The Quantum Theory of Measurement

Lecture Notes in Physics New Series m: Monographs Editorial Board

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Paul Busch

Pekka J. Lahti

Peter Mittelstaedt

The Quantum Theory of Measurement Second Revised Edition

Springer

Authors Paul Busch Department of Applied Mathematics The University of Hull Kingston upon Hull HU6 7RX, United Kingdom Pekka J. Lahti Department of Physics University of Turku SF-2o5oo Turku, Finland Peter Mittelstaedt Institute for Theoretical Physics University of Cologne D-5o937 Cologne, Germany Cataloging-in-Publication Data applied for. Die D e u t s c h e B i b l i o t h e k - C I P - E i n h e i t s a u f n a h m e Busch, Pa.l: The q u a n t u m theory of m e a s u r e m e n t / Paul Busch ; Pekka J. Lahti ; Peter M i t t e l s t a e d t . - 2. ed. - Berlin ; H e i d e l b e r g ; New Y o r k ; Barcelona ; Budapest ; H o n g Kong ; L o n d o n ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1996 (Lecture notes in physics : N.s. M, Monographs ; 2) ISBN 3-540-61355-2 NE: Lahti, Pekka J.:; Mittelstaedt, Peter:; Lecture notes in p h y s i c s / M ISBN 3-54o-61355-2 2nd Edition Springer-Verlag Berlin Heidelberg New York ISBN 3-54o-54334-1 1st Edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991,1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by authors Cover Design: Design ¢~ Production, Heidelberg SPIN: 1o54o939 55/3142-54321o - Printed on acid-free paper

P r e f a c e to t h e S e c o n d E d i t i o n Since the first edition of The Quantum Theory of Measurement appeared nearly five years ago, research into numerous areas of the foundations of quantum mechanics has been carried on at a breathtaking pace. The strangeness of the quantum world has continued to be a source of creativity, and it is becoming clear that new applications are to be expected in a not too distant future in fields such as 'quantum' communication, computation, or cryptography; applications that could not have been anticipated on the basis of classical physics. These developments, together with the increasing recognition of the ubiquity and, in many respects, fundamental importance of imperfect, or nonideal measurements, have made the need for a thorough understanding of the problems of measurement and an elaboration of measurement theory as an applied discipline ever more pressing. In the meantime this book has been accompanied by a treatise, coauthored by two of us, that takes up the latter demand: Operational Quantum Physics develops the theory and various applications of unsharp observables and their measurements. In view of this new text it was felt desirable that The Quantum Theory of Measurement should be made available again, but not without a revision of its scope in the context of the new developments. As a result, the central chapters, II, III and IV, have been substantially rewritten. The definition of the concept of objectivity has been reformulated so as to make clearly visible the distinction of its formal and interpretational components (Chap. II). This separation is then extended to all theorems which are now stated as formal results in the first instance. In this way, we hope, the whole theory is made accessible also to those readers who do not fully (or at all) share our philosophical inclinations. The most important and significant changes concern the elucidation of the various necessary objectification requirements - among them the pointer valuedefiniteness and pointer mixture conditions. A thorough understanding of their implications for the structures of measurement schemes has been achieved; and this has led to a completely comprehensive formulation of an insolubility theorem for the objectification problem, which includes unsharp object observables and unsharp pointers. As one consequence, the idea of unsharp objectification has now assumed a fairly sharp contour (Chap. III). The review of the various approaches and interpretational attempts at dealing with the objectification problem (Chap. IV) has been rearranged according to the simplified logical classification scheme offered by the insolubility theorem. As in the first edition, we have refrained from entering into detailed comparisons and evaluations of the various 'schools'. In fact it is becoming more and more obvious

vi

Preface to the Second Edition

in each of them that a lot of 'internal' questions are still open, so that it is too early for conclusive judgements. The kind of formal investigations needed for answering these questions is sketched out in some cases; apart from that we have restricted ourselves to short indications of the current state of the art, referring our readers to recent expert accounts of the various approaches. Many colleagues have encouraged us with their comments to embark on the work for this new edition. In particular we are indebted to Heinz-Jiirgen Schmidt and Reinhard Werner for posing critical questions about the previous edition. Sincere thanks go to our friends Gianni Cassinelli and Abner Shim0ny: it was in our collaborations with them that we envisaged the full scope of generality that has now been achieved.

Cologne, Huff, Turku April 1996

Paul Busch Pekka Lahti Peter Mittelstaedt

P r e f a c e to t h e First E d i t i o n The present treatise is concerned with the quantum mechanical theory of measurement. Since the development of quantum theory in the 1920s the measuring process has been considered a very important problem. A large number of articles have accordingly been devoted to this subject. In this way the quantum mechanical measurement problem has been a source of inspiration for physical, mathematical and philosophical investigations into the foundations of quantum theory, which has had an impact on a great variety of research fields, ranging from the physics of macroscopic systems to probability theory and algebra. Moreover, while many steps forward have been made and much insight has been gained on the road towards a solution of the measurement problem, left open nonetheless are important questions, which have induced several interesting developments. Hence even today it cannot be said that the measurement process has lost its topicality and excitement. Moreover, research in this field has made contact with current advances in high technology, which provide new possibilities for performing former Gedanken experiments. For these reasons we felt that the time had come to develop a systematic exposition of the quantum theory of measurement which might serve as a basis and reference for future research into the foundations of quantum mechanics. But there are other sources of motivation which led us to make this effort. First of all, in spite of the many contributions to measurement theory there is still no generally accepted approach. Much worse, a considerable fraction of even recent publications on the subject is based on an erroneous or insufficient understanding of the measurement problem. It therefore seems desirable to formulate a precise definition of the subject of quantum measurement theory. This should give rise to a systematic account of the options for solving the problem of measurement and allow for an evaluation of the various approaches. In this sense the present work may be taken as a first step towards a textbook on the quantum theory of measurement, the lack of which has been pointed out by Wheeler and Zurek (1983). In view of the difficulties encountered in the quantum theory of measurement many distinguished authors have considered the possibility that quantum mechanics is not a universally valid theory. In particular, the question has been raised whether macroscopic systems, such as measuring devices, are beyond the scope of this theory. Adopting this point of view would allow one to reformulate, and possibly solve, the open problems of quantum mechanics within the framework of more general theories. Such far-reaching conclusions should, however, be substantiated by means of a close chain of arguments. We shall try to spell out some of the arguments that endeavour to prove the limitations of quantum mechanics in the

viii

Preface to the First Edition

context of measurement theory. The resulting no-go theorems naturally entail a specification of the various modifications of quantum mechanics which might lead to a satisfactory resolution. At the same time they contribute to an understanding of those interpretations maintaining the universal validity of quantum mechanics. Next, we are not aware of the existence of a review of the measurement problem which takes into account the developments in the foundations of quantum mechanics over the past two decades. The operational language based on the notions of effects and operations, and the ensuing general concepts of observables and state transformers have proved extremely useful not only in foundational issues (as documented in the monographs of Ludwig (1983a,1987), Kraus (1983), or Prugove~ki (1986)), but also in applications of quantum physics in areas like quantum optics or signal processing (as represented by the books of Davies (1976), Helstrom (1973), or Holevo (1982)). These concepts must be regarded as the contemporary standards for the rigorous formulation of physical problems. They will be employed here for the precise definitions of operational and probabilistic concepts needed for uniquely fixing the notion of measurement in quantum mechanics and developing a formulation of the quantum theory of measurement general enough to cover the present scope of applications. The introduction of general observables has shed new light on the problem of macroscopic quantum systems and the question of the (quasi-) classical limit of quantum mechanics, thus providing a redefinition of the notion of macroscopic observables. In this way a new approach to the measurement p r o b l e m - unsharp objectification- has emerged in the last few years and will be sketched out in the course of our review. The failure of the quantum theory of measurement in its original form has led several authors to propose a modified conception of dynamics, incorporating stochastic elements into the SchrSdinger equation or taking into account the influence of the environment of a quantum system. In both cases the measuring process can no longer be described in terms of a unitary dynamical group. Hence the traditional theory of measurement should also be extended to cover nonunitary state transformations. The preceding remarks suggest that the incorporation of general observables and nonunitary dynamics into quantum measurement theory necessitates, and makes possible, an entirely new approach to this theory. We shall try to bring into a systematic order the new results obtained in the course of many detailed investigations, recovering the known results as special cases. In this way we shall hope to have established a systematic description of the quantum mechanical measurement process together with a concise formulation of the measurement problem. In our view the generalised mathematical and conceptual framework of quantum mechanics referred to above allows for the first time for a proper formulation of many aspects of the measurement problem within this theory, thereby opening up new options for its

Preface to the First Edition

ix

solution. Thus it has become evident that these questions, which were sometimes considered to belong to the realm of philosophical contemplation, have assumed the status of well-defined and tractable physical problems.

Cologne, June 1991

Paul Busch Pekka Lahti Peter Mittelstaedt

Contents I. 1. 2.

Introduction

..........................................................

The P r o b l e m of M e a s u r e m e n t in Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . .

1 1

Historical Account: I n t e r p r e t a t i o n s a n d Reconstructions of Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3.

Decision Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

II.

Basic Features of Quantum Mechanics .............................

7

1.

Hilbert Space Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.1. Basic F r a m e w o r k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

7

1.2. Tensor P r o d u c t and C o m p o u n d Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.3. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Probability S t r u c t u r e of Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.1. States as Generalised Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2. Irreducibility of Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3. Superselection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4. Nonobjectivity of Observables

18

.....................................

2.5. Nonunique Decomposability of Mixed States . . . . . . . . . . . . . . . . . . . . . . .

21

2.6. E n t a n g l e d Systems and Ignorance I n t e r p r e t a t i o n for Mixed States ..

22

III. T h e Q u a n t u m T h e o r y o f M e a s u r e m e n t

...........................

25

S u r v e y - T h e Notion of M e a s u r e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.

...................................

27

.

General Description of M e a s u r e m e n t

1.1. The P r o b l e m of Isolated Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.2. M e a s u r e m e n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Premeasurements ......................................................

31

2.1. P r e m e a s u r e m e n t s and State Transformers . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.2. U n i t a r y P r e m e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.3. Calibration Condition a n d Probability Reproducibility . . . . . . . . . . . . .

34

2.4. Reading of Pointer Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.5. Discrete Sharp Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.6. The S t a n d a r d Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

xii 3.

4.

Contents Measurement and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Two P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43

3.2. M e a s u r e m e n t Statistics I n t e r p r e t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.3. Statistical Ensemble I n t e r p r e t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Probabilistic C h a r a c t e r i s a t i o n s of M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . .

49

4.1. Statistical D e p e n d e n c e a n d Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2. S t r o n g Correlations Between Observables . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.3. S t r o n g Correlations Between Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.4. S t r o n g Correlations Between Final C o m p o n e n t States . . . . . . . . . . . . . . 4.5. First K i n d M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58

4.6. R e p e a t a b l e M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Ideal M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60

4.8. R ~ s u m ~ - A Classification of P r e m e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . .

63

I n f o r m a t i o n Theoretical Aspects of M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . . . . 5.1. T h e C o n c e p t of E n t r o p y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 65

5.2. T h e C o n c e p t of I n f o r m a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.3. I n f o r m a t i o n a n d C o m m u t a t i v i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. T h e Objectification P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73

6.2. Insolubility of t h e Objectification P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Classical P o i n t e r Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 77

6.4. R e g i s t r a t i o n a n d R e a d i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

7.

Measurement Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. T h e P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. A n Inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82

8.

L i m i t a t i o n s on Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

8.1. R e p e a t a b l e M e a s u r e m e n t s a n d Continuous Observables . . . . . . . . . . . . . 8.2. C o m p l e m e n t a r y Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 86

8.3. M e a s u r a b i l i t y a n d Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.

6.

9.

P r e p a r a t i o n a n d D e t e r m i n a t i o n of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

9.1. S t a t e P r e p a r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

9.2. S t a t e D e t e r m i n a t i o n Versus S t a t e P r e p a r a t i o n . . . . . . . . . . . . . . . . . . . . . .

90

Contents IV. Objectification and Interpretations 1. 2. 3. 4.

5.

V.

of Quantum

Mechanics

xiii .....

91

Routes Towards Solving the Objectification Problem . . . . . . . . . . . . . . . . . . . . Historical P r e l u d e - Copenhagen Interpretations . . . . . . . . . . . . . . . . . . . . . . . . Ensemble and Hidden Variable Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . Modifying Q u a n t u m Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 95 102 105

4.1. Operational Approaches and the Quantum-Classical Dichotomy . . . . 4.2. Classical Properties of the A p p a r a t u s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Moclified Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Changing the Concept of Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Many-Worlds Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Modal Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Decoherence via Environment-Induced Superselection . . . . . . . . . . . . . . 5.4. Algebraic Theory of Superselection Sectors . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Unsharp Objectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 107 109 111 113 116 123 125 127

Conclusion

...........................................................

131

.............................................................

139

Bibliography Author Notation

Index and References

..........................................

..................................................................

Subject Index

...........................................................

145 177 179

I. I n t r o d u c t i o n 1.1. T h e P r o b l e m of M e a s u r e m e n t in Q u a n t u m M e c h a n i c s An understanding of quantum mechanics in the sense of a generally accepted interpretation has not yet been attained. The ultimate reason for this difficulty must be seen in the irreducibly probabilistic structure of quantum mechanics which is rooted in the nonclassical character of its language. An operational analysis of the peculiarities of quantum mechanics shows that the interpretational problems are closely related to the difficulties of the quantum theory of measurement. It is the purpose of this review to spell out in detail these connections. The task of the quantum theory of measurement is to investigate the semantical consistency of quantum mechanics. Phrased in general terms, quantum mechanics, as a physical theory, and the quantum theory of measurement as a part of it, are based on a 'splitting' of the empirical world into four 'parts" (1) object systems S (to be observed), (2) apparatus ,4 (preparation and registration devices), (3) environments g (the 'rest' of the physical world which one intends to ignore), and (4) observers O. Depending on the type of interpretation in question, observers or environments may or may not be neglected in the description of the measuring process within the quantum theory of measurement. Providing that quantum mechanics is considered as a t~eory of individual objects, the most important questions to be answered by measurement theory are: (1) how it is possible for objects to be prepared, that is, isolated from their environments and brought into well defined states; (2) how the measurement of a given observable is achieved; and (3) how objects can be reestablished after measurements, that is, be separated from the apparatus. The underlying common issue is the objectit~cation problem; that is, the question of how definite measurement outcomes are obtained. We shall try to elucidate the status and the precise form of these questions. In Chap. II basic features of quantum mechanics are summarised which may be regarded as the root of the objectification problem. Chapter III is devoted to a systematic exposition of the quantum theory of measurement. Various solutions to the measurement problem proposed within a number of current interpretations of quantum mechanics will be reviewed in Chap. IV. Chapter V closes the treatise with our general conclusions. In the present chapter a decision tree will be formulated as a guide to a systematic evaluation of the various interpretations of quantum mechanics. A brief historical overview of these interpretations may serve as a first orientation, showing, in passing, the origins of the present approach.

2

I. Introduction

1.2. Historical Account: Interpretations and Reconstructions of Quantum Mechanics One may distinguish four or five overlapping phases in the development of research in the foundations of quantum mechanics. Early discussions among the pioneers (1927-1935) led to the well-known versions of the so-called Copenhagen interpretation. In the discussions between Bohr and Heisenberg [Bohr 28, Heis 27] and Bohr and Einstein [Bohr 49] the quantum theory of measurement was touched upon only in an informal way. It is only in the monographs of von Neumann [yon Neumann 1932] and Pauli [Pauli 1933] that one finds the first rigorous and explicit formulations of measurement problems in the manner in which they axe the subject of the present treatise. Reconsiderations of interpretational questions extending essentially from the 1950s to the 1970s were mainly motivated by attempts to explore the possibilities of establishing realistic interpretations of quantum mechanics considered as a universally valid theory. Much of this was anticipated in and taken up from the early works of von Neumann Iron Neumann 1932], Einstein, Podolsky, and Rosen [Eins 35], Schrhdinger [Schr5 35,36] and others. The London-Bauer [Lon 39] theory of measurement and its critique through the story of Wigner's [Wig 61] 'friend' are concerned with the possibility already pointed out by von Neumann and Pauli that the observer's consciousness enters in an essential way into the description of quantum measurements. Other denials of the possibility of realistic interpretations are formulated in the position that only a statistical interpretation of quantum mechanical probabilities is tenable [Bal 70, Eins 36, Maxg 36]. In this view quantum mechanics refers only to ensembles of measurement outcomes or of physical systems but does not lead to statements about properties of individual systems. On the other hand, hidden variable approaches aimed at restoring classical realism in quantum mechanics. These, again, are forced to render quantum mechanics as a statistical theory. Many of such attempts were refuted by a number of no-gotheorems like those by Gleason [Glea 57], Kochen and Speaker [Koch 67], or Bell [Bell 66] (see, e.g., [Giuntini 1991, Peres 1993]), leaving open up to now only nonlocal, contextual theories such as those of de Broglie [de Broglie 1953], Bohm [Bohm 52], or Bohm and Vigier [Bohm 54]. The 'many-worlds interpretation' developed by Everett [Eve 57], DeWitt and Graham [DeWitt and Graham 1973] offers one way of taking seriously quantum mechanics as a universal theory. We shall be very brief with our subsequent discussions of the early developments (Chap. IV) and refer the interested reader to the monographs of [Jammer 1966, 1974], and to the collection of papers edited by Wheeler and Zurek [Wheeler and Zurek 1983]. Reconstructions and generalisations of quantum mechanics (pursued systematically since the 1960s) have aimed at an understanding of the role of Hilbert space in quantum mechanics. One may distinguish three groups of approaches. (1) The quantum logic approach aims at an operational justification of the - gen-

