variables has been found in terms of countably additive states. For our purpose, the approach using quasi hidden variables in terms of finitely additive states is more appropriate in order to find connections with certain kinds of Bell inequalities. In this section, we will make the connection between the existence of quasi hidden variables and the Bell inequalities on orthomodular lattices ([~ulmannovi, 20021). We will consider Bell inequalities in a probabilistic formulation (see [pitowski, 19891). These Bell inequalities on quantum logics have been studied, e.g., in [Beltrametti and Mqczyhski, 1991; Pykacz and Santos, 1991; Pulmannova and Majernik, 1992; Pulmannovi, 1994a; D'Andrea and PulmannovB, 1994; DvureEenskij and Langer, 19951.

11 QUANTUM LOGICS AND PHYSICAL SYSTEMS Again, by a quantum logic we mean a a-complete orthomodular poset L. Recall that a mapping h : L1 -+ L2, where L1 and L2 are quantum logics is a a-homomorphism (of quantum logics) if (1) h(1) = 1, (2) h(at) = h(a)', and (3) if a = Vai for a sequence (ai)$ of pairwise orthogonal elements of L1, then h(a) = V h(ai). A a-homomorphism which in addition preserves all existing infima and suprema, is called a lattice a-homomorphism. A surjective a-homomorphism is an isomorphism if a 5 b ifF h(a) 5 h(b). If L is a logic, by a state on L we mean a countably additive state. By relaxing this condition to finite additivity, we speak on finitely additive states. The set of all states (the state-space of L) will be denoted by S(L). Let L be a logic and s a state. For any element a E L, the dispersion of s at a ~ . total dispersion of s may be defined as is the number s(a) - ~ ( a ) The

A state s is dispersion-free if a(s) = 0, i.e., if the range of s is (0, 1). Obviously, the latter notions make sense also for finitely additive states and we shall occasionally use them for this setup.

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Let L be a logic, and R, S two sets of finitely additive states. We say that R generates S if there is a a-algebra C on R such that (i) for every a E L, the map w o w(a) is C-measurable, (ii) for every s E S, there exists a probability measure p, on (R, C) such that for all a E L,

In accord with Chapter 1, let L be a logic and S a set of (finitely additive, in general) states on L. S is called order-determining if the inequality s(a) 5 s(b) for all s E S implies that a 5 b, and S is called rich if a -$ b implies that there is s E S with s(a) = 1,s(b) < 1. A system is a pair (L, S ) where L is a logic and S is an order-determining set of states. If (L, S) is a system, then a pure state is an element of S which cannot be written as a nontrivial convex combination of elements of S. If (L, S) and (L', Sf) are systems, a mapping f : L + L' is a system imbedding if f is a a-homomorphism of logics and (1) for every s E S, there exists sf E Sf such that sfo f = s. Observe that if f is an imbedding, then a 5 b iff f (a) 5 f (b). By definition, a system (L, S ) is a subsystem of (L', S f ) if there is a system imbedding of (L, S ) in (L' , St). Let us consider the following examples isis is, 20001. Example 1. Let L be a Boolean a-algebra. Let L possess an order-determining set of states. Consider the system (L, S(L)) associated with L. It can be derived from the LoomisSikorski theorem that there is a measurable space (R, C) and a set M of finite measures on (0, C) such that L is isomorphic to the quotient of C by the a-ideal {A E C : p(A) = 0 for all p E M}. This example has the following important special cases. ( l a ) . Let (R, C) be a measurable space. The system associated with it is the pair (C, S(C)). ( l b ) . Let (R, C, v) be a a-finite measure space. The system associated with it is the pair (L, S(L)), where L is the quotient of C by the a-ideal of sets of measure zero. Then S(L) can be identified with the set of the probability measures on C that are absolutely continuous with respect to v. Example 2. Let H be a separable Hilbert space, dim H 2 3. The system associated with it is the pair (L, S), where L is the logic of all closed subspaces of H (equivalently, of all orthogonal projections) and S is the set of all states on L. Examples ( l a ) and ( l b ) are related to classical mechanics. Example (2) corresponds to standard quantum mechanics. According to [Varadarajan, 1985, p.116-1171, if w is a dispersion-free state on a countably generated Boolean a-algebra L, then there is an atom p of L with

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w ( p ) = 1. So if L has no atoms, it has no dispersion-free states. Consequently, if

L is a logic with a countably generated Boolean a-subalgebra with no atom, then L has no dispersion-free states. It can is easily seen that a logic has a countably generated Boolean a-subalgebra with no atoms iff L admits an observable with the discrete spectrum empty. So we may conclude that while in Example ( l a ) all realvalued observables have nonempty discrete spectra, in Example ( l b ) there may be observables with empty discrete spectrum, like, e.g., the position observable. In standard quantum systems, if H is finite-dimensional, then all observables are discrete, while if dim H = oo, there are observables, like position and momentum observables, with continuous spectra. So the non-existence of dispersion-free states is not a criterion of whether a system is classical or quantum. There is an important difference between examples ( l b ) and (2). While in the first case all pure states are dispersion-free, in the second case there are pure states which are not dispersion-free. In some hidden-variable approaches, it is required that only that the pure states be generated by dispersion-free states. But this criterion cannot be applied to quantum mechanical systems with no pure states. The usual definition of a system that admits hidden variables is as follows. DEFINITION 130. A system (L, S) is deterministic if the set of dispersion free elements of S generates the entire S. DEFINITION 131. A system (L, S ) admits hidden variables if there is a set SHV such that SHv > S and (L, SHV)is a deterministic system. Recall that a logic L is concrete if L can be represented by a collection of subsets of a set [ ~ t a kand PulmannovL, 19911. By Theorem 48, a logic L is concrete if and only if it admits an order-determining set of dispersion free states. Observe that for dispersion free states, the notions of rich and order-determining system of states coincide. Recall further that the Jauch-Piron condition is satisfied for a, b if for every state s on L, if s ( a ) = s(b) = 1, then s ( a A b) = 1.