1.2 Historical Account

3

erally non-Boolean - structure of the lattice of the propositions of the language of a physical theory [Beltrametti and Cassinelli 1981, Jauch 1968, Mackey 1963, Mittelstaedt 1978, Piron 1976, Varadarajan 1985]. Measurement theory enters this approach only in an informal way in terms of postulates characterising propositions as properties of physical systems. In order to establish the formal language of quantum physics, one assumes that elementary propositions are value definite, that is, that there exists an experimental procedure - a measuring process - which shows whether the proposition is true or false. An essential presupposition is that this measuring process will lead to a complete objectification. The importance of the quantum logic approach for the present work lies in the fact that it supports the attempts at formulating a consistent realistic interpretation of quantum mechanics. (2) The operational approach takes as its starting point the convex structure of the set of (statistical) states representing the preparations of physical experiments. Measurement theoretical aspects are investigated primarily on the object level in terms of the notion of operation representing state changes induced by measurements [Davies 1976, Fou 78, Holevo 1982, Kraus 1983, Ludwig 1983]. The quantum theory of measurement presented in Chap. III is formulated in the spirit of the operational approach. (3) The algebraic approach emphasizes the algebraic structures of the set of observables and it exploits the formal analogy between classical mechanics and quantum mechanics, aiming, in particular, at convenient 'quantisation' procedures. One of its advantages is the great formal flexibility which allows for an elegant incorporation of superselection rules and other structural changes generalising quantum mechanics. Hence this approach offers a mathematical language for a discussion of the measurement problem in more general terms. As a survey and rather exhaustive literature guide the reader may wish to consult the monograph [Primas 1983]. Each of the so-called axiomatic approaches has deepened our understanding of the mathematical and conceptual structures of quantum mechanics. However, none of them led to a thorough justification of the ordinary Hilbert space quantum mechanics. Due to this fact, but also due to the success of Hilbert space quantum mechanics, many recent investigations in quantum mechanics have been done directly within the Hilbert space formulation of quantum mechanics. The present work is also written entirely within this framework. The recent revival of interest in foundational issues was encouraged during the 1980s due partly to advances in the formal and conceptual structures of quantum mechanics and also to new experimental possibilities and technological demands. This went hand in hand with new ideas on interpretations and on proposals for solving the objectification problem (see Chap. IV). Fundamental experiments have been performed and these have contributed to bringing the quantum theory of measurement closer to empirical testability. Quantum optical and neutron interferometry experiments on the wave-particle dualism, Einstein-Podolsky-Rosen and

4

I. Introduction

delayed choice experiments, macroscopic tunnelling, and mesoscopic quantum effects are some examples. Instead of trying to survey these important developments here, we shall simply refer to the many recent conferences devoted to them such as those in Baltimore 1994, Castiglioncello 1992, Cologne 1984, 1993, Erice 1989, 1994,

Gdafisk 1987, 1989, 1990, Helsinki 1992,1994, Joensuu 1985, 1987, 1990, Munich 1981, New York 1986, 1992, Nottingham 1994, Paris 1990, Prague 1994, Rome 1989, Tokyo 1983, 1986, 1989, 1992 or Vienna 1987. Various aspects of these developments are reflected in the monograph Operational Quantum Physics coauthored by two of the present writers [Busch, Grabowski, Lahti 1995] as well as in the recent books by Peres, Quantum Theory: Concepts and Methods, 1993, and Schroeck, Quantum Mechanics on Phase Space, 1996.

1.3. D e c i s i o n Tree

In the minimal interpretation, quantum mechanics is regarded as a probabilistic physical theory, consisting of a language (propositions about outcomes of measurements), a probability structure (a convex set of probability measures representing the possible distributions of measurement outcomes) and probabilistic laws. In addition, probabilities are interpreted as limits of relative frequencies of measurement outcomes, that is, in the sense of an epistemic statistical interpretation. It is well-known that the minimal interpretation has not been the only one proposed. Other interpretations were formulated earlier. We shall try to give a fairly systematic list of them along with a sequence of decisions to be made concerning the goals quantum mechanics could be desired to serve. The first decisive question to be answered is the one about the referent of quantum mechanics: measurement outcomes (the epistemic option) or object systems (the ontic, or realistic option)? The ontic answer maintains that quantum mechanics deals with individual objects and their properties. It is only here that the measurement problem arises. Following this branch, the second decisive question is the completeness of quantum mechanics, that is, the question of whether or not all elements of physical reality can be described by quantum mechanics. The first option leads to, and is motivated by, the consideration of hidden variable theories underlying the allegedly incomplete theory of quantum mechanics, which then is interpreted as a mere statistical theory about ensembles of objects. In the other option, that of maintaining the completeness of quantum mechanics and following a realistic interpretation, one is facing the phenomenon of nonobjectivity. Accordingly, quantum mechanical probabilities are objective in the sense of propensities, or potentialities, expressing tendencies in the behaviour of individual objects. Again, in the incompleteness interpretations there is no measurement problem: objectification is not at issue at all, since all properties are considered as real throughout but not as subject to quantum mechanics. However, as mentioned ear-

1.3 Decision Tree

5

lier, there is not much room for hidden variable theories, and the only ones that survived the known no-go statements do not really go beyond the formalism of quantum mechanics. Turning to the realistic interpretations maintaining the completeness of quantum mechanics, it must be said that these have not up to now produced generally accepted solutions to the measurement problems. Thus one is forced into a third decision about the range of validity of quantum mechanics: is quantum mechanics universally valid or only of limited validity? Some authors have concluded that quantum mechanics, originally devised as a theory for microsystems, cannot be extrapolated in a straightforward way to larger systems, such as measuring devices. It is argued either that more general theories need to be developed which allow for certain macroscopic quantities to be classical observables, or that the time evolution is not correctly described by the SchrSdinger equation. The reductionistic conviction is given up in these views. In some sense, the more challenging route is that which maintains the universality of quantum mechanics. It forces one to carefully reconsider the concepts of objectivity and objectification, a decision that is made in the many worlds interpretation, the modal interpretations, the decoherence theories and the unsharp objectification proposal. In our opinion no conclusive decision between these two options can be made at present. We shall therefore be content to provide a short systematic review of the various approaches to the measurement problem in Chap. IV, guided by the above discussion as summarised in the decision tree of Table 1.

6

I. Introduction

Table 1: DecisionTree: Interpretations of quantum mechanics and approaches to the objectification problem.

Quantum Mechanics Minimal Interpretation relative frequency of measurement outcomes (

REFERENT? ~

[ Statistical Interpretation only measurement outcomes

Realistic Interpretation properties of individual systems

- objectification problem excluded

C OBJECTIVITY/ OMPLETENESS ? ~

Incompleteness all properties objective hidden variables ensemble of objects

Completeness nonobjectivity

- objectification problem excluded

~BJECTIFICATION/~ UNIVERSALITY ? J

Limited Validity objectification

searched by modifying quantum mechanics:

superselection rules modified dynamics

Universal Validity cha//enging the concept of objectification: many-worlds interpretation modal interpretations decoherence approach unsharp objectification

II. Basic Features of Q u a n t u m M e c h a n i c s II.1. Hilbert Space Q u a n t u m Mechanics This section summarises the basic elements and results of quantum mechanics which are relevant to the quantum theory of measurement. It also serves to define our notations and terminology. The standard results quoted here can be found, for example, in the following monographs [Beltrametti and Cassinelli 1981, Davies 1976, Jauch 1968, Kraus 1983, Ludwig 1983, yon Neumann 1932]. We are also using freely the well-known results of the Hilbert space operator theory, as presented, for instance, in the book [Reed and Simon 1980].

II.l.1 Basic Framework The basic concepts of quantum mechanics are the dual notions of states and observables, both being defined in their most general forms in terms of operators acting on a Hilbert space. a) M a t h e m a t i c a l structures. Let 7-/be a complex separable Hilbert space with the inner product ('l'). An element ~ e 7-/is a unit vector if (~]~) - I I ~ II2 -- 1, and the vectors ~, ¢ E 7-/are orthogonal if (~I¢) = 0. A set ( ~ i ) C 7-/is orthonormal if the vectors ~i are mutually orthogonal unit vectors. If ( ~ i ) C T/is a basis, that is, a complete orthonormal set, then any ¢ E 7-/can be expressed as the Fourier series ¢ = ~'~(~i]¢)~i with ]1 ¢ II2 -- ~ ](~ill~))l 2" Any unit vector ~ E 7-/ determines a one-dimensional projection operator P[~] through the formula P[~]¢ - (~]¢)~ for ¢ e 7-/. We also use the bracket notation ]~)(~I for this projection. If {~oi} is a basis of 7-/, then the projection operators P[~i] are mutually orthogonal and P[~i] - I, where I is the identity operator on 7-/. Let/:(7-/) denote the set of bounded operators on 7-/. An operator A E/:(7-/) is positive, A _ O, if (~IA~) _ 0 for all vectors ~ E 7-/. Then the relation A > B, defined as A - B >_ 0, is an ordering on the subset of self-adjoint bounded operators. Let ~ be a nonempty set and ~" a a-algebra of subsets of ~ so that (~,~') is a measurable space. A normalised positive operator valued (POV) measure E " ~" --. /:(7-/) on (~,~') is defined through the properties: i) E(X) >_ 0 for all X e ~" (positivity); ii) if (Xi) is a countable collection of disjoint sets in ~ then E(UXi) E(X~), the series converging in the weak operator topology (a-additivity); iii) E ( ~ ) - I (normalisation). For any e o v measure E " ~ --, L(7-/) the following two conditions are equivalent: i) E ( X ) 2 - E ( X ) f o r all X E ~'; ii) E ( X M Y) = E ( X ) E ( Y ) for all X, Y E ~. Thus a positive operator valued measure is a projection operator valued (ev) measure exactly when it is multiplicative. Further, if the measurable space ( ~ , ~ ) is the real Borel space (R, B(R)), or a subspace of

8

II. Basic Features of Q u a n t u m Mechanics

it, then E determines a unique self-adjoint operator fR ~dE in 7-/. Here t denotes the identity function on R. Conversely, according to the spectral theorem, each self-adjoint operator A in 7-/defines a unique PV measure E : B(R) ~ / : ( 7 ~ ) such that A - f R tdE. If E is a PV measure on (R, B(R)) it shall be denoted as E A in order to explicate the unique self-adjoint operator A associated with it. The set of trace class operators on 7-/will be denoted as T(7-/), and T(7~) + consists of the positive trace one operators on 7-/. The trace, T ~ tr[T], is a positive linear functional on T(~/). The one-dimensional projections P[~] are positive operators of trace one. They are the extremal elements of the set T(7~) +. Indeed T(7-/) + is a convex set (with respect to the linear structure of T(7-/)), so that an element T e T(7-/) + is extremal if the condition T = wT1 + ( 1 - w ) T 2 , with T1,T2 e T(7-l) +, and 0 < w < 1, implies that T = T1 = T2. But T E T(7-/) + is extremal if and only if it is idempotent (T 2 = T), which is the case exactly when T is of the form P[7~] for some unit vector 7~ E 7-/. The set of extremal elements of T(7-/) + exhausts the whole set T(7-/) + in the sense that any T e T(7-/) + can be expressed as a a-convex combination of some extremal elements (P[~i]): T = ~ i wiP[7~i], where (wi) are suitable weights, that is, 0 < wi <_ 1, ~ i wi - 1, and the series converges in the trace norm topology. Such a decomposition can be obtained, in particular, from the spectral decomposition of T - ~'~i tiPi (as a positive, self-adjoint compact operator) since the spectral projections Pi (associated with the eigenvalues ti, which are poitive) are finite-dimensional. In that case T is decomposed into mutually orthogonal one dimensional projections P[~i], with its eigenvalues representing the weights wi (each appearing in the series as many times as given by the dimension of the eigenspace). For any POV measure E • j r __, £(~/) and any T e T(7-/) +, the mapping pTE . j r ~ [0, 1],X ~ pET(X ) "-- t r [ T E ( X ) ] is a probability measure. This follows from the defining properties of E and the continuity and linearity of the trace. For T - P[7~] we shall also use the notation p~. E If E -- E A, then pTE shall also be written as pA. We note that the decomposition of states, T - ~ wiP[~i], induces E the corresponding decomposition of the probability measures pET -- ~-~iw iP~. b) P h y s i c a l c o n c e p t s . In quantum mechanics a physical system S is represented by means of a complex separable Hilbert space 7-/s. The general structure of any experiment - a preparation of a system, followed by a measurement - is reflected in the concepts of states and observables. The states of a system S are represented by - and identified with - the elements of T(7-ls) +. We let S(7~s) denote the set of states of the system S. The usual notion of a state as a unit vector of 7-/s then refers to the extremal elements of S(7-/s). We refer to these states P[7~], and to the generating unit vectors 7~ E 7-/8, as vector states. They are often called also pure states. In the absence of superselection rules, all vector states are pure states (cf. Sect. 2.3)." Due to the linear structure of 7-ls, superpositions of vector states form new vector states; and any vector state

II.1 Hilbert Space Quantum Mechanics

9

can be represented as a superposition of some other vector states. The convexity of the set of states represents the possibility of preparing new states as mixtures of other states. The notion of an observable provides a representation of the possible events occurring as outcomes of a measurement. In this sense an observable is defined as and identified w i t h - a POV measure E : j c ~ £(7-ls), X ~ E ( X ) on a measurable space (~, ~ ) . Since the space (~, ~ ) describes the possible measurement outcomes of the observable E, we call it the value space of E. Usually, the value space of an observable E is simply (a subspace of) the real Borel space (R, B(R)), or some of its Cartesian products. The familiar notion of an observable as a self-adjoint operator in H s refers to a PV measure on the real line R. In that case, we shall also use the term sharp observable for both A - fR tdEA as well as the associated spectral measure E A. In general all PV measures will be called sharp observables. A pair (E, T) of an observable E and a state T induces a probability measure pE on the value space (~, ~') of E: pTE" 9v ~ [0, 1], X ~ p E ( x ) " = tr[TE(X)].

(1)

The minimal interpretation of these probability measures establishes their relation to measurements; in fact the very notion of measurement will be based on this interpretation (cf. Sects. III.2 and III.3). MINIMAL INTERPRETATION. The number pET(X ) is the probability that a measurement of the observable E performed on the system S in the state T leads to a result in the set X. This minimal interpretation is contained in any more extensive interpretation of quantum mechanics (cf. Table 1). In Chap. I we mentioned various approaches aiming at a reconstruction of the conceptual structures of quantum mechanics. We shall sketch out a combination of arguments coming from the quantum logic and convexity approaches in order to provide some motivation for the notions of states and observables which, in the above generality, may not be too well-known. Starting with the notion of sharp observables, it is a consequence of Gleason's theorem (Sect. 2.1) that states (as a-additive probability measures on the lattice of projection operators) are represented by elements of S(7-ls) [Varadarajan 1985]. Next, given S(7"ls) as the state space, one may ask for the most general notion of an observable compatible with the probabilistic structure of quantum mechanics. The requirement that any state T induces a probability measure on a measurable space (~, ~') of measurement outcomes brings about the notion of an observable as a state functional valued measure X H E x on (~,~'), where E x ( T ) represents the probability for an outcome in X in the state T. Assuming that the mapping E x extends to a linear functional on T(7-ls) and making use of the fact that the dual space of T(7-/s) is isomorphic to

10

II. Basic Features of Quantum Mechanics

£:(7-/8), one concludes that to any Ex there corresponds a positive operator E(X) such that E x ( T ) - trITE(X)]. The resulting mapping X ~-, E(X) is a POV measure. Finally, it turns out (as a slight extension of Gleason's theorem) that this general notion of an observable does not require a restriction, nor does it admit an extension of the notion of state as given above. In this sense we conclude that the concepts of states and observables as defined here are the most general ones compatible with the probabilistic structure of quantum mechanics. We note that the convexity of the set of states as well as the assumption of the linearity of the mappings Ex can be motivated by statistical characterisations of the assumptions inherent in the notion of state preparation [Kraus 1983, Ludwig 1983]. The positive operators in the range of a POV measure E (an observable) represent the events associated with the outcomes which may occur in a measurement of E. They are called effects. From the positivity and the normalisation of a POV measure it follows that effects are bounded by O and I, that is, O < E(X) < I. Hence their spectrum is a subset of [0, 1]. An effect E(X) is a projection operator (E(X) = E ( X ) 2) if and only if its spectrum is the two point set (0, 1). We let £(7-ls) = (A e f~(7-ls) " 0 <_ A <_ I) denote the set of effects. The interpretation of projection operators as properties of a physical system can be extended to a certain subset £p(7-ls) of £(7-ls). An effect A is a property if neither A < 11 nor ½I _< A, that is, the spectrum of A extends both above and below ½. In addition one includes O, I into the set of properties. This definition ensures that within £p(7-ls) the map A ~ A ± "= I - E is an orthocomplementation; that is, A _< B implies B ± < A ±, A ±± - A, and the greatest lower bound of A and A ± in the set of properties is O. A property A is sharp if A -- A 2, otherwise it is unsharp. The decisive difference between sharp and unsharp properties thus lies in the fact that an unsharp property A and its complement A ± are not orthogonal as AA ± ~ O. We say that an observable is sharp if all the effects in its range are projection operators, whereas an observable is unsharp if at least one effect in its range is an unsharp property [Busch, Grabowski, Lahti 1995]. The present approach to the notion of an observable offers a natural conception of joint measurability, called coexistence: a collection of effects (resp. observables) is coexistent if it (resp. the join of their ranges) is contained in the range of some observable. In the case of sharp observables coexistence is equivalent to commutativity. But in general coexistent pairs of observables need not commute. We emphasise that the generality inherent in the notion of observable described above is really needed in quantum mechanics. First of all, as just pointed out, this concept allows for the possibility of joint measurements of noncommuting observables, especially complementary observables, in full accordance with the uncertainty relations (see Sect. III.8.2). This option may also be decisive in an attempt to understand, in terms of quantum mechanics, the almost classical nature of

II.1 Hilbert Space Quantum Mechanics

11

macroscopic observables, like pointers of measuring devices (cf. Sect. IV.4.5). Further, POV measures provide an appropriate means for dealing with the unsharpness inherent in any real measurement, whatever the sources of the unsharpness may be - quantum mechanical nonobjectivity in microscopic parts of the apparatus, or simply imperfections in the construction of the device. Several examples of both types of sources, or of their interplay, are exhibited in detail in [Busch, Grabowski,

Lahti 1995]. For later use we explicate the structure of the probability measures pTE for discrete observables. An observable E is discrete if there is a countable subset Fro of the value space gt such that E ( f / \ fro) = O, the null operator. In that case the probability measures pTE are also discrete and p E ( x ) -- ~-~eXnno PTE({W}) • For a vector state PIT] one has p~E ({w}) = tr [P[~IE({w})] =

(~lE({w})~).