THEOREM 132. [ ~ e l i ~ a n n i19711 s , Let (L,S) be a system, S1 a generating set of states for S. Then L i s isomorphic (in the quantum logic sense) to a logic which i s represented by functions o n the set 0. If the states in S1 are dispersion free, these functions are characteristic functions of subsets of S1. Moreover, a necessary and suficient condition for the above isomorphism to preserve infimum a A b (or supremum a V b) i s the Jauch-Piron condition for this particular pair.

Remark. The preceding theorem implies that if a system (L, S) admits hidden variables, then L is a concrete logic. The converse need not be true. Indeed, a necessary and sufficient condition that a system (L, S), where L is a concrete logic admits hidden variables is that all states in S can be extended to states on a Boolean a-algebra of sets. Example ( l a ) is clearly deterministic. But in Example ( l b ) , if v is nonatomic, then every probability measure that is absolutely continuous with respect to v is

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also nonatomic, so it cannot be dispersion free. Thus L has no dispersion free states, hence (L, S(L)) does not admit hidden variables. In Example (2), the associate system (L,S) is deterministic ifE dimH = 1. If dimH 2 3, H has no dispersion free states [Alda, 19811. So in this case, (L, S) does not admit hidden variables. We see that the above definition does not distinguish classical and quantum systems. An interesting way to overcome this difficulty has been suggested in [Deliyannis, 19711. DEFINITION 133. A system (L,S) admits quasi hidden variables if the set of finitely additive dispersion free states of L generates S. The next lemma shows that a system (B, S) associated with a Boolean a-algebra B admits quasi hidden variables (see also [Deliymnis, 1971, Theorem 61). LEMMA 134. Let B be a Boolean a-algebra, and let s be a (a-additive) state on B. Then s is generated by finitely additive dispersion free states on B. Proof. Let Q denote the set of all finitely additive dispersion free states on B. For a E B, let & := {w E Q : w(a) = 1). By Stone's theorem, the family &,aE B forms a field of clopen sets on the (compact topological space) Q, which is isomorphic with B as a Boolean algebra. Write Co = {& : a E B), and define a function p : Co -+ [O,1] by p(6) := s(a), a E B. It is easy to see that p is a finitely additive probability measure on Co. Let C denote the a-feld of subsets of $2 generated by Co. We will show that p extends to a probability measure on C. According to the well-known theorems in measure theory (cf. [~almos,1962, Theorem A, Sec. 13]), we need to show that p is a-additive on Co, that is, p(Uz1Ai) = p(Ai) whenever a sequence {Ai)& of mutually disjoint sets from Co is such that Uz1Ai belongs to Co. So assume that Uihi = 6, for a countable set of pairwise disjoint sets hi,i E N,and for some b E B. Then we see that Viai 5 b. If Viai < b, then there is w such that w(b) = 1, and W(Q)= 0 for all i, which contradicts our suppositions. Therefore b = Viai, and p(6) = ~(Viai)= ~ ( a i= ) p(&i). So p extends to a a-additive measure on the a-algebra C generated by the algebra Co. Obviously, for every a E B, the mapping w I+ w(a) is measurable with respect to C. As a consequence of the previous considerations,

xi

xi

s, Let (L, S) be a system. Then (L, S) admits THEOREM 135. [ ~ e l i ~ a n n i19711 quasi hidden variables if and only if it is a subsystem of the system associated with a Boolean a-algebra. Proof. Assume that Q is a generating system of finitely additive states, and let

N be the a-ideal of subsets of Q on which all measures associated with the states

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s E S vanish. We define the mapping a I+ &, where &(w)= w(a) , and observe that orthocomplements are preserved. Moreover, a 5 b H & 5 b (the latter ordering is pointwise). Let a = Vai, where (ai)i is a sequence of pairwise orthogonal elements. So we have s(a) = C s(ai). Since each w E R is finitely additive, we have

X&~(W 5 )&(w). But h(w) -

hi (w))dpS(w) = 0

for all p,, s E S which, together with the above inequality yields & = C h i modulo

N. Further, let each w be dispersion free so that & is a characteristic function of a subset of R. Let a* denote the class h modulo N . Then * is an isomorphism which sends L into the Boolean algebra B of subsets of R modulo N. Since all measures which correspond to states in S vanish on N , they produce measures on B, so that s(a) = p,(a*). Conversely, if (L, S ) is a subsystem of a system associated with a Boolean ualgebra, then the result follows by Lemma 134. rn In [ ~ e l i ~ a n n i1971, s, Theorem 51 it is proved that a necessary and sufficient condition for the isomorphism a I+ a* to preserve an existing infimum (supremum) in L is the Jauch-Piron condition for this particular pair to be satisfied in all states generated by R. Indeed, all that we need to prove is that letting c = a A b so that ~~~nS={W:~(w)b(w)=i),wehavehni-~~~. But for every probability measure p on C , the map

is a state on L. Since &, b are 1 on & n b, we have m(a) = m(b) = 1, so m(c) as well. But E vanishes outside h n 6, so that

=

1

Therefore p,(& n b - E) = 0 for all p,, s E S. In particular, if all finitely additive states in R satisfy the Jauch-Piron property for a, b, the result follows. Notice that both Examples (la), (lb) admit quasi hidden variables. On the other hand, in Example (2) for H separable of dimension at least 3, there are no finitely additive dispersion free states on L ([Alda, 19811). Therefore (L, S ) does not admit quasi hidden variables.

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20 1

12 BELL INEQUALITIES In this section we assume that L is a lattice logic, and s is a finitely additive state on L. We will consider the following inequalities:

The inequalities (1) and (3) are called the inequalities of Bell- Wigner type, the inequalities (2) and (4) are called the inequalities of Clauser-Home type [pitowski, 19891. All these inequalities are satisfied by any finitely additive state on a Boolean algebra. Let s be a finitely additive state on a logic (in fact, all definitions and results apply well for an OML, thus we do not explicitly need the a-complete property of V in L). Recall that the a s is (i) subadditive if s(a V b) 5 s(a)

+ s(b) Va, b E L,

(ii) a valuation if s(a V b)

+ s(a A b) = s(a) + s(b) Va, b E L,

(iii) Jauch-Piron if s(a) = 1 and s(b) = 1 implies s(a A b) s (a) = 0 and s(b) = 0 implies s(a V b) = 0).