(2)

If the value space (f~,9v) of E is the real Borel space ( R , B ( R ) ) , then for a sharp observable A = fl~ LdEA one obtains A - Y~i aiE A ({ai}), where the numbers ai are the (distinct) eigenvalues of A. Let {9~j } be a basis of the i th eigenspace of A, so that A~ij - ai~ij for each j (the index j running from 1 to the dimension of the eigenspace, the degree of degeneracy of ai). The spectral projection E A ({ai}) can thus be written as ~-~.j P[~ij]. The bases {~ij} of the eigenspaces EA({ai})(7"is), i running over the disjoint eigenvalues of A, form a basis of 7-/s. Hence we get for the probability p A ( x ) that a measurement of the observable E A in the state T leads to a result in the set X:

pA(x) = E a~EX

PA({ai}) -- E

tr[TEA({ai})]-

a~EX

= E

(3) ij:a~EX

(~lEA({ai}) ~) =

a~EX

ij:aiEX

where the second line refers to the case of T - P[~].

II.1.2 Tensor ProduCt and C o m p o u n d S y s t e m s a) M a t h e m a t i c a l structures: tensor product Hilbert spaces. Let 7~s and 7-/A be complex separable Hilbert spaces. The Hilbert space tensor product of 7-/s and 7-/.4 is denoted as 7-/s®7-/,4. The linear span of the set {~®¢ : 9~ E 7-/s, ¢ E 7-/A} is dense in 7is @ 7-/.4, and the inner products of the involved Hilbert spaces are related as (~ ® ¢1~' ® ¢') = (9~1~')(¢1¢'). Moreover, if {~oi} and {¢k} are bases of 7-/s and 7-/A, then {~i ® Ck} is a basis of 7-/s ® 7-/~4. Any ~P E 7-/s ® 7-/.4 can then be expressed as V = Y~(~i ® Ckl~)~i ® Ck. The spaces/:(7is ® T/A), T(Tis ® 7-/A), etc., and, for example, a POV measure E :9r ~ / : ( T i s ® 7-/A) are defined in the usual way. In particular, any A E / : ( T / s )

12

II. Basic Features of Quantum Mechanics

and B-a E £(7-/-a) determine a bounded linear operator, their tensor product, A®B-a on 7~s ® 7-/-a via the relation (A ® B-a)(qo ® ¢) = Aqo ® B-he, qo E 7-/8, ¢ E 7-/-a. Also, for any T E T(7"ls) and T-a E T(7-l-a) their tensor product T ® T-a is a trace class operator on 7-/s ® 7-/-a. As is well known, the tensor product operators do not exhaust the respective operator sets. This important fact will be illustrated below in connection with the trace class operators. In the theory of compound systems the partial trace operation is particularly important. The partia/trace over the Hilbert space 7-/-a, say, is the positive linear mapping HT~ : T(7-ls ® 7t-a) ---, T(7"ls) defined via the relation

tr[n~ (W)A]

= tr[WA ® I-a],

(4)

where W E T(7-ls ® 7-l-a), A E £(7-ls), and I-a is the identity operator on 7-/-a. If {~i} C 7"/s and {¢k} C 7-/.4 are orthonormal bases, then HT~, (W) can be written as

® CklW j ® Ck>

(w) =

(5)

ijk

Here [~i)(~ojIis the bounded linear operator on ?-/sgiven by [~i)<~pj[(~)= (~j[~)~i, E 7-/s. The partial trace over ?'/s is defined similarly and denoted as IIns :

r(7 s ® As mentioned above, the operators of the form T ® T-a do not exhaust the set T(7-ls ® 7~-a). If W = T ® T.a, then T = HT~ (W) and T-a = HT~s(W), but, in general, W ¢ IIT~ ( W ) ® IIT~s(W). In particular, if W = P[@], then P[@] = HT~ (P[@]) ®HT~s (P[~]) if and only if @ = qa®¢ for some qo E 7-/8 and ¢ E 7-/-a. In that case also HT~ (P[@]) = P[qo] and HT~s(P[@]) = PIe]. This result demonstrates the important fact that the extremal elements of the sets T(Tls) + and Y(7-/-a)+ do not exhaust the set of extremal elements of T(7-ls ® 7-l-a)+. b) P h y s i c a l concepts: compound systems. Consider two physical systems S and .A, and assume that they are not identical. Let 7-/s and 7-/-a be the Hilbert spaces associated with S and A. The description of the compound system ,.q + ~4 is then based on 7-/s ® 7-/-a according to the usual ideas of quantum mechanics. The theory of compound systems investigates the connections between the descriptions of the systems S, ~4, and 8+.4. Here we recall only that any state W E S(7-ls®7-l-a) of S + ,4 uniquely determines the states of the subsystems S and A as the partial traces HT~A(W) and IIT~s(W), respectively. These states are called the reduced states. From now on we use the more suggestive notations

n s ( W ) := nT~ (W)

and

7~-a(W):= HT~s(W)

(6)

for these states. Thus, for example, TC.s(W) is the reduced state of S if W is the state of S + A. The fact that the reduced state ~ s ( W ) is uniquely determined

II.1 Hilbert Space Quantum Mechanics

13

by W means, in particular, that the system S can be identified as a subsystem of S + +4 in the sense that all the probability measures pTE referring to 3 can be obtained from the probability measures pE®l~ related to S + ,4; here E @ IA is the P O V measure X ~ E(X)@ I,a and W E S(7-ls @7"l,a) is such that T = R s ( W ) . We re-emphasise the important fact that given a system S + +4 in a vector state P[~], the subsystems S and jt are in vector states P[qa] and P[¢] if and only if P[~] is a product state and hence equal to P[~ ® ¢] = P[~] ® P[¢]. In general the states 748 (P[~]) and RA (P[~]), ~ e 7~s ® 7-/,4, II ~ II = 1, are no vector states. In such a case one says that the subsystems are entangled.

II.1.3 Dynamics The time evolution of an isolated system prepared (at time t - 0) in a pure state ¢0 is determined by the Schrbdinger equation

ihd¢(t) - H¢(t)

(¢(0) = ¢0),

(7)

where H is the Hamilton operator of the system. Equivalently, a system prepared (at time t - 0) in a state To undergoes the following state change:

T(t) = U(t)ToU* (t) =: lit(To),

(8)

where U(t) = exp(--~Ht), t e R, is the strongly continuous unitary group induced by the Hamiltonian (via Stone's theorem). The differential form of the above process is given by the yon Neumann-Liouville equation:

ih d T(t) = HT(t) - T(t)H = : / : [ T ( t ) ] .

(9)

It has been argued from various sides that part of the problem in understanding quantum measurements is rooted in the fact that the description of the measurement dynamics in terms of a unitary group may be too restrictive (cf. Sects. III.7 and IV.4.3). Accordingly, adopting this position, one should look for the most general representation of dynamics by means of families of mappings ))t on T(Tls). A fairly general class of state transformations ~;t are the positive, trace preserving linear mappings on T(7-ls). They are naturally obtained via the concept of reduced dynamics. In fact, let 7-/s ® 7~A be the Hilbert space of a compound system S + ,A, Ut be the dynamical (unitary) group, and T @TA some state representing the initial (t = 0) preparation; then the time evolution of 3, say, is obtained by application of the partial trace at every instant of time, yielding the following family of linear state transformations ~;t"

])t(T) := Us(blt(T ® TA)).

(10)

These mappings play an important role in the quantum theory of measurement.

14

II. Basic Features of Quantum Mechanics

In Sects. IV.5.3 and IV.4.3 we shall discuss two types of approaches to the measurement problem describing dynamics in terms of families of mappings ]2t applied to the compound system consisting of object system S and apparatus .4. In the first type of these approaches the system S + A is extended to a larger system including some environment E such that the unitary dynamics b/t of S + A + C gives rise to a nonunitary reduced dynamics Vt for S + .4. The second approach is more radical as it suggests that the ordinary unitary dynamics be modified to a more general dynamics Yt: a new linear term added to the von Neumann-Liouville operator £: breaks the purity of the unitary mappings, that is, it forces systems initially prepared in pure states to evolve into mixed states.

II.2. Probability Structure of Q u a n t u m Mechanics Much of the conceptual and interpretational problems of quantum mechanics is rooted in the irreducibility of its probability structure. We shall review those probabilistic aspects of the theory which are particularly important for the measurement problem.

II.2.1 States as Generalised Probability Measures A fundamental result in the mathematical foundations of quantum mechanics specifies the class of generalised probability measures on the set ~(7-/) of effects. Consider the probability measure pTE • jc ~ [0, 1] defined by an observable E " ~" ~/::(7-/) and a state T e 8(7-/). According to Equation (1.1), this measure is a composition of the POV measure E and the state-functional mT " /:(7-~) ---, C, A H mT(A):= tr[TA]. Since O < E(X) < I for all X e ~', the numbers pE(X) = m T o E ( X ) are, indeed, in the unit interval [0,1]. The measure property of E and the linearity and continuity of mT imply then that mTo E is a (Kolmogorov) probability measure. In fact the linearity and continuity of the state-functional guarantee that the mapping mT, when restricted to the set of effects C(7-/), is a generalised probability measure. That is, it has the following properties, i) mT(A) ~_ mT(O) -0 for all A E C(7-/) (positivity); ii) if (Ai) is a countable collection of elements of C(7-/) such that ~ Ai <__I, then m T ( ~ Ai) = ~ mT(Ai) (convergence in the weak operator topology; a-additivity); iii) mT(I) -- 1 (normalisation). We note that the requirement ~ Ai <_I in ii) is a natural generalisation of the condition of pairwise orthogonality of projections. Indeed, if (Ai) is a sequence of projection operators, then ~ Ai <_ I is equivalent to the condition that the Ai are mutually orthogonal, that is, Ai <_I - Aj for all i ~ j. Let m • E(7-/) ~ [0, 1] be a generalised probability measure. Then for any observable E " jr ~/::(7-/) the mapping m o E " ~" ~ [0, 1] is again a probability measure and the minimal interpretation can be applied to it. The question of whether there are generalised probability measures m other than those induced by the states is to be answered in the negative due to the following argument. It is easy

II.2. Probability Structure of Quantum Mechanics

15

to see that any m extends uniquely to a bounded positive linear functional on/:(7-/), which shall be called m again. Moreover this functional is normal; that is, for any norm-bounded sequence of positive operators (Ai) which is increasing, Ai <_ Ai+l, one has m(Ai) --. re(A), where A is the least upper bound and weak-limit of (Ai). It follows that m is of the form re(A) - tr[TA] for some positive operator T of trace one [Davies 1976]. Hence we have established the following result. THEOREM 2.1.1. For any generalised probability measure m" £(7-/) --, [0, 1] there is exactly one state T e S(7-l) such that m(A) - tr[TA] for all effects A e £(7-/). The state functionals mT are commonly used to express the expectation values of self-adjoint operators A. It has been argued that their additivity, m ( A + B) = m(A) + m(B), has no operational justification if A and B are non-commuting operators and correspond to non-coexistent sharp observables. The above reasoning offers a probabilistic motivation of the additivity in terms of the (a-)additivity of probability measures for sequences of effects the sum of which is less than I. Such collections of effects are coexistent, without necessarily being commutative. If one does not want to rely on the structure of effects and POV measures in order to obtain the statement of Theorem 2.1.1, one can still take recourse to Gleason's theorem [Glea 57] which determines the structure of generalised probability measures defined only on the projection lattice 7~(~/) rather than on E(7-/). The a-additivity requirement is essentially weaker in this case since the condition ~;~i Pi <_ I for a sequence of projections implies PiPk - 0 for i ¢ k, so that it can only hold for mutually commuting projections Pi. THEOREM 2.1.2 (GLEASON). Let m " 7a(7-l) - , [0, 1] be a generalised probability measure. If the vector space dimension of 7t is at least 3, there is exactly one state T e S(7-l) such that m(P) = tr[TP] for all projections P e 73(?-l). This shows that for dim7-/> 3 the probability measures on ~(7-/) are precisely the extensions of those on 7v(7-/). Gleason's theorem, or its variant referring to the set of effects, has several important implications in quantum mechanics. First of all, it specifies the probabilistic content of the state concept. It also ensures that the notions of observables and states as POV measures and positive trace class operators, respectively, are the most general measures compatible with the probability structure of quantum mechanics. The correspondence between generalised probability measures mT and states T is one-to-one and onto, and it preserves the natural convex structures of the two sets.

II.2.2 Irreducibility of Probabilities Quantum mechanics is an irreducibly probabilistic theory in the following sense: it is impossible to decompose the generalised probability measures into dispersionfree (that is, {0, 1}-valued) ones, simply because the latter do not exist [Beltrametti and Cassinelli 1981, Jauch 1968]. This important fact is a consequence of Gleason's

16

II. Basic Features of Quantum Mechanics

theorem (if dim7-/>_ 3), or its variant referring to the generalised probability measures on £ (7-/). In classical probability theory, the dispersion-free probability measures on a measurable space (~, ~-) are exactly the extremal elements of the convex set of all probability measures on (~,~'). Moreover it is possible to decompose any probability measure into (continuous/discrete) convex combinations of dispersion-free probability measures. In quantum mechanics the extremal elements of the set of states S(7-/) are precisely the vector states, and none of them corresponds to a dispersion-free generalised probability measure. In fact for any vector state P[~] there exists an effect A (for example, a projection A = PIe] with ¢ neither collinear nor perpendicular to ~) for which 0 ~ (~]Ag~) ~ 1. This fact rules out the possibility of an epistemic interpretation of the quantum mechanical probabilities as measures of the ignorance about which properties actually pertain to the system under consideration. Indeed a {0, 1}-valued measure on P(7-/), if it existed, could be understood as a 'truth value' assignment, or value attribution, determining which properties are real or absent. According to Gleason's theorem such assignments do not exist globally. Hence the following important question arises and will be investigated in the remaining part of this chapter: which properties can be claimed as being actual or absent if the system is in a given state? Equivalently, which observables have definite values in a given state? Such properties, or (the values of) such observables shall be called objective. The question, as it stands, is not yet formulated with sufficient precision in order to admit a definite answer. In the literature one finds indeed a variety of formalisations which yield different sets of observables that can be regarded objective in a given state. We shall not enter into a discussion of the various distinct approaches and their consistency here but return briefly to them in Sec. IV.5.2 where their connections with certain interpretations of quantum mechanics become apparent. Instead we shall present shortly the 'orthodox' concept of objectivity which forms the basis of our account of the problem of objectification. Our definition of the objectivity of an observable E will be linked to the possibility of decomposing a state T into other states which yield dispersion-flee Edistributions. This feature is more restrictive than the mere possibility of decomposing pTE into {0, 1}-valued measures on the value space (f~, ~ ) of E, which is always given. Hence we insist that the dispersion-free measures involved arise from some states in 8(7-/), and that these states establish a decomposition of T. This condition is violated, in particular, in the case of the dispersion-free generalised probability measures on :P(7-/) that are known to exist in a two-dimensional Hilbert space: while these {0, 1}-valued measures are sufficient to recover all quantum states as convex combinations, they do not correspond to states in S(7-/). Therefore they are commonly regarded as 'hidden' states in the sense of not being physically accessible or preparable.

II.2. Probability Structure of Quantum Mechanics

17

We investigate next the conditions under which for one given observable E a probability measure pE . jr ~ [0, 1], T E S(?-/), admits a decomposition into dispersion-free probabilitymeasures associated with states. Let T = ~ wiP[~i] be a decomposition of T into vector states (~i) (~i e 7-/, ]] ~i ]] - 1) with weights (wi), O<wi<_l,~wi-1. Then for a n y X E J C ,

pE(x) = Z wipE (X) = Z wi((PilE(X)cPi)"

(1)

Now assume that (~IE(X)~) e {0,1} for all ~ . This holds true if and only if E ( X ) v i = ¢Pi or E(X)(pi = O, which implies that each P[(Pi], and hence T, commutes with all E(X). The converse implication holds true whenever E is a sharp observable: then T E ( X ) = E ( X ) T holds exactly when all ~Pi are eigenstates of E, in which case pTE = ~ wipe,E is a decomposition into dispersion-free probability measures. If T is not a mixture of E-eigenstates, one can still always find such a mixture which has the same E-statistics. As an illustration, let A = ~-'~iaiEi be a discrete sharp observable, with Ei - ~"~kEik being a decomposition of the spectral projections Ei into 'smaller' projections (including the possibility of having only one k-value). Then for any state T we consider the new state

T(A) := Z EikTEik = ~-~' tr[TEi]Ti. ik

(2)

i

Here the sum ~' runs over a l l / for which tr[TEi] ~ O, in which case Ti =

(tr[TEi]) -1 ~-]~kEikTEik. Thus T (A) is decomposed into states Ti giving dispersion-free probability measures for A. Furthermore, T and T (A) yield the same A-statistics: This consideration suggest exploring the implications of the requirement that with respect to some given observable E all (physically distinct) states have a decomposition into states with dispersion-free E-distributions. This question addresses the issue of superselection rules.

II.2.3 Superselection Rules In this book quantum mechanics is primarily understood as a theory of systems for which there is no apriori restriction of the set of observables. Accordingly, a system S is called a proper quantum system if all bounded self-adjoint operators in £(7-/s) represent (sharp) observables. In this case the set O of observables of S is said to be unrestricted. In general S may have a restricted set of observables, particularly in the presence of a superselection rule. A system S possesses a superselection rule if there exists a sharp observable Z which commutes with all other observables F E O. Such a superselection observable is called a classical observable of S. Occasionally

18

II. Basic Features of Quantum Mechanics

it has been suggested that the objects of classical physics are to be characterised as systems the observables of which are all coexistent. Accordingly one could consider a classical system as a quantum system all observables of which commute with each other. The possible restriction of the set of observables suggests the introduction of an equivalence relation on the set of states. Two states T1 and T2 are equivalent with respect to a set 50 of observables if these states assign the same probability E for each E E 59. In that case measures to each observable of (9, that is, if pE _ PT2 we denote T1 ~ o T2, or simply T1 ~ T2 if there is no ambiguity with regard to the set O. For a given set of observables this relation is, indeed, an equivalence relation on the set 8(7-/) of states. As an example, the state T A of Eq. (2) is equivalent to T with respect to A according to Eq. (3). No two different states from S(?/) are equivalent with respect to a/1 observables unless the set O of observables is restricted. In the presence of a discrete superselection rule only those vector states represent pure states which are eigenvectors of the superselection observable Z. All other vector states are equivalent to mixtures of pure states. It follows that all states are equivalent, with respect to all observables, to some mixture of eigenstates of Z. States which are equivalent with respect to the set of all observables of a system shall be called physically equivalent of simply regarded as identical. This means that the set of observables O of a system S determines the set of states, So(7-/), as the family of all equivalence classes with respect to (.9. Thus we find the following answer to the question posed at the end of the preceding subsection: THEOREM 2.3.1. Given a system S with the sets O, S o(Tl) of observables and states, respectively, the following are equivalent:

(a) Z is a discrete classical observable; (b) all states admit a convex decomposition into states whose Z-distributions are dispersion-free. In this sense a classical observable can be thought of as having objective values in all states.

II.2.4 Nonobjectivity of Observables As we have pointed out in Sect. 2.2, the irreducible probability structure of quantum mechanics has an important corollary for the interpretation of the basic probabilities p E ( x ) of the theory. Even in the case of pure states P i l l , the probabilities ( ~ [ E ( X ) ~ ) are, as a rule, neither 0 nor 1. In general it is not consistent to assume that in a state P[~] the system S possesses the property E ( X ) or its complement property I - E ( X ) = E(12 \ X ) = E ( X ' ) , that is, that the value of E in a state P[~] is either in X or in X', respectively. This fact constitutes the nonobjectivity of the observable E in the state P[~].

II.2. Probability Structure of Quantum Mechanics

19

In order to make this statement more precise, we consider first a proper quantum system 8 and a sharp observable A of S. Let X E B(R) be any value set of A such that O ~ E A ( x ) 7£ I. For the following consideration it is convenient to introduce the term 'reality'. DEFINITION 2.4.1. For a sharp observable A of the system 8, the property E A ( x )

is a real property of S in a state T if a measurement of A leads to an outcome in X with probability equal to one: p A ( x ) = 1. Thus a property E A (X) is real exactly when E A ( X ) T = T. Especially for T - P[9~] this is equivalent to EA(x)~p -- 9~. Now one can conceive of the following situation. Suppose it is known that either E A ( x ) or E A (X') is a real property of the system S but it is not necessarily known which one is actually the real property. This means that the system is known to be either in a 1-eigenstate Tx of E A ( x ) or in a 1-eigenstate Tx, of E ( X ' ) , with the subjective probabilities wx, wx,, respectively. In this case S is described by a state operator T - w x T x + w x , T x , , with the additional information that the ignorance interpretation (Sec. 2.6) applies to this particular decomposition of T. Such a situation can be represented by associating with the system a collection { (wx, Tx), (wx,, Tx,)}, which shall be called a Gemenge. The state T shall be said to represent this Gemenge. In order to afford a convenient terminology, we introduce the term 'objectivity'. DEFINITION 2.4.2. Let A be a sharp observable of the system $. The property E A (X) is objective in the state T if the following conditions are fulfilled:

(1) T admits a decomposition T = w x T x + w x , T x , into states Tx, Tx, for which (x)

= 1 -

(x');

(2) T represents the Gemenge { (wx, Tx ), (wx,, Tx, ) }. We shall see shortly that there are different ways of preparing a mixed state T and that one cannot in every case apply the ignorance interpretation to a given decomposition of T. Hence the term objectivity contains more information about the system's state than just its predictive content which could be tested by performing measurements on that system alone: in order to ascertain that a property is objective in the state T, one needs to know the system's history, that is, the way T has been prepared. Let E A ( x ) be objective in a state T, so that, in particular, T - w x T x ÷ w x , T x , . It follows that w x - p A ( x ) , wx, -- p A ( x ' ) , and E A ( X ) T E A ( X ) p A ( X ) T x , E A ( X ' ) T E A ( X ') - pAT(X')Tx,. Therefore,

T - E A ( X ) T E A (X) + E A ( X ' ) T E A (X')

(4)

whenever E A ( x ) is objective in the state T. Condition (4) is, in turn, equivalent to the fact that T commutes with E A ( x ) .

20

II. Basic Features of Q u a n t u m Mechanics

In accordance with Definition 2.4.1 we say that a sharp discrete observable A is objective in a state T if all of the properties E A ({hi}) are objective. This is the case only if the state T is a mixture of eigenstates of A. In addition, this mixture must admit an ignorance interpretation. If T is a pure state, T - PIca], then the objectivity of E A ( x ) in this state implies, as a consequence of Eq. (4), that either EA(x)~o - ~ or EA(x)~o -- O, that is, either E A ( x ) or E A ( x ') is real in the state P[~] itself. Hence E A ( x ) is nonobjective in any pure state of a proper quantum system which is not an eigenstate of A. Experimentally, this nonobjectivity becomes manifest in the socalled interference effects. Define a state T~ as follows:

T~ "= E A ( X ) P M E A (X) + E A (X')P[~]E A (X').

(5)

Further, let B be another sharp observable. If the observable A were objective in the state T - P[~], then on the basis of equations (4) and (5) one would get PvB (Y) = P ~ (Y)"

(6)

However, without this objectivity assumption one finds the following relationship between the probability measures pB and pTB • B PvB (Y) = PT~(Y) + 2Re[(EA(X)~[EB(y)EA(X')v)].

(7)

The interference term 2Re[(EA(X)v[E B (y)EA(X~)v)] is zero if V is an eigenvector of E A ( x ) . Otherwise it can be made nonzero by a suitable choice of B; in particular, such a B must not commute with A. If the system 8 is not assumed to be a proper quantum system, then the objectivity of E A ( x ) in a vector state P[~o] leads to the equivalence P[~] ~ o T~, with T~ as given in (2) and O representing the set of observables of 8. If ~ is not an eigenvector of E A ( x ) , then the interference term (7) will not vanish unless the set of observables is restricted. As stated implicitly in Theorem 2.3.1, an observable A is objective in a/1 vector states P[~] exactly when A is a classical observable. In particular, all observables of a classical system are objective. The notion of the objectivity of a property can be generalised for effects as well as for discrete effect valued observables. These generalisations are needed later on. DEFINITION 2.4.3. An effect E E E(7-/) is rea/in a state T ff tr[TE] - 1. The reality of an effect in a state implies that this effect has eigenvalue 1 and that the state in question is a 1-eigenstate of the effect, E T - T. DEFINITION 2.4.4. Let E l , E 2 , . . . be a collection of effects such that each Ei has eigenvalue 1 and ~Ei = I. Then the Ei and the observable generated by them are objective in a state T if T represents a Gemenge of 1-eigenstates of the Ei.

II.2. Probability Structure of Quantum Mechanics

21

The objectivity of an effect Ei in a state T entails that T - E~I)TE~ 1) + ( I -

E i ) ( I ) T ( I - Ei) (1), where, for example, E~ 1) is the spectral projection of Ei associated with the eigenvalue 1. More generally, we note the following important connections, which will be important in the analysis of the objectification requirement (Secs. III.2.4 and III.6). THEOREM 2.4.1. Let E l , E 2 , . . . ( E E i = I) be a collection of effects and T a state. If T is a mixture of 1-eigenstates of the Ei then this mixture can be expressed as T = Z

Ei(1)T E i(1) .

(8)

Furthermore, the (non-normalised) components of this mixture s a t i s ~

E~I>TE~ 1> -~ E:/2TE:/2,

(9)

(where E~/2 denotes the square root of Si) so that the mixture (8) can also be written as

Thus all the relations (8)-(10) hold whenever the effects El, E2, • .. are objective in the state T.

II.2.5 Nonunique Decomposability of Mixed States The objectivity of an observable in a state T is linked to the possibility of decomposing this state in'a certain way. In addition this decomposition is required to admit an ignorance interpretation. However the set of states S(7-/) possesses a fundamental structural feature which makes the ignorance interpretation of mixed states highly problematic: any mixed state T of 8(7/) admits infinitely many decompositions into vector states P[~] (cf. [Beltrametti and Cassinelli 1981]). To see how 'bad' the situation is, one may ask which vector states P[~o] can occur as components in some decomposition of T, that is, for which P[~] there exist w E (0, 1) and T t E 8(7-/) such that

T = wP[~o] + (1 - w)T ~.

(11)

The answer is as follows: it is precisely the unit vectors ~ in the range of the square root of T which give rise to such a decomposition [Had 81]. The nonunique decomposability of mixed states in quantum mechanics is quite in contrast to classical probability theory, where a decomposition of any probability measure into extremal elements is unique. The reason for this difference is that in the classical case all extremal elements of the set of probability measures are {0, 1}-valued measures and therefore mutually disjoint: for two {0, 1)-valued measures to be distinct, there must exist a set on which one assumes the value 1 and the other one the value 0. By contrast, in 8(7-/) there are plenty of pairs of vector states which are not mutually orthogonal, so that there is room for convex decompositions of mixed states into nonorthogonal extremal elements, besides the orthogonal decomposition(s) induced by the spectral resolution.

22

II. Basic Features of Quantum Mechanics

II.2.6 Entangled Systems and Ignorance Interpretation for Mixed States The nonunique decomposability of mixed states bears severe implications for the interpretation of such states in quantum mechanics. In fact, generally a mixed state T, with a decomposition T = ~ w~P[~], does not admit an ignorance interpretation according to which the system S prepared in state T would actually be in one of the component states P[~i] with the subjective probabilities wi. The above result (11) at once makes such an interpretation problematic. However, this question deserves to be studied in greater detail since it is of foremost importance within measurement theory. In order to decide on this issue, let us review the possibilities of preparing mixed states in quantum mechanics. Consider a sequence of vector states ~i, i - 1, 2,..-, together with a sequence of weights wi, 0 <_ wi <_ 1, ~ wi = 1. The vectors ~i may or may not be mutually orthogonal. Assume that a system S is prepared in such a way that it is known to be in one of the states P[~i] with subjective probability wi. This knowledge is represented by assigning to S the Gemenge {(wi, P[Ti])" i = 1, 2 , . . . }. This Gemenge determines a mixed state T = ~ wiP[Ti] which represents the probabilistic content of the Gemenge. Due to the knowledge about how the state T was produced one is allowed to apply the ignorance interpretation to the above decomposition. Note, however, that the properties P[~i] are objective in the above Gemenge only if the vector states ~i are mutually orthogonal. Such a Gemenge situation may occur if a preparation instrument, for instance, an accelerator, does not work completely accurately, but it prepares systems in states ~1, T2, "" ", say, with the a priori probabilities wl, w2,..., which depend on the construction of the accelerator and which can be determined separately. Thus, if S is prepared with such an instrument, then 8 is to be described by T - ~ w~P[~], which is now the relevant decomposition of T. Whether this type of preparation procedure can also be described within quantum mechanics as a possible physical process will not be analysed here (cf. Sect. III.9). Another type of situation in which the ignorance interpretation is applicable arises in the presence of a discrete superselection rule. Such a rule implies that any state T has a unique decomposition into convex components Ti := EiTEi/tr[TEi], where Ei are those spectral projections of the underlying superselection observable for which tr [TEi] ~ O. This case will be of importance in Sect. III.6. A different way of preparing a physical system in a mixed state may occur when the system is part of a compound system. Consider a compound system S + A associated with the Hilbert space 7-/s ® 7-/.4. If S + A is (prepared) in a vector state ~, then the subsystems S and ,4 are (prepared) in the reduced states 7~s (P[~]) and 7~.4 ( P I l l ) which are vector states P[~] and P[¢], say, if and only if P[~] -- P[~] ® P[¢]. Now in a typical (idealised) measurement situation the state 7~s (P[~]) has a natural decomposition 7~s (P[~]) - ~ wiP[~i] into an orthogonal system of eigenvectors of the measured observable A = ~ aiP[~i]. Hence the

II.2. Probability Structure of Quantum Mechanics

23

(necessary) condition of objectivity (4) is satisfied. However, if P[~] is a pure state, then due to the entanglement inherent in that state, the observable A cannot be regarded as objective, and the ignorance interpretation cannot be applied to the reduced state of S. Let us see under which conditions the ignorance interpretation can be applied to a subsystem S of a compound system S + ,4 if the latter is described by a vector state • E 7-/s ®?-/A which is not of the product form ~®¢. Consider a biorthogonal (or polar) decomposition [Jauch 1968] of ~, =

® ,,.

(12)

Now ~ s ( P [ ~ ] ) - ~ wiP[~i]. Assuming that the ignorance interpretation holds for this decomposition of the state of S amounts to the same as assuming that S + .A is actually in one of the states P[~i ® ¢i]. In fact, under the ignorance interpretation, S is actually in one of the states P[~i], with the probability wi; but S being a subsystem of S + ,4, then this larger system must actually be in a vector state of product form P[~i ® ¢~], with the probability wi. Therefore, the state P[~] must represent the Gemenge { (wi, P[~i ® ¢~]) ) from which it follows readily that P[¢~] - P[¢i]. Clearly this is excluded if • is a pure state. Otherwise, it would follow that the set of observables of S + A is restricted such that P[~] is equivalent to To = ~ P[~i] ® I.AP[~]P[~i] ® I~, that is, P[~] ~ To. This argument can be rephrased in terms of the observables A = ~ aiP[~i] and A ® I.A = ~ a~P[~i] ® Ix. Indeed, the ignorance interpretation applies to the above mixture ns(P[~]) exactly when A is objective in that state. This is then equivalent to the objectivity of A ® Ix in P[~]. The ensuing equivalence P[~] ~ T~, is in accordance with Eq. (4). Note that a similar reasoning holds if the roles of S and ¢4 are interchanged. Moreover, it is then evident that the objectivity of the observable A.n = ~ a~P[¢~] entails that of A. In Sect. III.6 we shall apply these arguments to explore the implications of the objectification assumption on the pointer observable PA and the corresponding quantity I ® PA. The above reasoning does not presuppose the identification of the observables A of a system S with the observables of the form A ® 1.4 pertaining to a compound system S + A; rather this correspondence follows insofar as the objectivity of the observable A of the subsystem is concerned: this objectivity is inherited exactly by A ® Ix (and functions thereof), and vice versa. The transfer of the objectivity of A to A @I is in accordance with the corresponding transfer of the reality of properties: if S is in an eigenstate T of Ei [so that Ei is real or absent in T], and if S is part of a system S + ,4 in state W [so that T = ~s(W)], then W is an eigenstate of E~ ® I. In short: E i T = T implies Ei ® I W = W, and EiT = 0 implies Ei ® I W = O. These features of the reality concept developed here are found to be in contrast to the modified notions offered by the so-called modal interpretations (Sect. IV.5.2).

24

II. Basic Features of Quantum Mechanics

A particular situation arises when the polar decomposition (12) is not unique and the reduced mixed state 7~s (P[~]) is degenerate. Then the assumption of the ignorance interpretation for an observable corresponding to one of the orthogonal decompositions of the object state would again lead to the contradiction mentioned above. However since in the case of a degenerate mixed object state there are infinitely many orthogonal decompositions of this state, one could even try to assign the values of several observables (corresponding to the different decompositions) to the system. From this somewhat stronger assumption one can derive equations, which were shown to be violated experimentally, in agreement with quantum mechanics [Bus 93]. A well-known example illustrating this situation is given by the singlet state • = ~ { ® ¢ - - ~ ® } of S +,4 consisting of two spin- ½ objects.