=

1 (equivalently,

Observe that the following implications hold: (ii) (iii) and (ii) @ (i). If s is dispersion free, then (i), (ii) and (iii) are equivalent. Recall that a, b E L are compatible if a = (a A b) V (a A b'), written a tt b. Recall also that L is a Boolean algebra iff a tt b holds for all a, b E L. The element =(a, b) = (a A b) V (a' A b) V (a A b') V (a' A b') is called the lower commutator of a, b, and a tt b iff =(a, b) = 1. The upper commutator is defined by iZE(a, b) = (a V b) A (a V b) A (a V b') A (a' V b') = =(a, b). Recall finally that by the Foulis-Holland theorem, if a, b, c are elements of L such that a tt b, a tt c, then (a, b, c) is a distributive triple, that is, x l A (x2 V x3) = (XI A 52) V (XI A ~ 3 )xi , E {a, b, c), i 5 3. PROPOSITION 136. [ ~ u l r n a n n o vand ~ Majernfk, 19921 Let L be a lattice logic and s be a finitely additive state on L. The following statements are equivalent: (i) s is subadditive; (ii) s is a valuation; (iii) the inequalities (1) hold.

Pave1 Ptdk, Sylvia Pulmannov6

202

-

Proof. (i) =+(ii): Define aAb := (a V b) A (a A b)'. Using the Foulis-Holland b, we obtain theorem and the fact that (a' V b') * a and (a' v b') (a V b) A (a' V b') = (a A (a' V b')) V (b A (a' V b')). By the property (i), = s((a A (a' V b')) V (b A (a' V b')))

s(aAb)

5 s(a (a' v b')) V (b A (b A (a' V b') = s(a) - s(aA b) + s(b) - s(a A b).

+

This entails that s(aAb) = s(a V b) - s(a A b) 5 s(a) s(b) - s(a A b), i.e., s(a V b) s(a A b) 5 s(a) s(b). Replacing a, b by a', b', we obtain the converse inequality, and hence s is a valuation. (ii) =+(iii) If s is a valuation, then s(a) s(b) - s(a A b) = s(a V b) 5 1, hence the inequalities (1) hold. (iii) & (i): If the inequality (1) holds, then for very a, b, s(at) s(bf) - s(a' A b') 5 1,which yields 1-s(a)+l-s(b)-(1-s(aVb)) 5 1,hence s(aVb) 5 s(a)+s(b).

+

+

+

+

We now study the inequalities of type (2). Following [pykacz and Santos, 19911, we consider the function S(a, b) = s(a)

+ s(b) - 2s(a A b).

It can be checked by a direct computation that the inequalities (2) are equivalent to the inequalities S(a, b) S(b, c) S(c, e) L S(a, e), These inequalities hold iff S is a metric on L. Put d(a, b) := s(aAb). It is easy to check that s is a valuation iff S = d.

+

+

PROPOSITION 137. Let L be a lattice logic and let s be a (finitely additive) state on L. Then s is a valuation iff p = s(aAb) is a metric on L.

Proof. First we prove that for any a, b, c E L, aAb 5 (aAc) V (cAb). Put e = a A b A c. Clearly a A (a A b)' 5 (a V c) A e'. Using orthomodularity, we prove that (a V c) A e' 5 (aAc) V (cab) (see [Pulmannovb and Majernik, 19921 for the details). Similarly, b A (a A b)' 5 (b V c) A e' 5 (aAc) V (cAb). Hence aAb

=

5

(a (a b)') V (b A (a A b)') (aAc) V (cab).

Now s is a valuation iff it is subadditive, which yields p(a, b) 5 p(a, c)p(b, c), hence p is a metric. Conversely, if p is a metric, then the inequality p(a, b) 5 p(a, 0) d(0, b) yields s(a V b) - s(a A b) 5 s(a) s(b). Replacing a, b by a', b', respectively, we see that the converse inequality holds, and hence s is a valuation.

+

+

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The above considerations yield the following result. THEOREM 138. Let L be a lattice logic and let s be a (finitely additive) state on L. The following statements are equivalent: (i) the inequalities (2) hold; (ii) S is a metric; (zii) S = p; (iv) s is a valuation; (v) p is a metric; (mi) the inequalities (1) hold.

Proof. We have already seen that the inequalities (2) hold 8 S is a metric. If S is a metric, then S(a, b) 5 S(a, aVb)+S(avb, b) implies s(a)+s(b)-2s(a~b) 5 s(a)+ s(avb) -2s(a)+s(a~b)+s(b) -2s(b). Consequently, s(a)+s(b) - s ( a ~ b ) 5 ~ ( a v b ) . Replacing a, b by a', b', this yields the converse inequality. Hence S = p, and p is a metric. The remaining statements follow by Proposition 136 and Proposition 137.

rn THEOREM 139. Let L be a lattice logic and let S be an order-determining set of states on L. If every state s E S satisfies Bell inequalities of the type (I), then L is a Boolean algebra.