III. The Quantum Theory of Measurement Survey-

T h e N o t i o n of M e a s u r e m e n t

The purpose of measurements is the determination of properties of the physical system under investigation. In this sense the general conception of measurement is that of an unambiguous comparison: the object system S, prepared in a state T, is brought into a suitable contact - a measurement coupling- with another, independently prepared system, the measuring apparatus from which the result related to the measured observable E is determined by reading the value of the pointer observable. It is the goal of the quantum theory of measurement to investigate whether measuring processes, being physical processes, are the subject of quantum mechanics. This question, ultimately, is the question of the universality of quantum mechanics (see Chap. I). In classical physics all observables are objective in any state, that is, they always assume well-defined though possibly unknown values. Moreover, it is possible in principle to measure them without in any way changing the observed system. Hence the measurement outcome is nothing but the value of the observable before as well as after the measurement. On the other hand, in the case of quantum mechanical systems for any observable there exist states in which the observable is not objective (Sect. II.2.4). In that case the reading shown by the apparatus cannot refer to an objective value of the observable before the measurement. Furthermore, it is not evident that a measurement may be such that its outcome refers to an objective value of the observable after the measurement. In this situation the question arises of how to explain and interpret the occurrence of a particular value of the pointer observable. Answering this question amounts to fixing the notion of measurement. The minimal requirement to be fulfilled by a measurement is the probability reproducibility condition (Sect. 1.2). This condition stipulates that the probability measure pE is 'transcribed' into a probability measure for the pointer observable in the apparatus state reached after a suitable measurement coupling. In the quantum theory of measurement for sharp observables the probability reproducibility requirement is equivalent to a still more elementary postulate, the calibration condition: if the observable to be measured is real, then the measurement should exhibit its value unambiguously and with certainty. We shall see (Sect. 2.3) that in the quantum theory of measurement for sharp observables the calibration and probability reproducibility conditions are equivalent. While the calibration requirement is generally applicable only to sharp observables (allowing for objective values), the probability reproducibility condition may

26

III. The Quantum Theory of Measurement

be and will be taken as the minimal content of the notion of measurement for arbitrary observables. In the way outlined above this minimal concept of measurement is precisely what is needed as the operational basis of the minimal interpretation of quantum mechanical probabilities (cf. Sect. II.l.1). In Sect. 2 the measurement theoretical implications of the probability reproducibility condition are worked out, thus providing the basis for an analysis of the foundations of the probability interpretation to be carried out in Sect. 3. There it will be shown that the probability pE(x) represents the potential occurrence of a measurement result in X and is realised in the course of a measurement in the form of a distribution of definite pointer values in the following sense: if the same E-measurement were repeated sufficiently many times under the same conditions (characterised by T), then in the long run the relative frequency of the occurrence of the measurement results in X would approach the number pE(x). The connection between probability and frequency presupposes that the apparatus does assume a definite pointer value in the course of a measurement. The question of how this pointer objectification is achieved (in view of the nonobjectivity of the measured observable) constitutes the first part of the so-called objectification problem in the quantum theory of measurement and will be discussed in Sect. 6. The second part of this problem - value objectification - is related to a question touched upon above. A particular pointer reading refers to the object system prior to measurement only if the measured observable was objective before the measurement. Where the observable was nonobjective the question arises as to what happens to the system in the course of the measurement. In the context of the quantum theory of measurement it follows that in general a state change is unavoidable. Attempts to minimise this irreducible 'disturbance' lead to the concept of ideality of a measurement. We shall see that this property requires another one: repeatability. A repeatable measurement will force the system into a state in which the pointer reading X refers to an objective value of the measured observable. This shows that the existence of repeatable measurements is necessary for realistic interpretations of quantum mechanics. For such measurements, pointer objectification entails value objectification via strong value-correlation. The probabilistic and information theoretical prerequisites for formulating the objectification problem (Sect. 6) are investigated in Sects. 4 and 5 where the notions of ideal, first kind, and repeatable measurements and their relationships are also analysed. To complete and summarise this survey, Sect. 1 provides an outline of the basic steps of any quantum physical experiment: state preparation and measurement, the latter including registration and reading. We shall also point out the difficulties encountered in an attempt to describe these steps within the quantum theory of measurement. In its conventional formulation measurement theory is restricted to the consideration of measurements which preserve the identity of the object system and the measuring apparatus. Aiming at a description of the measuring

III.1. General description of measurement

27

apparatus as a quantum mechanical system, measurement theory is a part of the theory of compound systems with its own specific questions: how the constitution and preparation of isolated objects is achieved; what types of couplings between the object system and the measuring apparatus may serve as measurements of a given observable of the object system (Sect. 7); how to determine a result on the apparatus; and how to relate a measurement outcome to the measured observable (Sects. 2 through 6); which limitations to the measurability of physical quantities are to be taken into account (Sect. 8). Finally, the problem of state preparations is revisited in Sect. 9.

III.1. General Description of Measurement III.l.1 The Problem of Isolated Systems Measurement theory should be able to account for the process of preparation of object systems prior to measurements performed on them. The peculiar nature of quantum mechanical compound systems causes appreciable difficulties in explaining the existence of isolated (that is, dynamically and probabilistically independent) systems, in other words, the existence of objects. It may turn out that, strictly speaking, objects can be constituted only in a classical environment since otherwise the entanglement with the environment would persist. In fact let S + C be an isolated compound system undergoing a unitary dynamical evolution as described in Sect. II.1.3. The probabilistic independence of S and $ requires the state of S + $ to be of product form: W = T ® Tc. But this form cannot be preserved for all times unless the systems are dynamically independent, that is, Hint = O, or hot = L/~ ® bit~. Therefore, starting with a vector state preparation of the form ~ @ ¢, the dynamical evolution will inevitably lead to an entangled state • = U(~ ® ¢), a state which is not of product form. Moreover, this entanglement will persist once the subsystems become dynamically independent. The correlation inherent in such a state manifests itself in the existence of pairs of observables A and Ac which are strongly correlated (Sect. 4). The quantum nature of these correlations is due to the purity of the state • and shows itself in the nonobjectiuity of A and AE (Sect. II.2.6). Such pairs of observables are constructed by making use of the biorthogonal decomposition of the state ~, = ~)~i'Ti ® ~?i. Here {~/i} and {rli} are (or can be extended to) orthonormal bases of the Hilbert spaces of S and C and, as such, define pairs of discrete sharp ! observables A - E aiP["/i], A~ = ~] aiP[~i ] which are in fact strongly correlated but not objective in ~. It follows that the system S does not admit an exhaustive description independently of its environment $. On the contrary, any manipulation performed on t~ will in general lead to changes on S. In this situation the only way to break the entanglement seems to be the introduction of superselection rules (Sect. 6). Some authors even conclude that quantum mechanics, rather than being a universal theory, needs to be extended into

28

III. The Quantum Theory of Measurement

a more general theory, in particular, because it does not allow for the description of classical systems. If one intends to maintain the universal validity of quantum mechanics, then one seems to be facing the conclusion that both the preparation of objects and their restoration after their interaction with a measuring apparatus are at best only approximately realisable. On the other hand, there are superselection rules, such as the symmetrisation rules for compound systems consisting of identical systems, which seem to be valid to a high degree of accuracy. In such cases there is a tendency to incorporate statements concerning the existence of superselection rules into quantum mechanics and to be content with the (possible) universality of the thus extended theory. In our opinion it cannot be decided at present whether the notion of isolated quantum systems really requires the environment to be purely classical. In nature there are only very few interactions 'available', so that it may as well be sufficient to have only 'a few' superselection rules (referring to classical observables of the environment) in order to avoid entanglement. This possibility can only be decided on the basis of a (not yet existing 'final') theory of the fundamental interactions. These facts sharpen the difficulties in understanding the objectification of measurement outcomes. Firstly, the formation of correlations between object system and apparatus is necessary for obtaining information in a measurement; but at the same time, objectification requires the restoration of the independence of the systems after the measurement. In the simplest conceivable case, the measurement interaction between an object system S and a measuring apparatus ,4 leads to a state • of the form given above such that the observable to be measured (A) and the pointer observable (An) are strongly correlated. The objectification problem (Sect. 6) is then nothing but the question of how the pointer observable can assume a definite value (pointer objectification) which would indicate a value of the measured observable (value objectification). In the present simple model case, the process of objectification, accompanied with the reading of the actual result, would leave the system S + ,4 in one of the product states .'yi ® ~/i. It is worth recalling that the pioneers of quantum mechanics were well aware of these problems. There are detailed formal expositions in [Paul/ 1933], [Schr5 35], or Iron Neumann 1932] and various proposed remedies, like von Neumann's recourse to the observers' consciousness, Heisenberg's 'cut', or Bohr's stressing the priority of classical concepts for the description of measurement devices. On the other hand, Einstein, Podolsky and Rosen [Eins 35] tried to base an argument against the completeness of quantum mechanics just on the existence of entangled quantum systems. Some of these, as well as the more recent approaches, will be briefly discussed in Chap. IV. III.1.2 M e a s u r e m e n t Let us consider an observable E of the object system S. The designing of a measurement of this observable requires specifying a measuring apparatus .,4 with Hilbert

III.1. General description of measurement

29

space ~/4 and a pointer observable P4 of ,4 to be correlated with E. It may be useful in the first instance to allow the value space (f~4, Jc4) of the pointer observable P4 to differ from that of the measured observable. In that case one needs a (measurable) function f : f~.4 --* f~ correlating the value sets of the two observables. We call this function a pointer function. Let T E S(7~s) and T.4 E 8(7-/4) be the initial states of S and ,4. The initial state of S + A is then T ® T4, since we assume that prior to measurement S and ,4 are both dynamically and probabilistically independent of each other. A measurement coupling between S and A shall be described as a state transformation T ® T4 ~ V(T ® T.4) of the compound system S + A. The final state V(T®T.4) of S + A determines uniquely the final states of the subsystems S and ,4 as the reduced states 7~s (V(T ® T4)) and 7~4 (V (T ® T4)), respectively. Hence, in particular, the probability measure of the pointer observable P4 in the final state of the measuring apparatus 7Z~ (V(T ® T.4)) is completely determined. It is convenient to collect the basic ingredients of the description of a measurement into a 5-tuple (7-l.4, P4, T.4, V, f}, which we shall call a measurement scheme for the system S. It should be noted that the term apparatus must not always be understood in a literal sense. In most measurement theoretic models one includes only some microscopic part of the macroscopic device into the description, which could be referred to as a probe system. Hence when speaking of an apparatus we shall leave open the possibility that its 'pointer' is an observable of a quantum mechanical probe system. As pointed out in the Survey, the minimal interpretation of the probability measures pE requires the probability reproducibility condition as the first basic condition which makes a measurement scheme (7-l.4, P4, T.4,17, f) qualify as a measurement of E: the pointer observable P4, with the function f, and the final state 7~.4 (V (T ® T4)) of the measuring apparatus must reproduce the probability measures pE, pE(x) = PT~(V(T®T~)) P~ (f-X(X)) (1) for any value set X E ~-, and for all possible initial states T of S. We observe that the pair (P4, f) defines another pointer observable PA/ [on the value space (f~, 9r)] via the relation PIA(X ) = P 4 ( f - x ( X ) ) , X E ~'; hence the probability reproducibility condition (1) can equivalently be written as

pE(X ) _ t'n.a(V(T®T.a)) ..P1 (X)

(2)

for all X E 9~, T E S(7-/s). This shows that the value spaces of the measured observable and the pointer observable can always be identified by an appropriate resealing of the pointer. If this is the ease we shall simply write {7-/4, P4, T4, V) instead of (7-/4, P.4, T.4, V, ~), where ~ denotes the identity function on ~. The measurement schemes (7-/4, P4, T4, V, f) and (7-/4, P~, T4, V), which lead to the same observable, are equivalent in a sense to be explained later.

30

III. The Quantum Theory of Measurement "

An alternative way of identifying the value spaces of the measured observable and the pointer is provided by the measurement scheme (7-/A, P,4, TA, V). In that case, Eq. (1) determines an observable E with the value space (12A,JzA), that is, -

PT~.4(V (T®TA))

(3)

for any XA E ~'~4 and all T E S(7-ls). The relation between the original observable E a n d / ~ i s / ~ / = E. If such a relation holds between two observables E a n d / ~ , then we say that E is a refinement of E, or E is a coarse-grained version of E. Since E(~') C E(JzA), the two observables are coexistent. The second basic condition for (7-/A, PA, TA, V, f) to qualify as an E-measurement is the requirement that the measurement should lead to a definite result. We take this objectification requirement to entail, first of all, the pointer objectification: the pointer observable should have assumed a definite value at the end of a measuring process. According to Sect. II.2.4, the pointer observable should thus be objective in the final state of the probe. In addition it may happen that also the value objectification takes place, that is, that the measured observable assumes a definite value. We shall investigate the objectification requirement in Sect. 6. A measurement scheme (7-/A, P,4,T,4, V, f) satisfying the probability reproducibility condition (1) will be called a premeasurement of E, in order to emphasise that a measurement of E has to fulfil, in addition, the objectification requirement. The probability reproducibility condition determines whether a measurement scheme is a premeasurement of a given observable. One may wonder whether there are any weaker relations between measurement schemes and observables which could qualify as criteria for more general notions of measurements. Such criteria should still reflect the idea that a 'measurement' of an observable E provides information concerning the initial state T of the object system. Since that information is encoded in the probability measure pTE, the weakest conceivable condition is the following: a measurement scheme is called an E-discrimination if for any two states T , T ' of $, pE # pE, implies P~.a(V(T®T.aj) P.4 P.a (V(T'®T.a))" This requirement, ~ PTe.4 which was formulated by Fine [Fine 69] [though in a different terminology and for discrete sharp observables only], is surely fulfilled for any premeasurement of E. But it will not in general single out any particular observable as the one measured by the given measurement scheme. In fact a discrimination of any informationally complete observable (of. Sect. 9) is a discrimination of any other observable as well. One could go on to define an E-filter as an E-discrimination with the property that for any two states T, T' ' pE -- pE, if and only if PT~(V(T®T~)) P~ P~ = PT~(V(T'®T.a))" Still this definition does not specify just one observable E as the one determined by a filter. An E-filter is an F-filter for any observable F which is informationally equivalent to E, i.e., for which pTF -- pF, iff pT~ -- pE, for any pair of states T, T'. By contrast, the concept of premeasurement based on the probability reproducibility condition (1) is just sufficient to define, in a unique way, the measured observable.

III.2. Premeasurements

31

III.2. P r e m e a s u r e m e n t s III.2.1 P r e m e a s u r e m e n t s a n d S t a t e T r a n s f o r m e r s It is a basic result of the quantum theory of measurement that for each observable of a physical system there are premeasurements, and even unitary premeasurements (Sect. 2.2). In order to formulate these results and to study some further properties of premeasurements we assume that the map T ® TA ~ V (T ® TA) induced by the measurement process preserves the convex structure of the set of states. In other words, we assume from now on that the state transformation V in the premeasurement (7-/A, P.4, TA, V, f} is a trace-preserving positive linear mapping, that is, a trace-preserving operation V : T(7-ls ® TIA) --+ T(Tls ® TI.4). Such a premeasurement shall be denoted Ad. An explicit construction of a premeasurement for discrete sharp observables has been known since von Neumann's work [yon Neumann 1932]. In its present most general form the mentioned existence result is due to Ozawa [Oza 84]. Important intermediate contributions are summarised in the monographs [Davies 1976] and

[Kraus 1983]. Any premeasurement M of an observable E determines a state transformer, that is, an operation valued measure ZM " 9r ~ C.(T(7-ls)) through the relation

(, ®

®

®

for all X E ~, T E T(7-ls). The state transformer Z ~ summarises all the features of the premeasurement j~i that pertain to the object system S. It reproduces the observable E via the equations pE (X) = tr [:/~ (X)T]

(2)

for all X E ~, T E S(?-ls). Further, it gives the non-normalised final states Z ~ ( X ) ( T ) of S, on the condition that the measurement has led to a result in X. In fact, for any observable A of $, the number tr[AZ~(X)(T)]/pE(X) is the conditional expectation of A, given that the pointer reading has been in the value set X (so that pET(X) ~ 0); hence the normalised final state is Z~ (X) (T) /pE (X). In particular, ZM(f~)(T) = Tis (V(T ® TA)) represents the state of S after the measurement but before reading the result. This interpretation of the state transformer ZM presupposes, however, that the objectification has been ensured. The relation between measurement schemes and induced state transformers is many to one. This is a reflection of the fact that different apparatus may effect the 'same' measurement, or that for one apparatus there may be different ways of carrying out the pointer reading. Therefore it is convenient to identify the class of measurement schemes yielding the same state changes for the object (and thus the same measured observable): any two measurement schemes A/t1 and A/i2 of

32

III. The Quantum Theory of Measurement

E called (operationally) equivalent if the induced state transformers are the same, that is, Z ~ I - Z~2. This holds, in particular, in the case of the premeasurements (U~, P~, T~, V, f) and (U~, P~, T~, V). The relation between state transformers and the measured observables is many to one as well. This corresponds to the possibility that different measurements of one and the same observable may have quite different effects on the measured object. The observable induced by a state transformer Z is called the associate observable of Z, and any state transformer inducing a given observable E is called E-compatible. The present framework offers thus a notion of measurement that is wider than the one quite commonly used since the appearance of von Neumann's book: it does not identify the term 'measurement' with 'repeatable measurement'. At this point it is possible to give a simple demonstration of the fact that a quantum mechanical measurement must induce some state changes if it is to provide any information at all. In fact assume that there is a state transformer Z which leaves unchanged all states T of 8, in the sense that :T(~)(T) - T. Let T = P i l l , and take any X e jc. Then Z ( ~ ) ( P I l l ) = Z(X)(PImp]) + ~ ( ~ \ X ) ( P I l l ) gives rise to a convex decomposition of P I l l . This implies Z(X)(P[~]) - A(X)P[~], where due to the linearity of the map Z(X) the function A does not depend on ~. The associated observable E is then given by E ( X ) - A ( X ) I , and pTE -- A is independent of T. An observable of this form will be called trivial We conclude that no premeasurement leaves unchanged a/1 states of the object system unless the measured observable is trivial. An important example of a state transformer can be constructed for a discrete observable, that is, a POV measure E on (the power set of) a finite or countable set ~ - {wl,w2,...}. We denote E({wi}) - : Ei. Then the following defines an E-compatible state transformer: for all X C ~ and all T, let

Z(X)(T) := Z E~/2T E~/2.

(3)

wiEX

This state transformer shall be called the L/iders transformer. In the case of discrete sharp observables (where E~/2 = E~) it will be found to be uniquely singled out by the requirement of ideality (cf. Sect. 4.7). Model realisations of unitary measurement schemes inducing the Liiders transformer will be given in Sects. 2.5 and 2.6. More generally, for any observable E and any value set X one may define the associated Liiders operation acting on a state T as

+ x ( T ) := E ( X ) 1/2 T E ( X ) 1/2.

(4)

This form of a state transformation plays an important role in formulating necessary conditions for the pointer objectification (cf. Sects. 2.4 and 6.1). Note also the

III.2. Premeasurements

33

appearance of a Liiders operation in Eq. (1), which was chosen there in order to make the argument of the partial trace operation a manifestly positive state operator.

III.2.2 Unitary Premeasurements It might seem natural to assume that the measurement mapping T ® T~4 ~-* V(T ® TA) would preserve the extremal points (vector states) of S(7-ls ® 7-l.4), so that V would be a pure operation. Since the structure of pure operations is completely known [Davies 1976], one could explore further the corresponding premeasurements. Rather than following this option (cf. [Belt 90]), we consider the more restricted class of premeasurements generated by unitary maps. Most treatments of the quantum theory of measurement in the literature are based from the outset on unitary premeasurements, for which the measurement coupling V : T(7-ls ® 7-l.4) --* T(?-ls ® 7-/,4) is unitary, that is, V(W) = UWU*, W E T(?-ls®?-l~4), for some unitary operator U : 7"/s®7-/.4 --* ~/s®~/~4. In addition, it often occurs that the pointer observable can be taken to be a sharp observable and that the apparatus is initially in a vector state ¢. Moreover, the pointer observable may always be so chosen that it has the same value space and scale as the measured observable (so that the pointer function is the identity). For unitary premeasurements with these properties we use the notation M u := (7-l,4,PA, ¢, U), with the understanding that P,4 is a PV measure on (f~, jr). If A/tu is such a premeasurement of an observable E, the probability reproducibility condition assumes the simple form

(7~JE(X)~) = (U(7~ ® ¢)[I ® P.4(X)U(~ ® ¢))

(5)

for all X E 9r and for all unit vectors 7~ E ~/s. Since any mixed state can be expressed as a a-convex combination of some vector states, it follows that (5) entails the probability reproducibility condition for all states. From the outset, the notion of state transformer is more general than that given above. A state transformer is a (normalised) operation valued measure Z : ~" --* £(T(7-ls)) on a measurable space (f~, ~') [Davies 1976]. Any such state transformer determines a unique observable E " j r ~ L(7-/s) via the relation tr[TE(X)] tr[Z(X)(T)] for all X E jr, T E T(?-ls) (cf. Eq. (2)). The observable E is the associate observable of Z. As will be seen shortly, every observable E with a value space (~,~') is the associate observable of at least one state transformer. Such state transformers are called E-compatible. In fact for each observable E there is a unique family of E-compatible state transformers Z. A state transformer Z" jr ~ f~(T(7-ls)) is completely positive if all the operations Z(X), X E ~', in its range are such. An operatio n • : T(7-ls) --. T(7-ls) is completely positive whenever its canonical extension • ® ~ to T(7"ls ® C n) is positive for all n - 1, 2 , . . . (5 denotes the identity operation). An equivalent characterisation states that ¢ is completely positive if and only if it can be represented

34

III. The Quantum Theory of Measurement

as ~(T) = ~,~ AnTA~, T E T(7-/8), for some countably many bounded linear operators An on 7-/8 [Davies 1976]. Since completely positive state transformers are induced by unitary premeasurements [Kraus 1983, Oza 84], it is important to recognise that for each observable E there exist E-compatible completely positive state transformers. Indeed let (Xi)ieI be a countable partition of f~ into disjoint sets from $- and (Ti)iex a collection of states. Then Z(X)(T) ~iEI t r [ T E ( X MXi)]Ti defines an E-compatible state transformer which is completely positive. It may also be noted that the Liiders operation and state transformer introduced in Eqs. (3), (4) are completely positive. Consider next a premeasurement M of an observable E. The state transformer Z ~ induced by this premeasurement is completely positive whenever the measurement mapping V, or at least its restriction V I~r(Us)®[T~], is completely positive [Bus 90a] We summarise the above discussion as follows: =

THEOREM 2.2.1. For any observable E there exists a unitary premeasurement M u

such that <~]E(X)T> -

(U(T ® ¢)]I ® PA(X)U(~ ® ¢))

for all X E 9r, ~ E H s , ][ ~o ]1= 1. A premeasurement M ' of an observable E is equivalent to a unitary premeasurement M u of E if and only ff the induced state transformer ~ , is completely positive. An E-compatible state transformer is completely positive exactly when it is induced by a premeasurement M ' of E for which V'IT(~s)®[T~ ] is completely positive.

III.2.3 Calibration Condition and Probability Reproducibility It was pointed out in the Survey that under certain circumstances the probability reproducibility condition (1.1) can be reduced to a seemingly more elementary requirement. In fact a minimal demand a measurement should fulfil is that it should exhibit unequivocally what is the case; that is, if an observable has a definite value, a measurement of that observable is supposed to determine its value. This requirement will only be meaningful if the observable can assume definite values; that is, for each X E 3c the effect E ( X ) should have eigenvalue 1. Therefore we restrict our attention to sharp observables in formulating the non-probabilistic calibration condition. A measurement scheme M will be said to satisfy the calibration condition with respect to a sharp observable E if for any X E $- and any state T E S(7-ls) the relation E ( X ) T - T implies I ® P . ~ ( f - I ( X ) ) V ( T ® T~) - V ( T ® TA). This will be the case if M is a premeasurement of E, since then pET(X) -- 1 is equivalent to pI®P.a V(T®Ta) ( f - l ( X ) ) - 1; but the converse is also true [Bus 96a]. THEOREM 2.3.1. Let E be a sharp observable of the system S. A measurement scheme M is a premeasurement of a sharp observable E in the sense of the probability reproducibility condition with respect to E i[ and only if M satist~es the calibration condition with respect to E.

III.2. P r e m e a s u r e m e n t s

35

This result shows that the probability reproducibility condition represents one straightforward way of generalising the calibration requirement from PV measures to POV measures.

III.2.4 R e a d i n g of Pointer Values The recovering of the probabilities pE(x) as relative frequencies of pointer values requires that such pointer values do become realised at the end of a measuring process. This will be the case if a measurement scheme satisfies the calibration condition and the observable to be measured has itself a definite value prior to measurement. If the measurement coupling V is generated by a unitary map, then the objectification problem arises, which consists of the incompatibility of the unitary measurement dynamics with the occurrence of definite pointer values in cases where the measured observable is indeterminate. In spite of the determinism of the unitary time evolution, the very fact that quantum mechanics ascribes probabilities to measurement outcomes seems to entail that the object as well as the apparatus undergo indeterministic state changes that can be described as stochastic processes. We shall deal with the objectification problem as the problem of explaining the possibility of such stochastic processes in greater detail in Sect. 6. Here we develop the tools needed for describing the state changes of S and A conditional upon the reading of a definite pointer value. In Sect. 2.1 we have already introduced the state transformer Z associated with a premeasurement M of an observable E. The idea of a measurement as a stochastic process is represented by Z in the sense that for each initial state T of S, Z(X)(T) is the (non-normalised) state of ,S after the measurement conditional upon reading an outcome (pointer value) in f - 1 (X). The probability for this transition is tr[Z(X)(T)] - pE(x). As shall be explained shortly, the requirement of objectification makes it necessary that the final apparatus state conditional upon reading an outcome in f - l ( X ) is similarly given by TiA(I®PfA(x) 1/2 V(TQTA)I®PSA(X)I/2). We introduce the following notations for the normalised final component states of S and A:

Ts(X,T)

"= p E ( x ) - I

Tis(I ® PfA(x) 1/2 V(T ® TA) I ® PfA(x) 1/2)

-- RE(x)-Iz(X)(T),

T a(X,T) "- pE(X)-I Ti.a(I @ PfA(x) 1/2 V(T @TA)I @ P~(X) 1/2) =

pE(x)-I P~(X) 1/2 T4,.A(V(T @TA)) P~(X) 1/2,

(6b)

whenever pE (X) ~ 0. If pE (X) - 0, it is occasionally convenient to use Ts (X, T) 0 and T.a(X, T) = O. Reading measurement outcomes amounts to discriminating between the elements of a finite (or, as an idealisation, countable) set of alternative pointer values. In order to formulate this idea in the most general context of an E-measurement

36

III. The Quantum Theory of Measurement

¢~4, we introduce the notion of a reading scale as a countable partition of the value space of the pointer observable, ~A = Uf-l(Xi), induced by a countable partition of the value space of the measured observable, ~ = UXi, Xi E ~, Xi N Xj = 0 for i ~ j. Such a reading scale will be denoted T~, and we let I denote its index set. In general I = N, or occasionally I = Z. If I = ( 1 , . . . , N ) , we shall call T~ a/inite reading scale. The notion of a reading scale will be important in studying the correlation properties of measurements (Sect. 4), and it will turn out crucial to the formulation of the objectification requirement (Sect. 6). Indeed it appears that the registration and reading of measurement outcomes are carried out necessarily with respect to a given reading scale. A reading scale :R determines discrete, coarse-grained versions of the pointer observable P.4 and the measured observable E:

P~ "i ~ Pi "= P A ( f - l ( x i ) ) , i e I,

(7)

E n : i ~ Ei := E(Xi), i e I .

(8)

The P~-value i refers to the pointer reading f-I(X~) which, in turn, is correlated to the value set Xi of the measured observable E. If E is discrete, there is a natural (finest) reading scale T£ such that E = E n and PA = P~. The premeasurement A/[ and the reading scale ~ define another premeasurement A/[T~ - (7-/A,P ~ , TA, V) determining the discrete observable E T~. The corresponding conditional final component states of S and A, given in Eqs. (6a), (6b), shall be denoted

Ts (i, T) := Ts (Xi, T),

(9a)

T~4(i, T) := TA (X~, T),

(95)

The interpretation that TA(i, T) is the state assumed by A on the condition that the pointer observable has its value in f - l ( X i ) follows if the pointer objectification is postulated. Indeed the objectivity of the pointer in the state T ~ (V(T ® TA)) -TA (f~, T) entails that this state represents a Gemenge of 1-eigenstates of the Pi. According to Theorem II.2.4.1, the (normalised) eigenstates are TA (i, T), and TA (~, T) is a mixture of these states. The objectification requirement implies thus the following two conditions on the premeasurement ~ [ n . First, the pointer observable P ~ must be real in the state TA(i, T). Clearly this is the case if the pointer is a sharp observable. In order to leave open the possibility of more general pointer observables, we shall refer to this requirement as the pointer value-definiteness condition for ,4 with respect to a reading scale T~, written in either of the following equivalent forms:

tr[T~(i,T)Pi] = l (whenever pET(Xi) ¢ O), P~ T.4(i, T) = TA(i, T),

(10) (11)

III.2. Premeasurements

37

for all i E I and all initial states T of S. Second, the state TA(f~, T) must be a mixture of the pointer eigenstates TA(i,T). We call this demand the pointer mixture condition for .4 with respect to a reading scale T~:

7~A(V(T ® TA)) = ~ p E ( X i ) T A ( i , T )

(12)

for all initial states T of S. This property of a premeasurement is connected with the associated state transformer [Bus 96b]. THEOREM 2.4.1. Let A4 be a premeasurement of an observable E and T~ any reading scale. For any initial state T of the object system, a) implies b) and c)"

a)

Ts(i,T) . Ts(j,T) = 0

for i ~ j;

b) T~A (V(T ® TA)) = EPT(Xi)TA(i, E T) c)

PiTA(i, T) = TA(i,T)

for all i;

for all i.

If A4 is a unitary premeasurement A4u, then a) and b) are equivalent for any initial vector state T = P[~] of S. The pointer mixture condition b), together with the pointer value-definiteness condition c), ensure a decomposition of the final apparatus state in terms of pointer eigenstates; this means that the final apparatus state 7~A (V(T @T~t)) is conditionalised with respect to the reading scale 7~ [Cas 84]. The mutual orthogonality a) of the component states of S, Ts(i, ~). Ts(j, ~) O, corresponds to the optimal distinguishability of these states, which is mandatory for making unique inferences from the pointer readings on properties of the measured object. These orthogonalities are realised in an important subclass of measurements, the repeatable measurements, which will be discussed in Sect. 4. A repeatable measurement drives the object system S into an eigenstate of the measured observable E, so that the following eigenstate conditions holds:

EiTs(i, T) - Ts(i, T) for each i and T such that pE(Xi) ~ O. In that case one can infer from a pointer reading i that the measured observable has a value in Xi after the measurement. It is interesing to note that the implication b) =~ a) does not hold in general if the measurement is not unitary or if the initial pointer state TA is not pure. This becomes evident in the following examples. Let ?/s and 7-/A be two-dimensional, and let {~1, ~2} C ~'~S, {¢1, ¢2} C ~'~S, and {¢1, ¢2 } C 7-/.4 be orthonormal bases. Consider two unitary premeasurements generated by the unitary maps U, U' and the same pointer observable 1 ~-~ P1 P[¢1], 2 ~-. P2 - P[¢2], where

V((cl~l + c2~2) ® ¢) - c1~1 @ ¢1 + c2~2 ® ¢2, u'

+

® ¢) -

1¢1 ®

+

®

(14)

38

III. The Quantum Theory of Measurement

The measured observable E is given by E1 - P[Th], E2 - P[7~2] in both cases, and the reduced apparatus state is

Ti,4(U(T ® TA)U*) - TiA(U'(T ® TA)(U')*) = Icll P[¢1] + Ic212 P[¢2]

(15)

(where TA = P[¢]). We define a nonunitary state transformation V for S + A as a mixture of the maps (14):

V(T @ TA) = AU(T @ T,a)U* + (1 - )~)U'(T @ T.a)(U')*,

(16)

with 0 < A < 1. Taking the same pointer observable as above, the resulting measurement scheme is again a premeasurement of E. The reduced apparatus state TiA(V(T @ TA)) coincides with (15). Hence it commutes with the Pi so that b) of Theorem 2.4.1 is fulfilled. But one finds

Ts(i, T) = )~P[7~i] + (1 - )~)P[¢i]

(i = 1, 2),

(17)

so that Ts(1,T).Ts(2,T) ¢ 0 unless (Th 1¢2 ) = (¢1 17~2) = 0. Thus a) is violated in general. Next let 7-/s and 7{.4 be two- and four-dimensional, respectively, with the same basis systems for 7-/s as given above. Let {¢, ¢'}, {¢1, ¢2, ¢3, ¢4} be orthonormal systems of 7-/A. Consider a unitary mapping

V((clTh + c2~2) ® ¢) = c17h ® ¢1 + c2~2 ® ¢2,

V((Cl~l "-~c2~2)

@ ¢')

(18)

-- Cl~)l ® ¢3 "~- c2~22 ® ¢4.

Let a unitary measurement scheme Ad be defined by U, the initial apparatus state TA = AP[¢] + (1 - A)P[¢'], 0 < A < 1, and a two-valued pointer observable PA with/91 = P[¢1] + P[¢3], P2 = P[¢2] + P[¢4]. The measured observable is again i ~ Ei = P[Th], and the states Ts(i, T) are the same as in the preceding example, i.e., as given in (17). Hence a) is again violated while b) is trivially fulfilled.

III.2.5 Discrete Sharp Observables The concepts introduced so far shall be illustrated in the important case of a discrete sharp observable A - ~ aiEA({ai}). Let g be the (finite or infinite) number of the eigenvalues of A and {7~ij} be a complete orthonormal set of eigenvectors of A, AT~ij -- aiThj, where for each i the index j runs from 1 to n(i), the dimension of the eigenspace E A ({ai})(7{s) (finite or infinite). We may now construct all unitary premeasurements A/Iv of A with a particular choice of the pointer observable. To this end we consider a measuring apparatus (or probe system) ,4 with a Hilbert space 7-/~4 of dimension equal to N, the number of distinct eigenvalues of the measured observable. Let {¢i : i = 1,... , N } be an orthonormal basis of 7-/,4, and define a pointer observable as the discrete nondegenerate quantity A~4 = ~ a~P[¢d with the same eigenvalues as A. The unitary premeasurements of A with such pointer observables can then be completely characterised [Belt 90].

III.2. P r e m e a s u r e m e n t s

39

THEOREM 2.5.1. A measurement scheme (7-/.a, AA,¢,U) with a discrete nondegenerate pointer observable AA - E ff aiP[¢i] is a unitary premeasurement of the discrete sharp observable A - ~-~ij aiP[~j] if and only if U is a unitary extension of the mapping ~ij ® ¢ ~ ¢ i j ® ¢i (19)

where {¢ij } is any set of unit vectors in 7-ls satisfying the orthogonality conditions (¢ij[¢ik) = 5jk for all j, k = 1 , . . . , n(i), for each i - 1 , . . . , N, and dim {~ij®¢} ± = dim {¢ij ® ¢i } ±. The unitary premeasurements of A thus specified shall be called minimal since the pointer observable is minimal in the sense that it is just sufficient to distinguish between the eigenvalues of A. It is straightforward to determine the state transformer associated with a minimal unitary premeasurement A/I~ := (~.4, A.4, ¢, U) of A. Let ~ = Y~ Cij~Oij be the initial vector state of S, so that U(~ ® ¢) = ~ cij¢ij ® ¢i ij

(20)

is the final state of $ + J[. We define A

-

-½

¢ij

(21)

,

J whenever p~A (ai) ~ 0, and 7i - 0 otherwise. Then (20) assumes the form

i

and the final states of S and A are: A

7~s (P[U(~ ® ¢)]) - E P~ (ai)P[q'i],

(23)

i .....