Proof. By Theorem 138, every state s E S is subadditive. It suffices to prove that a tt b for all a, b E L. By subadditivity we obtain s((a V b) A b') = s(a V b) - s(bl) 5 s(a) for every s E S, and since S is order-determining, this gives us (aVb)Abl = aAb'. By orthomodularity, b' = ( a ' ~ b ' ) ~ ( a ~ b = ) ~(a'Ab1)V(aAb'), b' which entails that b' * a. As this holds for all a, b E L, we conclude that L is a rn Boolean algebra. COROLLARY 140. If the system (L, S), where L is a lattice logic, satisfies the inequalities (1)for every s E S , then (L, S ) admits quasi hidden variables. The converse need not be true. Let (Bl, Sl) and (B2,S2) be the systems associated with Boolean a-algebras, and let L := B1/B2 denote their horizontal sum. Then B1/B2 is a lattice logic, where the supremum Vai corresponds to the supremum taken in Bj if all ai's belong to Bj, j = 1,2, otherwise the supremum is 1. If s is a state on L, then the restrictions sj := s/Bj, j = 1,2 are states on Bj, j = 1,2, respectively, and s corresponds to the couple (sl ,s2), where s(a) = sj(a) iff a E Bj. Conversely, if (p1,pz) is a couple of states on Bj, j = 1,2, we can form a state p on L by defining p(a) = pj (a) ifF a E Bj, j = 1,2. It can be easily seen that a state s = (sl, sz) on L is subadditive, hence Jauch-Piron, ifF both sl and s z are subadditive, and at least one of them is faithful l. Indeed, if neither of sl, s z is faithful, then there are elements a1 E B1, a2 E Bz, both different from 1 with sl(a1) = 1,sz(a2) = 1. But a1 A a2 = 0, which contradicts Jauch-Piron property. It follows that L has no finitely additive dispersion free Jauch Piron state unless one of B1, B2 is {O,l). But L admits quasi hidden variables. Indeed, consider the tensor product B1 QD Bz. The set of product states sl @s2(al@az)= sl(al)s2(a2) is order-determining for B1QDBz (see, e.g., [Aerts and Daubechies, 19781). Moreover, there is an imbedding j : B~+BZ + B1 QD B2 such that j(a) = a QD 1 if a E B1 ' ~ e c a l lthat a state s is faithful on L if s(a) = 1 8 a = 1.

Pave1 Ptbk, Sylvia Pulmannovd

204

and j(a) = 1@I a if a E B2. Further, the states (sl,sz),si E Si,i = 1,2 are in a one-bone correspondence with the product states sl @ s 2 . It follows that j is a system imbedding. An interesting example of this type was found in isis is, 20001, where the observables of position Q and momentum P in the standard quantum mechanics are considered. Since Q and P are complementary ([Ptik and Pulmannovb, 1991]), it can be shown that the smallest sublogic of L(H) containing their ranges B1 := R(Q) and B2 := R ( P ) is isomorphic to the horizontal sum of B1 and B2. Now let us consider the inequalities of type (3), i.e., for a state s and all a, b E L, let us suppose that

If we put c = 0, we obtain the inequality (I),hence (3) implies that s is subadditive. Recall that a subset J of a lattice logic L is a p-ideal if (i) a, b E J imply a V b E J; and (ii) a E J,b E L imply (a V b') A b E J. There is a one-toone correspondence between the set of all pideals and the set of all congruence relations on L. If 8 is the congruence corresponding to J, then a = b(8) iff aAb E J, and J = {a E L : a E O(8). Let ii denote the congruence class corresponding to a E L, and let L / J denote the set of all congruence classes. Then L / J (the quotient of L corresponding to J ) is an OML, and the mapping q5 : L -+ L / J , q5(a) = a (the quotient mapping) is a a-homomorphism. b) : a, b E L), Let Jc be the smallest pideal which contains the set {-(a, where m ( a , b) = (a V b) A (a' V b) A (a V b') A (a' V b') is the upper commutator of a, b. It is well-known that the quotient L/Jc is a Boolean algebra. Moreover, by [Marsden, 19701, (10)

Jc=

{C

E L : c 5 V ( m ( a j , b i ) ) , a i ,bi E L , i 5 n). isn

Another characterization of Boolean quotients can be obtained by means of prime ideals. Recall that a proper lattice ideal J is prime if a A b E J implies a E J or b E J. In a lattice logic, an ideal J is prime iff for all a E L, either a E J or a' E J . It is straightforward to see that a proper ideal J of L is prime iff J is a pideal and the quotient L / J is (0, 1). Let J be a prime ideal of an OML L. Define p : L t {0,1) by p(a) = 0 if a E J and p(a) = 0 otherwise. If a I b, and a $! J, then we must have b E J . this yields p(a V b) = p(a) +p(b). Moreover, p(a) = 0 = p(b) implies p ( a b) ~ = 0, hence p is a dispersion free finitely additive Jauch-Piron state (valuation) on L. Conversely, if s is any dispersion free valuation on L, then s-I (0) is a prime ideal of L. Recall that for any ideal J, the quotient L/ J is Boolean iff J is the intersection of all prime ideals that contain it [~almbach,19831. Consequently, Jcis the intersection of all prime ideals of L. Evidently, the notions of a pideal and prime ideal make sense also for a-ideals, i.e. the ideals J such that ai E J for all i E N implies ViENaa E J . The quotient of a lattice logic with respect to a a pideal is a lattice logic. obviously, the quotient L / J is a Boolean a-algebra iff the ideal J contains all upper commutators.

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Recall that a state s on L is a superposition [~aradarajan,19851 of a set S of states on L if for any a E L, t(a) = 0 Vt E S implies s(a) = 0. We then have the following characterization of the inequalities of type (3). THEOREM 141. Let (L, S) be a system, where L is a lattice logic. Denote by J, the ideal generated by all upper commutators in L. Let s E S. The following statements are equivalent: (i) the inequalities (3) hold; (ii) s(ZET(a, b))

=0

for all a, b E L;

(iii) s/Jc = 0, where s/Jc denotes the restriction of the state s to the ideal Jc; (iv) s is a superposition of a set of finitely additive dispersion free Jauch-Piron states on L; (v) there is a Boolean a-algebra B, a surjective a-homomorphism and a state 5 on B such that 3 o I#J = s;

I#J

: L + B,

(vi) the inequalities (4) hold.

Proof. (i) 3 (ii): Assume that the inequalities (3) hold, and put c = a'. By an easy computation we obtain s(b) = s(a A b) s(af A b). Consequently,

+

+

1 = s(b) s(bf) = s((a A b) V (a A b') V (a' A b) V (a' A b') = s(=(a, b)).

The latter equality yields s(ZET(a, b ) ) = 0. (ii) =+ (iii): Assume that (ii) hold. Then for all a, b we have s(a V b) = s((aV b) A a) +s((aV b) Aa') = s(a) +s((aV b) A a' A b') +s((aV b) A a' A b) = s(s) +s((aV b) Aa' A b) 5 s(a) +s(b). This proves that s is subadditive. Using (lo), we obtain for any c E J,, s(c) 5 s(Vi<, com(ai, bi)) 5 x i < , s(EEE(ai, bi)) = 0. (iii) & (iv): We have s-'(0) >J,, and J, is the intersection of all prime sil(0), where the intersections goes over all ideals of L. Consequently, s-I (0) finitely additive dispersion free Jauch-Piron states on L. Hence s is a superposition of all sa7s. (v): If s is a superposition of a set I of finitely additive dispersion free (iv) s;l(O), and every s i l ( 0 ) is a prime ideal Jauch-Piron states, then s-l(O) 2 s;1(0) is a pideal, and J 2 J,. Let 9 be the smallest a-p on L. Hence J = ideal containing J. Then J contains all commutators, and therefore B := LIJis a Boolean a-algebra. If I#J : L + B is the quotient mapping, then I#J is a surjective ahomomorphism. In addition, the mapping S : B + [O, 11defined by 3(I#J(a))= s(a) is a a additive state on B. (v) =+ (vi): Since 3 o I#J = s and the inequalities (4) hold for 3 on B , it easily follows that the inequalities (4) hold for s on L. (vi) & (i): Putting d = a in the inequalities (4), we obtain inequalities (3).

>U

UaEI

UaEI

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Pave1 PtAk, Sylvia Pulmannovi

The following theorem shows that there are close connections between Bell inequalities of type (3) and quasi-hidden variables. THEOREM 142. Let L be a lattice logic, and s be a (a-additive) state on L. The state s is generated by finitely additive dispersion free Jauch-Piron states on L if and only if one of the conditions of Theorem 141 is satisfied. Proof. If s is generated by the set R of all finitely additive dispersion free JauchPiron states on L, then there is a a-algebra C of subsets of R, which makes the mappings w ct w(a), a E L, measurable, and there is a probability measure p on C such that for every a E L,

It is clear that the equality w(a) = 0 for all w E R implies s(a) = 0. Hence s is a superposition of R. Conversely, suppose condition (v) of Theorem 141 holds. Let fi be the set of all dispersion free finitely additive states on B, and let 3 be the state on B corresponding to s. Since 3 is a probability measure on B, by Lemma 134, or by applying the Choquet-Bishopde Leeuw theorem [~helps,19661,

for a suitable probability measure p. This yields

4.1

=

Sy

R(a)dll(w),

where Cl is the set of the finitely additive dispersion free states w on L defined by ~ ( a= ) b(4(a)), a E L. Thus p is the measure on R induced by ji. rn

Remarks: 1.The Bell-type inequalities presented in the previous section were generalized in [Dvure~enskijand Langer, 19951 in the following way: By a Bell-type inequality of order n is understood any inequality of the type

which holds for a (finitely additive) state s on a logic L (and for all a1 . . . , an E L), where f (I) is a real coefficient, s(AiEI ai) is a correlation (or a joint probability) of the set of events {ai : i E I} in the state s, a l , . . . , a n E L, and N := {1,2,. . . ,n}. In other words, the function Sf(s) := CrcN f (I)s(AiEI ai) is a linear combination of s(AiEI ai), and s(AiEI ai) can be represented as a joint probability of ai(i E I ) in the state s. In [~eltramettiand Mqczyhski, 1991] and [Langer and Mqczyliski, 19951, it has been proved that the inequalities (11)are equivalent to the inequalities

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for every state s on a Boolean algebra. It was shown in [Dvure~enskijand Langer, 19951 that Bell-type inequalities of order 2 are equivalent with the inequalities (1) (resp. (2)), and the inequalities of order 3 are equivalent with the inequalities (3) (resp. (4)), while higher order Bell-type inequalities are equivalent with Bell-type inequalities of the type (3). 2. Let us finally step into the realm of universal algebra. Let us see how notions of quantum logical theory may be linked with universal algebra. Let OML denote the variety of orthomodular lattices. Recall that by a famous result of universal algebra [ ~ r a t z e r19861, , a subclass Q of the variety OML is a quasivariety if and only if Q is closed under the formation of subalgebras, products, ultraproducts, isomorphic algebras and it contains the trivial algebra. A quasivariety Q is a variety if it is closed under the formation of epimorphic images. It should be noted that several varieties and quasivarieties of OMLs related to quantum logical theory have been r Navara, 1987; Godowski, 1981; Carrega et al., 2000; considered in [ ~ i n d e and De Sirnone et al., 2001; Bruns et al., 1990; Matouavsek, 2004; Mayet, 1986; Mayet et al., 2000; Haviar and KonGpka, 20001. For X I , xz, . . . ,xn, y1, y1, . . . ,yn, let tc be a term of 2n variables defined as follows: n

tc(x11~21... > X n > Y l ~ Y l ~ - -:= - ~ V-(xilyi). yn) i=l Put N B Q := {L E OML : Vxl, X Z , . . . ,x,, yl, yl, . . . ,yn E L : ( t c ) L ( x ~ , ~-z-,- , x ~ , Y I , Y I , . . - ,#Y~~L) ) U I , where I is the class of all one-point orthomodular lattices, and the indices indicate to which L the element belongs. Recall that for an orthomodular lattice L, the following statements are equivalent: (i) L has a nonzero Boolean quotient, (ii) L has a proper prime ideal, (iii) there is a subadditive state on L. Since every prime ideal contains the ideal Jc defined by (lo), it can be easily seen that L E N B Q iff one of the equivalent conditions (i)-(iii) is satisfied. PROPOSITION 143. [D'Andrea and PulmannovL, 19941 (i) every free orthomodular lattice belongs to NBQ. (ii) The variety BA of Boolean algebras is a proper subclass of NBQ. (iii) The class NBQ is a proper quasivariety of OML.