7~A(P[U(~ ® ¢)]) - E

~/pA(ai)pA(ak) (Yil'~k)I¢k)(¢il.

(24)

ik

The final component states are now simply Ts(i, ~) = P[~/i] and T.a(i, ~) = P[¢i], and the state transformer Zu induced by A/I~ has the form

Zu(X)(P[~])-

E i:aiEX

A

p~(ai)P['~i]-

E

*

KiP[q°lKi'

(25)

i:aiEX

with Ki := Y~z [¢iz)(~iL[ E £(7-/s). If V and U' are two different unitary extensions of (19), then the resulting A-measurements are equivalent, that is, the state transformers Zu and Zu, are the same.

40

III. T h e Q u a n t u m T h e o r y of Measurement

Next we describe various important types of the minimal unitary premeasurements of A as well as some further unitary premeasurements induced by them. There are three particularly interesting choices of the set {¢ij} in Theorem 2.5.1. First, the set {¢ij} may be an orthonormal system of unit vectors. In that case also the vectors ~/i are orthogonal (for each ~o E ~/s) and the decomposition (22) is biorthogonal (for each ~o e ?'/s). The natural decompositions (23) and (24) of the reduced states of S and ,4 are then the spectral ones:

T~s(P[U(~o ® ¢)1)

=

E p ~ A (ai)P[~/i],

(26)

i

T~.4(P[U(~o ® ¢)]) =

A EP~(ai)P[¢i].

(27)

i

In Sect. 4.4 we shall see that this choice of the generating vectors ¢ij corresponds to an A-measurement Ad~ which gives rise to strong correlations between the final component states ~/i and ¢i of S and A. Another, more restrictive possibility is that the set {¢ij} is a complete orthonormal set of eigenvectors of A such that A¢ij = ¢ij. Then for each ~o E 7-/s, also A'~i = ai~/i. It turns out (Sect. 4.3) that this choice of the set {¢ij} is characteristic of measurements which lead to strong correlations between the values of the measured observable and the pointer observable. Finally, the set {¢ij } may be chosen as {~oij}. A characteristic feature of the related A-measurements is their ideality (Sect. 4.7). Since this type of (pre)measurements is most often discussed in the literature, we list some of their properties here. Let UL be a unitary operator on 7-/s ® 7{.4 which has the restriction ® ¢) =

(2s)

®

The final states of S and A are then

ns(P[UL(~O ® ¢)])

= E

P~A(ai)P[qOa,],

(29)

i

T~A(P[UL(~O® ¢)1)

=

E p ~ A (ai)P[¢i],

(30)

i

where ~oa, :- p~A (hi) - 1 / 2 Y~j cij~oij, if case the state transformer ZL is

ZL(X)(P[qo]) = E i:ai6X

p~A (hi) 7~ O, and

A p~(ai)P[~oa,]-

E

~oa, = 0 otherwise. In this

EA({ai})P[~°lEA({ai})"

(31)

i:ai6X

The unitary mapping (28) in the premeasurement (7-/4, A.4, ¢, UL) of A was already discussed by von Neumann [von Neumann 1932], whereas the state transformer (31) induced by that measurement was studied in greater detail by Liiders [Liid

III.2. Premeasurements

41

51]. In the literature this measurement is called sometimes a Liiders measurement, sometimes a von Neumann-Liiders measurement. Its operational, probabilistic, and information theoretical characterisations will be reviewed in Sects. 4 and 5. We choose to call this particular premeasurement (7"IA,AA, ¢, UL) of A its L/iders (pre-)measurement and the induced state transformer the L/iders state transformer. There are other classes of unitary premeasurements of (a degenerate) A that can be obtained from Theorem 2.5.1 by means of the following method. Let B be any refinement of A, so that A = f(B) (or E A = E s,y) for some function f. If A/I~ is a premeasurement of B, then (7-lA, An, ¢, U, f) is a premeasurement of A (cf. Sect. 2.1). In particular, if the premeasurement of B is a Lfiders measurement, then the resulting A-measurement will be called a yon Neumann (pre-)measurement. Accordingly the induced state transformer will be referred to as a yon Neumann state transformer. In this view Liiders measurements are but a special class of von Neumann measurements. They turn out to be the ideal ones (Sect. 4.7). Especially, if B is a maximal (meaning nondegenerate) refinement of A, then the resulting von Neumann state transformer of A is

z,A(x)(P[~]) -

~

~ P[~jIP[~IP[~j] = z ~ ( / - ~ ( x ) ) ( P I l l )

i:ai EX

-- Z

(32)

j

Z

EB({bij}) P[qp]EB({bij})"

alEX f(bij)=ai

For the sake of clarity, we have indicated here the observable associated with the state transformer in question. This measurement was also discussed by von Neumann [yon Neumann 1932]. It should be noted that in general the final state of the object system S after a v o n Neumann measurement of A is not the same as the state reached after a Liiders measurement of A. III.2.6 The Standard Model Suppose that one intends to measure an observable A of the system S by coupling it to some observable B of another system, the apparatus ,4, through an interaction

U -- ei~A®B,

(33)

where ~ is an appropriate coupling constant. Using the spectral decomposition of A, one may write U in the form

U = /R EA(da) ® eia~B"

(34)

Then an initial state 7~® ¢ of the compound system is transformed into (35)

42

III. The Quantum Theory of Measurement

where we have defined ¢ ~ := eia~B¢. If the coupling U is to serve its purpose one needs to choose the initial apparatus state ¢, the pointer observable PA, and a pointer function f such that (7-/.4, Px, ¢, U, f) constitutes a premeasurement of A. With any specific choice of ¢, PA, and f, the probability reproducibility condition always defines the observable E actually measured by this scheme. To evaluate this condition, one first determines the final state of the apparatus,

nA(P[U(T®¢)])

=

/R p~A(da) P[¢~al •

(36)

In view of the coupling constant A (# 0) it is convenient to introduce a pointer function f(x) = A-Ix. Then the measured observable takes the form

E(X) = /I~ p¢~. P~ ()~X) E A (da).

(37)

The structure of the effects E(X) show that in general the actually measured observable E is not the observable A, but an unsharp version of it [Busch, Grabowski, Lahti 1995]. The question then is which choices of ¢ and P x would possibly yield E = E A, that is,

/R P¢~a P~ ()~X) EA(da) --

EA(x)

for all X e B(R).

(38)

This is the case exactly when for (EA-almost) all a E R, P¢~aP'4()~X) -- Xx (a), with Xx denoting the characteristic function of the set X. Since (¢)~a ]PA(AX)¢xa ) (dp]e-ia~BpA(AX)eiaABdp), one may expect that an 'optimal reading' can be obtained by choosing the pointer PA to be conjugate to B in the sense of covariance: =

-

(39)

The general form of the state transformer defined by this measurement can also be given explicitly. Its structure depends on that of the pointer observable Px. Writing PA()~X) = fax Ix )( xl dx, with a formal resolution of identity, one obtains

Z(X)(T) - f~ Kx T K~ dx,

Kx - / R (x I¢~a ) EA(da).

(a0)

x

Without using this formal expression, one can directly confirm that the measurement outcome probabilities for E, as well as for A, are the same both before and after the measurement: the measurement does not alter these probabilities. It is instructive to work out two special cases of this model. Consider first a discrete observable A - ~ akE A, and assume that the set of eigenvalues of A is closed. As the apparatus take a particle moving in one-dimensional space, so that

III.3. Measurement and Probability

43

~A = L2(R), and couple A with its momentum HA: U = exp (-iAA ® HA). Since the momentum generates translations on the position, it is natural to choose the position QA conjugate to HA as the pointer observable. An initial state ~a @ ¢ of the compound system S + A is transformed into ~--~kEA~ @ ¢~ak. In the position representation (for ,4) one has ¢~ak (X) -~ ¢(X- Aak). Assuming that the spacing between the eigenvalues ak is greater than ~ and that ¢ is supported in ( - ~ , ~), then the pointer states ¢~ak are supported in the mutually disjoint sets )~Ik, where Ik -- ( a k - 2~, ak + 2~)" Introducing yet another pointer function g such that g ( I k ) - {ak} for each k, and g((Uklk)') C {ak" k - 1,2,..-}', then E({ak}) -- E

(¢£a' I EQA()~Ik)¢~,a, ) E A -

E A,

(41)

for each k, which shows that the observable measured by this scheme is indeed A. The state transformer of this measurement is the Lfiders state transformer. In the case of a continuous observable, such as position, the above model amounts to measuring not the position but an unsharp version of it. In fact taking A = Q and adopting the respective spectral representations, one has U(9~ ® ¢)(q, x) = ~ ( q ) ¢ ( x - ~q). Then I ¢ ( q ' - Aq)l2 X),x (q')dq' EQ (dq) - Xx * e(Q),

E(X) - / /

(42)

where the function (Xx * e)(y) = f X x ( x ) e ( y - x)dx is the convolution of the characteristic function Xx with the confidence function e(x) "- AI¢(-Ax)] 2. Since e cannot be a delta-function, the measured observable E is never the sharp position but rather an unsharp one. The ensuing state transformer IQ is also easily obtained:

:£Q (X)T = / x Kq T Kq dq III.3. Measurement

Kq "- V/-f ¢ ( - A ( Q - q)).

(43)

and Probability

III.3.1 Two Problems

According to the minimal interpretation the number pE(x) is the probability that a measurement of the observable E performed on the system S in the state T leads to a result in the set X. The probability reproducibility condition requires that the probability measure pE is transcribed into the probability distribution of the measurement outcomes in a given E-measurement A4"

p

(x) - PT~.a(V (T®T.a)) ( f-1 ( x ) ) .

(1)

The question arises as to whether it is possible to justify, within the quantum theory of measurement, a statistical interpretation of these probability measures in the following sense: if a measurement of an observable E were repeated, under

44

III. The Quantum Theory of Measurement

the same conditions, a sufficient number of times, then the relative frequency of outcomes in the set X would approach the number pT E (X). This question addresses two problems. First, there is the general conceptual problem of connecting the probability measures, defined formally in the theory, with the notion of relative frequency [vFra 79]; here one is asking for a precise formulation and justification of a measurement statistics interpretation for the quantum mechanical probabilities [Cas 89]. Second, one is facing the measurement theoretical problem of giving a quantum mechanical description of the whole process of collecting the measurement statistics. Carrying out this task, which was envisaged by Everett [Eve 57], enables one to prove a quantum mechanical version of the strong law of large numbers, thus providing a justification of the probability interpretation as a statement concerning (approximately) real properties of a sufficiently large ensemble of systems. In this way a statistical ensemble interpretation is obtained for quantum mechanics. The solution to the second problem involves, strictly speaking, the formalisation and analysis of all the physical preconditions for the probability interpretation. In fact one goal of measurement theory is to determine to what degree of accuracy it is possible to 'repeat the same measurement under the same conditions sufficiently many times'; thus it is necessary to investigate the possibilities of preparing 'many' identical and independent (object and apparatus) systems in the same state. Some of these questions, especially the problematics of considering identical particles as independent (in view of the symmetrisation superselection rule), were studied by Ochs [Ochs 80]. In the present context it will be sufficient to survey briefly the answers to the two problems mentioned.

III.3.2 Measurement Statistics Interpretation The measurement statistics interpretation refers to the following intended empirical content of the minimal interpretation of the probability measures pT E" if a measurement of E in a state T has been repeated n times, and the result f - l ( X ) has occurred v times, then limn-.c~ v/n = pET(X)" The difficulties encountered in giving a precise formulation of this idea are due to the facts that relative frequencies are not probabilities, and probabilities need not be relative frequencies. Nevertheless a formal justification of this interpretation can be based on the strong law of large numbers, and it amounts to what has been called the modal frequency interpretation of probability [vFra 79]. When considered within the context of measurement theory, this interpretation allows one to recover all the probabilities pET(X), X E jz, T E S(7-/s), as the relative frequencies of the measurement outcomes obtained in an E-measurement f14 [Cas 89]. The measurement statistics interpretation of the probabilities pE(X) with respect to A4 constitutes a family of discretised pointer observables P ~ from which the probabilities

(2)

III.3. Measurement and Probability

45

for any Xi (in 7~) and for each reading scale 7~ can be obtained as relative frequencies. That is, for each T and for each 7~ there is a sequence FTn of P~-outcomes such that for each i, n

relf(i, FTn)

:=

lim 1 E x { , ,(FTn(j)) = pt'T~.a ~ (V (T®T.a ) ) ({i}) n-.oo n

(3)

j----1

The fact that for each T the family { ( n , FT~) • n a reading scale} forms a good family of special frequency spaces guarantees the consistency of this interpretation [vFra 79]. It is to be stressed that for any state T the possible measurement results in an E-measurement A/[ with respect to a reading scale 7~ are the elements of a sequence FTn of P~-values. They correspond to the values f - l ( x i ) of the pointer observable PA which, in their turn, correspond to the values Xi of the measured observable E. It is another question whether the occurrence of a result i means also that the pointer observable has the value f - 1 (Xi) after the measurement, or even that E has the value Xi after the measurement. Such qualifications depend on some further properties of the premeasurement A/[ (Sects. 2.4, 4, 6, and IV.5.2). We summarise this discussion as follows. THEOREM 3.2.1. Let A4 be a premeasurement of an observable E. For any state

good

T the

o eque.cy

as-

space (gtA , 3cA,Pl~(V(T®T~))), P~ so that, in particular, for any X E jz there is a sequence F~ of P~-values such that s o c i a t e d with the probability

pE(x)

-- P ~P~ A(V(T®T~))

(f-l(x)) = relf(X, rT~)

(4)

We may now illustrate the above measurement statistics interpretation of the probability measures pE in the case of a unitary premeasurement A/[~ of a discrete sharp observable A - ~ aiE A as described in Sect. 2.5. All the involved probability measures are discrete, and a natural (finest) reading scale is the one given by the eigenvalues of the pointer observable AA: 7~ = U{ai}. (Here we assume that the set {ai : i - 1 , . . . , N} of eigenvalues of the pointer observable AA is closed.) With respect to this reading scale we have AA - A ~ , and for each initial state 9~ of S there is a sequence F~ of pointer eigenvalues such that A (x)

-

A.a

-

relf(X,

(5)

for any X E B(R). It is interesting to observe that in the present case of a discrete sharp observable A one sequence F~ per initial state 9~ of S suffices to recover all the probabilities p A ( x ) as relative frequencies.

46

III. The Quantum Theory of Measurement

III.3.3 Statistical Ensemble Interpretation We now turn to the second problem of providing an ensemble interpretation of the quantum mechanical probabilities. In formulating such an interpretation, one considers n runs of the same measurement, performed on n identically prepared copies of the object system $, as one single physical process to be described by quantum mechanics. Regarding this theory as universally valid and complete, one would expect it to be able to predict that in a large system consisting of n equally prepared systems S the relative frequency of any outcome after a measurement would be almost equal to the corresponding quantum mechanical probability. Hence, let S (n) be an n-body system consisting of n identical copies of S: S (n) - $1 + . . . + Sn. The associated Hilbert space is the tensor product Hilbert space 7-/(n) - 7-/1 ® . . . ® 7-/n. A measurement scheme A/[ for S (= $1 = . . . . Sn) can now be extended to a measurement scheme A4 ('0 for S (n) by forming the n-fold tensor products of the constituents of A//. In order to collect the statistics, one needs to fix a reading scale /'4. This gives rise to a discretised measurement scheme A4 n,(n). A typical outcome sequence is ~ - (/1,"" ,ln), with lk 6 I, k - 1 , . . . , n. Denote the spectral projection of Pi - P A ( f - l ( Z i ) ) associated with the eigenvalue 1 as p(1). We assume that the premeasurement A4 fulfils the pointer value-definiteness reqirement [Eq. (2.11)], so that P(I)TA(i,T ) = TA(i,T). (This is the case, for example, when the pointer observable PA is sharp.) Let p ( n ) : - p[:) @ . . . @ p(1) for all ~. One may define for each i 6 I a relative frequency operator [DeWitt and Graham 1973, Har

68] = E S: with the eigenvalues

1 =

n

n

(7)

j=l

Let T °O (~, T) - TA(ll, T ) ® . . . @TA(ln, T).denote the final component state of the n-fold apparatus .,4(n). The eigenvalue equation

F(n)T('0 (f., T)

-

gn)T('~) (f., T)

(8)

shows that the relative frequency of the pointer value i [or f - l ( X i ) ] corresponds to a real property in the final component state of the a p p a r a t u s , a property which is given by the eigenvalue f~n)(~) of t(i n) . The statistical ensemble interpretation of probability, which is to be justified here, states that if one performed a large number of E-measurements, with a fixed reading scale 7~, on systems $ equally prepared in state T, then the relative frequency of the outcomes i, would approach the probability pET(Xi). Hence probability is related again to the situation after the measuring process, that is, to the final states T (n) (f~, T) and T (n) (f~, T) of S (n) and ,4 (n) , respectively.