208

Pave1 P t i k , Sylvia Pulmannovi

Proof. (i) Let F be a free orthomodular lattice over a nonempty set X . Let Jc be the commutator pideal in F . Then the quotient mapping h : F -+ F/Jc maps F into BA, and F / J , is a free Boolean algebra over h ( X ) (h(X) # 0). (ii) Obviously, BA NBQ. On the other hand, let F2 denote the free o r t h e modular lattice over the set {x, y} [ ~ e r a n19851. , Then F2 belongs to NBQ, but it does not belong to BA. (iii) From the definition of NBQ, we immediately see that NBQ is characterized by the set of quasiidentities V n E N, tc(xl, 3 2 , . . .,xn, yl, y1,. . . ,yn) FE 1 & 0 E 1. Therefore NBQ is a quasivariety. To show that N B Q is not a variety, consider the OML M 0 2 (the horizontal sum of two four-point Boolean algebras), which is rn a homomorphic image of Fz, but it is simple. Let us denote by the symbol Bell the class of orthomodular lattices L satisfying the following property: L E I or there is a state s on L such that (L, s ) satisfies one of the equivalent conditions of Theorem 141. The proof of the following statement is then easy. THEOREM 144. The class Bell coincides with the class NBQ, hence it is a proper quasivariety in O M L . BIBLIOGRAPHY [Aerts and Daubechies, 1978) D. Aerts and I. Daubechies. Physical justification for using tensor product t o describe two quantum systems as one joint system, Helv. Phys. acta 51, 661-675, 1978. [Alda, 19811 V. Alda. On 0-1 measure for projectors, 11, Aplikace Matematiky 26, 57-58, 1981 [Alfsen and Schulz, 19791 E. Alfhen and F. Schulz. On non-commutative spectral theory on Jordan algebras, Proc. London Math. Soc 37, 497-516, 1979. [Bell, 19601 J. S. Bell, On the problem of hidden variables in quantum mechanics Reviews of Modern Physics 38, 447-452, 1960. [Beltrametti and Cassinelli, 19811 E. Beltrametti and G. Cassinelli. The Logic of Quantum Mechanics, Addison-Wesley, Reading, Mass. 1981. [Beltrametti and Mgczyhski, 19911 E. Beltrametti and M. Mwzyriski. On a characterization of classical and nonclassical probabilities, J. Math. Phys., 32, 1280-1286, 1991. [Beran, 19791 L. Beran. On finitely generated orthomodular lattices, Math. Nachrichten 88, 129-139, 1979. [Beran, 19851 L. Beran. Orthomodular Lattices, Algebraic Approach, D.Reidel, Holland, 1985. [Binder and Navara, 19871 J. Binder and M. Navara. Quantum logics with lattice state spaces, Proc. Anate. Math. Soc. 100, 688-693, 1987. [Birkhoff and von Neumann, 19361 G. Birkhoff and J. von Neumann. The logic of quantum mechanics, Ann. Math. 37, 823-843, 1936. [Brabec, 19791 J. Brabec. Compatibility in orthomodular posets, Cas. p&t. mat. 104, 149-153, 1979. [Brabec and Pt&, 19821 J. Brabec and P. Ptik. On compatibility in quantum logics, Found. Phys 12, 207-212, 1982. [Bruns and Greechie, 19901 G. Bruns and R. Greechie. Blocks and commutators in orthomodular lattices, Algebra Universalw 27, 1-9, 1990. and Kalmbach, 19731 G. Bruns and G. Kalmbach. Some remarks on free orthomodular lattices, Proc. Univ. Houston, Lattice Theory Conf., J. Schmidt ed., Houston, 397-403, 1973. et al., 19901 Bruns, G., Greechie, R., Harding, J., Roddy, M., Completions of orthomodular lattices, Order 7 , 67-76, 1990.

un runs runs

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[Poguntke, 19831 W. Poguntke. Finitely generated ortholattices, the commutator and some apllications, Colloquia Math. Soc. J. Bolyai 99, Contributions to Lattice Theory, Szeged, 1980, pp. 651-666. North Holland, Amsterdam, 1983. [Primas, 19811 H. Primas. Chemistry, Quantum Mechanics and Reductionism, Ledure Notes in Chemistry 24, Springer-Verlag, Berlin, 1981. [Prugovecki, 19711 E. Prugovecki. Quantum Mechanics in Hilbert Space, Academic Press, London, New York, 1971. [Ptik, 19811 P. Ptik. Realcompactness and the notion of observable, Journal London Math. Soc. 23, 534-536, 1981. [Ptik, 19981 P. Ptik. Some nearly Boolean orthomodular posets, Proc. Amer. Math. Soc. 126, 2039-2046, 1998. [Ptik and Pulmannovi, 19911 P. P t i k and S. Pulmannovi. Orthomodular Structum as Quantum Logics, Kluwer Academic Publishers, Dordrecht, Boston, London - VEDA, SAV, Bratislava, 1991. [ P t i k and Pulmannovi, 19941 P. P t i k and S. Pulmannovi. A measuretheoretic characterization of Boolean algebras among orthomodular lattices Conzmentationes Mathematicae Universitatis Carolinae 35, 205-208, 1994. [ P t i k and Rogalewicz, 1983al P. P t i k and V. Rogalewicz. Measures on orthomodular partially ordered sets, J. Pure Appl. Algebra 28, 75-80, 1983. [Pt& and Rogalewicz, 1983bl P. P t i k and V. Rogalewicz. Regularly full logics and the uniqueness problem for observables, Ann. Inst. H. Poincare' 38, 69-74, 1983. [ P t i k and Weber, 20011 P. P t i k and H. Weber. Lattice properties of splitting subspaces in an inner product space, Proc. Amer. Math. Soc. 129, 2111-2117, 2001. [Pulmannovi, 19781 S. Pulmannovi. Joint distributions of observables on quantum logics, Int. J. Theor. Phys. 17, 665-675, 1978. [Pulmannovi, 19801 S. Pulmannovi. Relative compatibility and joint distributions of observables, Found. Phys. 10, 641-653, 1980. [ ~ u l m a n n o v i1981al , S. Pulmannovi. On the observables on quantum logics, Found. Phys. 11, 127-136, 1981. [Pulmannovi, 1981b] S. Pulmannovi. Compatibility and partial compatibility in quantum logics, Ann. Inst. H. Poincark, Ser. A 34, 391-403, 1981. [Pulmannovi, 19851 S. Pulmannovi. Commutators in orthomodular lattices, Demonstratio Math. 18, 187-208, 1985. [Pulmannovi, 19881 S. Pulmannovi. Uncertainty relations and state spaces, Ann. Inst. H. Poincare' 48, 325-332, 1988. [Pulmannovi, 1994a) S. Pulmannovi. Bell inequalities and quantum logics, in: The Interpretations of Quantum Theory: where do we stand? edited by L. Accardi, Encyclopedia Italians, Roma 1994. [Pulmannovi, 1994bl S. Pulmannovi. Quantum measurements and quantum logics, J. Math. Phys. 35, 1555-1572, 1994. [Pulmannavi, 20021 S. Pulmannovi. Hidden variables and Bell inequalities on quantum logics, Found. Phys. 32, 193-216, 2002. [Pulmannovi and Dvureknskij, 19801 S. Pulmannovi and A. DvureEenskij. Stochastic processes on quantum logics, Rep. Math. Phys. 18, 303-315, 1980. [Pulmannovi and Dvureknskij, 19851 S. Pulmannovi and A. DvureEenskij. Uncertainty principle and joint distributions of observables, Ann. Inst. H. Poincare', Ser. A 42, 253-265, 1985. [Pulmannovi and Majernik, 19921 S. Pulmannovd and V. Majernik. Bell inequalities on quantum logics, J.Math.Phys. 33, 2173-2178, 1992. [Pulmannovi and Stehhiovi, 19861 S. Pulmannovi and B. Stehlikovi. Strong law of large numbers and central limit theorem on a Hilbert space, Rep. Math. Phys. 23, 99-107, 1986. [Pykacz, 19991 J. Pykacz. On Bell type inequalities in quantum logics, in The concept of Probability edited by E.J. Bitsakis and C.A. Niwlaides, Kluwer, Dordrecht, 1989. [Pykacz and Santos, 19911 J. Pykacz and E. Santos. On Bell inequalities, J. Math. Phys., 32, ~. 1287, 1991. [Rarnsav. 19661 A. Rarnsav. A theorem on two commuting- observables, Indiana Univ. Math. J. 15, ii7-234; 1966. [ E d e i , 19981 M. E d e i . Quantum Logic in Algebraic Approach, Kluwer Academic Publishers, Dordrecht, 1998.

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[Ri&anovi, 19981 Z. FtieEanovi. Order-topological separable complete modular ortholattices admit order continuous faithful valuations, Proc. Amer. Math. Soc. 126, 231-237, 1998. [Fbgalewicz, 1984al V. Rogalewicz. Remarks about measures on orthomodular posets, Cas. p h t . mat. 109, 93-99, 1984. [Rogalewicz, 1984b] V. Fbgalewicz. On the uniqueness problem for quite full logics, Ann. Inst. H. Poincare'41, 445-451, 1984. [Riittimann, 19851 G. Riittimann. Expectation functionals of obsembles and counters, Rep. Matrh. Phys. 21, 213-222, 1985. [Riittimann, 19771 G. Riittimann. Jauch-Piron states, J. Math. Phys.18, 189-193, 1977. [Ruymgaart, 19831 F. Ruymgaart. A note on the concept of joint distribution of pairs of observables, Sankhya, Ser. A 45, 38-43, 1983. [Sarymsakov et al., 19831 T. Sarymsakov, S. Ayupov, D. Khadziev, and V. Chilin. Ordered Algebras (in Russian) Filial AN, Tashkent, 1983. [Schultz, 19741 F. W. Schultz. A characterization of state space of orthomodular lattices, J. Comb. Theory A 17, 317-325, 1974. [ ~ e g a l19471 , I. Segal. Postulates for general quantum mechanics, Ann. Math. 48, 930-948, 1947. [Sherstnev, 19681 A. N. Sherstnev. On boolean logics, Uchen, Zapiski KGU 128, 48-62, 1968. (Russian) [Svozil, 19981 K. Svozil. Quantum Logic, Springer, Singapore 1998. [Tkadlec, 19931 J. Tkadlec. Partially additive measures and set representations of orthoposets, J. Pure Appl. Algebra 86, 79-94, 1993. [Urbanik, 19611 K. Urbanik. Joint probability distributions of obsembles in quantum mechanics, Studia Math. 21, 117-133, 1961. [Urbanik, 19851 K. Urbanik. Joint distributions and compatibility of obsembles, Demonstratio Math. 18, 31-41, 1985. [~aradarajan,19851 V. S. Varadarajan. Gwnaetry of Quantum Theory, Springer-verlag, New York, Berlin, Heidelberg, Tokyo, 1985. [Weber, 19941 H. Weber. There are orthomodular lattices without non-trivial group valued states; a computer-based construction, J. Math. Anal. Appl. 183, 89-94, 1994. [Zisis, 2000] M. Zisis. Approximate hidden variables, Found. Phys. 30, 971-2000, 2000.

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HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

215

QUANTUM LOGIC AND PARTIALLY ORDERED ABELIAN GROUPS David J. Foulis and Richard J. Greechie 1 INTRODUCTION

Our purpose in this expository article is to give an account of the connection between quantum logics1 and partially ordered abelian groups. In what follows, we use the word "logic" in the sense of a partially ordered algebraic structure L that could be interpreted as a semantic model for a formal symbolic logic L. This usage is customary in the literature of algebraic logic [Halmos, 19621, and it is consistent with the nomenclature of Birkhoff and von Neumann in their seminal article on the logic of quantum mechanics [~irkhoffand von Neumann, 19361. Our emphasis is on the mathematical structure of L rather than on the interpretation of its elements as propositions pertinent to a physical system. An account of L and C from the point of view of logic as an instrument of reasoning, as well as more comprehensive treatment of the general notion of a quantum logic, can be a et al., 20041. found in [ ~ a l l Chiara We begin, in Sections 2-5 below, with a brief sketch touching on some of the basic notions of orthodox quantum mechanics (QM). By orthodox QM, we mean nonrelativistic Hilbert-spaced-based QM without superselection rules and for which the observables are represented by projection-valued (PV) measures. Readers already knowledgeable about quantum physics can skim these sections rapidly to familiarize themselves with our notation. The relevant quantum logic (QL) for orthodox QM is the complete atomic orthomodular lattice P(4j) of projection o p erators on a Hilbert space Jj. The QL P(4) is a subset of the partially ordered abelian group G(4) of Hermitian (i.e., bounded self-adjoint) operators on 4. A comprehensive and authoritative account of the standard quantum logic P(4j) in the context of orthodox QM can be found in [~eltramettiand Cassinelli, 19811. In Section 6, we turn our attention to the much more general and flexible positiveoperator-valued (POV) measures for which the corresponding QL is the secalled effect algebra IE(4) of all positive semidefinite Hermitian operators on 4j that are dominated by the identity operator. Evidently P(4) C IE(4) C_ G(5). As "quantum-logical propositions", effect operators in IE(4) can manifest "fuzziness" or "un~harpness'~, while the projection operators in P(4) can be regarded as l ~ follow e common usage in which the term "quantum logic" refers not only to a logic associated with a quantum-mechanicalsystem, but indeed to any physical system whatsoever.

David J. Foulis and Richard J. Greechie

216

"sharp," even though their "truth values" might be subject to statistical fluctuations. For an exposition of POV measures and the contemporary quantum theory of measurements, see [ ~ u s c het al., 19911. The mathematical structures of P(Ej), IE(fj), and G(4) are so rich and the quantum-mechanical interpretations of these structures are so intriguing that one cannot resist the temptation to formulate and study more general triples P C E c G as abstractions or analogues of P(4) C IE(4j) c G(4). Thus, in Sections 7 through 16 we conduct an abstraction process, motivated and guided by suitable examples, that will ultimately result in the notion of a so-called "CB-triple" P c E c G consisting of a regular orthomodular poset P, an effect algebra E, and a CB-group G, i.e., a partially ordered abelian group enriched by a so-called "compression base." The notion of a CB-group is both very general and mathematically attractive, and may be considered as an appropriate basis for the general study of quantum logics. In Sections 17 and 18, we introduce and study a class of CB-groups, called archimedean RC-groups, for which a spectral theory has been developed. Because of the expository nature of this article, we give prooh only when we regard them as being illuminating or when they could be difficult to locate in the existing literature. Background material on Hilbert spaces can be found in [Halmos, 19981, a source for basic facts in regard to a-fields, measurable functions, and so on, is [Halmos, 19501, and [Halmos, 19631 is an authoritative exposition of the theory of Boolean algebras-all three of these classic references were authored by P.R. Halmos. The monographs of R.V. Kadison and J.R. Ringrose ri ad is on and Ringrose, 19831 provide comprehensive expositions of both the basic and advanced theory of operator algebras, and the treatise of G. Emch [ ~ m c h 19721 , is an excellent reference for applications of operator algebras to physics. In our treatment of partially ordered abelian groups, we follow the monograph [~oodearl, 19851 of K.R. Goodearl. For general lattice theory, the classic [~irkhoff,19791 of G. Birkhoff is the standard reference.

2 ORTHODOX QUANTUM MECHANICS Orthodox quantum mechanics (QM) is founded on the assumption that there is a correspondence S -+ fi assigning a complex separable Hilbert space fi to a QMsystem S . Let (. , be the inner product2 on f j and denote the additive abelian group of all bounded Hermitian operators on 4j by G(fi). The tenets of orthodox QM speclfy that bounded (real) observables for S are represented by operators A E G(fj), (pure) states for S are represented by unit vectors E 4, and the expectation value of an observable A when the system S is in the state $J is given by (A$J,$J). DEFINITION 1. If $J is a state vector in 4 , i.e., ll$Jll = 1, and W denotes the system of real numbers, we define the expectation mapping w $ : G(4j) -+ IR by a)

$J

2 ~ h y s i ~ i susually ts use the Dirac bra-ket notation (. I .) for the inner product.

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Quantum Logic and Partially Ordered Abelian Groups

w+(A) := (A$, $) for all A E G ( J ~ ) . ~ The expectation value w+(A) is usually interpreted as the limit as n + oo of the arithmetic mean of n independent measurements of the bounded observable A on the QM-system S , or replicas thereof, in the fixed state $. The expectation mappings w+ are used to organize the Hermitian group G(4) into a partially ordered abelian group [Goodearl, 19851 as follows: if A, B E 6 ( 4 ) , then by definition A 5 B iff4 w$(A) 5 w+(B) for every state vector $JE 4. In fact, under the partial order 5, G(4) is an order-unit-nomed real Banach space with the identity operator 1 E G(4) as the order unit [Alfsen, 1971, page 691. F'urthermore, the order-unit norm coincides with the uniform operator norm 11 . 11 on G(fi), and therefore, for A E G(a), (1) IlAll = inf{m/n

I 0 5 m E Z, 0 < n E Z,

and - m l 5 nA 5 m l ) ,

where, as usual, Z is the ring of integers. If 0 is the zero operator in G(*), then (by a slight abuse of language), observables A E G(4) with 0 5 A are called positive. We note that the norm of a positive bounded observable is the maximum expectation value of that observable for all state vectors, i.e.,

The positive operators A in G(4) are exactly the operators of the form A = B2 for B E G(4). Indeed, if B E G(4), then for every state vector $,

whence 0 5 B2. Conversely, if 0 5 A E G(4), then there is a unique operator and A = E G(4) such that 0 5 [ ~ i e s and z Nagy, 1955, page 2651.

a

If A E G(4), then 0 5 A', and one defines IAl := A- := i(IAI - A). Then

a, A+

:= l(IAl 2 +A), and

Thus, every bounded observable is the difference of two bounded positive observables. Suppose that A1 5 A2 5 . . . is an ascending sequence of bounded observables. Then the sequence is bounded above in norm (i.e., there is a real number P with llAnll 5 P for all n E M := {1,2,3, ...)) iff it is bounded above in G(4) (i.e., there is a bounded observable B E G(4) such that A, 5 B for all n E M). Furthermore, if the sequence is bounded above in either sense, then by Vigier's theorem [Riesz and Nagy, 1955, page 2631, there is a uniquely determined bounded observable A E G(4) such that, for every $ E 4, IIA,$ - A611 + 0 as n + oo, i.e. the sequence A1, A2,... converges to A in the strong operator topology. But, if A, -t A in the strong operator topology, then A, + A in the weak operator 3 ~ h notation e := means "equal by def