III.3. Measurement and Probability

47

For notational simplicity, we let pi stand for tr[T,a(gt, T)Pi], and we write

p~n) = Pll "... "Pl,~. The expectation and variance of the frequency operator F (n) in the state T(~n) (f~, T) are

Exp(F(n);T(n)(~,T))

- tr [T(n)(~,T) F (n)] - Pi,

Var(F(n);T(n)(~,T))

- tr[T(n)(~,T)(F(n)-pi)2 = E

( f } ) 2 n) (i) - Pi

p~n)

] -

(9b) 1 - P i (1 - Pi). n

The expressions on the right hand sides are verified by induction with respect to n. The probabilities pi are thus recovered, in the limit of large n, as relative frequencies of the pointer values i. In other words, the uncertainty about the pointer value of the individual apparatus system is turned into a (nearly) determinate property of a large ensemble of apparatus, namely, the (almost) real value of the relative frequency observable. This result can be extended to the object system as follows. There is a unique observable E of which the present measurement scheme A/[ is a premeasurement. The connection between A/I and E is specified by the probability reproducibility relation (1), which implies pi - pE(xi). Similarly, the scheme A/I (n) qualifies as a premeasurement of the observable E (n) = E ® - . . @ E of an ensemble $(n) of n systems S in the state T (n) - T @... ® T. Hence the object probabilities pE(Xi) are tied equally well to the (almost) real frequency values of the ensemble of apparatus. THEOREM 3.3.1. Let A/I be a premeasurement of an observable E such that the pointer value-definiteness condition is fulfilled. For any state T E S(7-ls), any reading scale 7~, and all Xi E T~, one has

Exp(F(n) ; T (~) (Ft, T)) = pE (Xi), lim Var(F(n); T(n)(~,T))

- O.

(10a) (10b)

n----~ OO

According to this theorem the relative frequency of the pointer value i after a premeasurement A/[ 7e'(n) on an ensemble of n systems S in the state T (n) approaches the probability pE(Xi) in the limit of large n. At this point it is important to recall that the approximation of the object probabilities pi by relative frequencies is itself a probabilistic statement, this time involving probabilities about the large ensemble of measuring apparatus. This is a reflection of the fact that the concept of probability cannot be reduced to that of relative frequency. But what is essential to the individual interpretation of quantum mechanics advocated in this text is the fact that on the ensemble level one obtains statements involving probabilities close

48

III. T h e Q u a n t u m Theory of Measurement

to unity, so that the corresponding properties can be ascertained to be 'almost' real to any degree of confidence. To see this more clearly, let us turn the limit statement (10b) into the form known as Bernoulli's theorem. We note first that the eigenvalues fi(n) (g) are degenerate since according to Eq. (7), for any permutation

f~n)(iv(g)) - f~n)(g). Therefore the spectral projection of F (n) associated with an eigenvalue f~n)(g)is )-~ P(~) ~(t) =: n(n) ..[t] . Here the sumr(g) of a sequence g one has

mation runs over all permutations r which do not permute identical elements of g among themselves; and [g] denotes the class of all sequences resulting from such permutations of a sequence g. For a positive number e we define .-

.

.E .

[e] "(°)

(Ii)

[t]:lf~=)(e)-pil_<e

One can show that the statement Var(F(n); lim tr[T (n) (~,

n---4(X)

T(n)(~,T)) ~ 0 implies

T) p(n)] = 1.

(12)

Thus for any positive e the probability for the frequency being close within c to the intended probability pi approaches one as n ~ c~. In this sense the frequency is an approximately real property of a large ensemble of apparatus. One may wonder whether a more specific statement could be achieved, in the sense that for some of the spectral projections n(n) the expectation value would approach unity. "~[~1 However, one can show that maxt {tr[T (n) (~ , T)n(n)] ..it] j } ~ 0 as n ~ c~. Hence it cannot be maintained that in the/inite ensembles a definite value of the frequency would be approached in this somewhat stronger sense. Yet in the context of intinite ensembles the frequency may assume the status of a real property. Indeed, based on the work of Coleman and Lesniewski [Cole 95], Gutman [Gut 95] has constructed projections for an infinite ensemble of spin-~1 systems which represent probabilities for the individual members as real properties in the sense of frequencies for the ensemble. The reduction of probabilities to properties can thus be achieved without any problematic limiting procedures if the notion of infinite ensembles is accepted. Provided that the reduced state T(~) ( ~ , T ) of A can be understood as the description of the Gemenge { (pET(Xi),Ta(i,T))" i e I}, then Theorem 3.3.1 is in fact an important result. It then shows that the number pET(Xi ) can be interpreted as the probability for i to be the actual pointer value in the state T (n) (~, T) of ,4. Hence this interpretation of the probability pET(Xi ) would follow from the theory itself, and it would not have to be added as an independent hypothesis. It must, however, be emphasised that this reasoning rests on the pointer mixture assumption, T ('~) (~, T) - ~ pET(Xi)T(n) (i, T) and on the applicability of the ignorance interpretation to this decomposition of the reduced apparatus state. The above equality is not fulfilled in general but may be realised under certain conditions [cf.

III.4. Probabilistic Characterisations

49

Theorem 2.4.1]. Still, the Gemenge interpretation for T(n) (~, T), suggestive as it might appear, will still be facing the nonobjectivity argument of Sect. II.2.6 (see also Sect. 6). A theorem similar to the above one was proved earlier by Finkelstein [Fin 62], Uartle [Uar 68] and Graham [Grah 73] and generalised mathematically by Ochs [Ochs 77]. However, instead of referring to the reduced state of the object system or the apparatus after the measurement, these authors prove an analogue of Equation (10b) for a vector state preparation prior to the measurement. The relevance of their result to measurement theory therefore seems to be rather limited except when interpretations without objectification are considered (cf. Sect. IV.5). We finally come to consider the realistic interpretation of the probabilities pE(x) [Mitt 90]. Consider again a premeasurement A4 of E with a reading scale 7~, and assume that the final component states Ts (i, T) are eigenvectors of E n, that is, EiTs(i,T) -- Ts(i,T). Then the probabilities pE(Xi) can also be attached to the object system after the measurement, that is, pE(Xi) -- tr[Ts(~, T)E(Xi)] for each i E I and for any T. Accordingly, the number pE(Xi) could also be interpreted as the probability for Xi being the actual value of the measured observable E n in the final state of the object system, provided that the Gemenge interpretation of T(n) (~, T) = ~ pE(Xi)T(n)(i, T) is justified. Similar remarks apply equally well to the measurement statistics interpretation discussed above.

III.4. P r o b a b i l i s t i c C h a r a c t e r i s a t i o n s of M e a s u r e m e n t s

A premeasurement qualifies as a measurement if it satisfies the objectification requirement. This requirement refers to the fact that a measurement leads to a definite result, so that one should be able to 'read the actual value' of the pointer observable PA and to deduce from this the value of the measured observable E. Accordingly the objectification requirement can be divided into two parts, pointer objectification and value objectification, referring to the objectivity of the pointer observable PA and the measured observable E in the final states of ,4 and S, respectively. The operational prerequisites of pointer objectification have been discussed in Sect. 2.4. Value objectification can be achieved through the pointer objectification via strong correlations. The corresponding statistical dependence between E and PA is not guaranteed by the probability reproducibility condition alone. Rather they constitute additional restrictions on the structure of the premeasurements Az[ of E, which turn out to be closely related to the measurement theoretical notions mentioned earlier (ideal, first kind, repeatable measurements). In order to give appropriate formulations of these connections, it proves useful to begin with a general study of some probabilistic properties of premeasurements. Such properties of measurements can be characterised solely in terms of the involved probability measures, with no further interpretational assumptions.

50

III. T h e Q u a n t u m

T h e o r y of M e a s u r e m e n t

A p r e m e a s u r e m e n t A/t of an observable E brings the object-apparatus system into a state V ( T ® TA). The possibility of transferring information from Jt to S rests on the fact t h a t this state entails statistical dependencies between quantities pertaining to these systems. Accordingly, three types of correlations inherent in the state V ( T ® TA) are of special interest for characterising the measurement: i) correlations between the measured observable and the pointer observable; ii) correlations between the corresponding values of these observables; and iii) correlations between the final component states of the two subsystems. It is helpful to recall some basic notions and facts concerning the relation between statistical dependence and correlation. III.4.1

Statistical

Dependence

and Correlations

Let ]2 be a probability measure on the real Borel space ( a 2 , B(R2)), and let ]21 and ]22 be the marginal measures of ]2 with respect to a particular Cartesian coordinate system: for X, Y E B ( R ) , ]21 (X) - ]2(X x R ) ,

(1)

]22(Y) - ]2(R x Y).

These marginal measures correspond to the coordinate projections (random variables) lrl • (x,y) H x and 7r2 • (x,y) ~-+ y in the sense t h a t #i = #~', i - 1,2. Assume t h a t the expectations and the variances of #i, i - 1, 2, are well defined and finite: ei = f xd#i(x), a i2 _ f ( x - ei)2d#i(x), and let e12 - f xyd#(x, y). The (normalised) correlation of the marginal measures #1 and #2 in # is then defined aS"

P(#1,#2;#)

:=

f (X -- £ 1 ) ( Y

-- £2)

O"1 (T2

d#(x,y) -

£12 -- £1£2 6r 1 (72

(2)

(whenever a l ~: 0 ~: a2). From Schwarz' inequality one gets - 1 __ P(#I, #2; #) _< 1. The marginals #~,#2 are uncorrelatecl if p ( # ~ , # 2 ; # ) - 0 (that is, £12 = £ 1 £ 2 ) , strongly correlated if P(#I, ]22; ]2) - - 1 (that is, £12 - - ( 1 £ 2 - - ffiff2), and strongly anticorrelated if p(]21, ]22; ]2) - - 1 (that is, (le2 - e 1 2 - ala2). The strong correlation conditions can equally be written in terms of the coordinate projections 7rl and 1r2" p(Trl, 71"2;]2)

--

+1

iff

ffl

7rl = --(zr2 - e2) +

£1 --" e + 0 71"2

(]2-a.e.),

(3a)

O"2 p ( z h , zr2;

#)

-

-1

iff

(71 71"1 - -

(7r2 - - £2) -[- £1 --" ~ - o 71"2

(]2--a.e.).

(3b)

dr2

(Here we have introduced the function l+ "y ~ ~ + ( y ) " - + ~ ( y - e2) -t- (1). A case of special interest arises when the marginals ]21 and ]22 have the same (finite) first and second moments so t h a t el = e2, al - a2. T h e n one has: P(]21, ]22; ] 2 ) -

+1 iff

iff 7rl - ~r2

P(]21, ]22; ]2) - - 1

(4a)

(12 - - £2 _[_ 612,

(]2-

a.e.),

iff e12 - e21 - a 2, iff lrl - - l r 2 + 2el

(4b)

(]2- a.e.).

III.4. Probabilistic Characterisations

51

The notion of correlation can be applied to quantify the degree of mutual dependence of the marginal measures. In order to avoid dealing with unnecessary complications, we assume that #1 and #2 are no {0, 1}-valued measures; equivalently, we let al ~ 0 ~ a2. ~tl and ~2 are independent if # = #1 x ~t2. Otherwise, #1, #2 are dependent. They are completely dependent if there is a (measurable) function h " R --, R such that # ( X × Y) = # 2 ( h - l ( x ) n Y) for X, Y E B(R). That is, the marginal measure #2 suffices to determine the whole measure #. The relation of complete dependence is symmetric with respect to the two marginals only if h is bijective. This is the only case of concern here. It is evident that the statistical independence of #1, #2 implies p(#l, #2; #) - 0. However, the latter condition is not sufficient to ensure their independence. On the other hand, Eqs. (3a,b) show that strong (anti)correlation entails complete dependence, the dependence being given by the affine function ~_~. Indeed, the condition ~rl = g+ o 7r2 ( ~ t - a.e.) implies that # ( X × Y) = 0 for all X and Y for which ~ : I ( x ) N Y - ~. Thus, in particular, for any X and Y, # ( X ' x i~:1 ( X ) n Y) = # ( X x ~_1 ( X ' ) n Y) = 0, where X ' = gt \ X. The additivity properties of # allow one then to confirm that for any pair X, Y, #2(g~_1(X) n Y) = # ( X x Y), that is, #1 and P2 are completely dependent with i±. By a direct computation one can verify that the converse implication holds true whenever h is an affine function. Therefore we have: P(#I, #2; #) = +1

iff

#1, ~2 are completely dependent

with h(y) = ay + b, a > O, P(#1,#2; #) = - 1

iff

#1, #2 are completely dependent

with h(y) = ay + b, a < O.

(hb)

In both cases the constants are a = :1=hi~a2, b = £ 1 - a £ 2 , SO that h = l+. This shows that the concept of strong (anti)correlation captures the idea of complete statistical dependence only in those cases where the dependence is given by an affine function. This can always be achieved by an appropriate rescaling of the marginal measures in question. It may be remarked that the notion of Shannon information (Sect. 5.2) offers a scale-independent quantification of correlations so that (in discrete cases) maximal correlation information is equivalent to complete statistical dependence. Consequently this measure does not distinguish between strict correlation and anticorrelation, a difference that will be crucial in the subsequent considerations. For this reason we shall restrict our attention to the correlation quantity as defined in this section.

III.4.2 Strong Correlations Between Observables According to the probability reproducibility condition, in an E-measurement the initial E-outcome distribution is recovered from the final PA-outcome distribution.

52

III. The Quantum Theory of Measurement

In addition to this basic requirement, a measurement may also establish complete statistical dependence between the measured observable and the pointer observable after the measurement; that is, the observables E and PA/ may become strongly correlated in the final object-apparatus state V(T ® T,4). In order to avoid any technical complications in the formulation of this correlation, we assume that the value space of E is the real Sorel space, (f~,~') -- (R,B(R)). Then for any state T E S(7-/s) the map #" X x Y ~ tr[V(T ® T,a)E(X) ® PA/(Y)]

(6)

= tr[Z(Y)(T)E(X)] = tr[Z(X)(Z(Y)(T))] extends to a probability measure on (R2, B(R2)) [Berberian 1966, Dav 70]. The marginal distributions are

#1" X ~ tr[Z(R)(T)E(X)],

(7a)

#2" Y ~ tr[TE(Y)].

(75)

Denoting the correlation of #1 and #2 in # as p(E, PIA; V(T ® TA)), we say that the premeasurement A4 of E produces strong observable-(anti)correlation in the state T if this number equals 1 (-1). According to (5), this occurs exactly when the probability measures (Ta) and (Tb) are completely dependent:

tr[Z(X)(Z(Y)(T))]

= tr[Z(g~:l(x)N Y)(T)]

(8)

for all X, Y E 8(R). A special case of this is the one with g+ (y) - y, that is,

tr[Z(X)(Z(Y)(T))] = tr[Z(X N Y)(T)]

(9)

for all X, Y, which, if valid for all states T, expresses the repeatability of the measurement (Sect. 4.6). Hence any repeatable measurement leads to strong observablecorrelations. Clearly, the repeatability condition (9) is not necessary for strong observable correlation; it suffices to have the complete dependence (8) with g+ (y) ~~2( y - e2) + el. Using the minimal unitary measurement model of a discrete sharp observable one may easily exhibit examples where (S) is satisfied without (9) being true. Condition (9) implies, in particular, the equality of the marginal measures #1, #2 of Eqs. (Ta,b): for all X,

vf(x)

--

v 7~s(V(T®T.4)) ( X ) .

(10)

This corresponds to the first-kind property of the measurement (Sect. 4.5). In general, it may occur that these marginal measures coincide irrespectively of whether

III.4. Probabilistic Chaxacterisations

53

(9) holds; in that case conditions (4a,b) give the relevant characterisations of strong (anti)correlations. Again, it is obvious that condition (10) is weaker than condition (9); in particular, the equality of the marginal measures need not lead to strong observable correlation. T h a t the first-kind property (10) does not imply the repeatability (9) nor strong observable-correlations can be illustrated by means of the standard model of a measurement of unsharp position E: X ~ E(X) = Xx * e(Q) (Sect. 2.6). In this E for each qo but the repeatability condition (9) model one has p~E = P~s(P[U(~®¢)]) is never fulfilled and the (positive) correlation p(E, QA; U(qo® ¢)) is always strictly less than 1. While repeatability entails complete dependence with h = L and thus also strong correlations, the converse implication does not in general hold. As an illustration we consider a minimal unitary premeasurement of A = ~-~'~ieIiP[qoi] induced by the map

and an affine function h on I. Such a function can easily be defined if I = N or I = Z; for example, h(j) - j + 1 or h(j) - 2j will do in both cases. Introducing the pointer observable PA = ~JP[¢j], the joint probability measure (6) assumes the form # ( i , j ) - Icjl2dii,h(j). Thus A and P.a are strongly dependent, whereas (9) is not fulfilled unless h(j) - j. One easily calculates p(A, PA; U(qo ® ¢)) - 1 for h(j) = j + 1 or h(j) = 2j, and the choice h(j) = - j for I = { - g , - g + 1 , . . . , N} or I = Z gives rise to strong anticorrelation, in accordance with Eqs. (5). In the case of a premeasurement M R induced by a reading scale 7~, it is possible to formulate necessary and sufficient conditions for strong observable-correlations. We let E n : i ~ Ei, P~ : i ~ Pi, and I : i ~ 2"i be the 7~-induced coarse-grainings of E, PAy, and 27, respectively (see Sect. 2.4). For any T e T(Tls) and all i,j e I, one now has # ( i , j ) = tr[2"i(27j(T))], ] z l ( i ) - tr[2"(I)(T)Ei], and ] z 2 ( j ) - t r [ T E j ] . To determine the correlation, one needs to calculate the expectations and variances:

c12 -- Z ij#(i,j) = Z ij tr [27i(Ij(T))], ij ij

(12b)

el = E i#1 (i) = Z i tr[Z(I)(T)E,], i

i

e2 = E j # 2 ( j ) = J

Ej

(12c)

tr[TEj],

J

a2 = Z i2tr[ Z(I)(T)Ei] - ( Z i tr [ Z ( I ) ( T ) E ' ] ) 2 i

a2 = Zj2tr[TEj]J

(12a)

(12d)

i

(Zjtr[TEj]) J

2.

(12e)

54

III. The Quantum Theory of Measurement

Assume that A/In has the property (10) for any i E I and any T E T(7-/s),

tr[Z(I)(T)Ei] = tr[TEi].

(13)

Then, indeed, el = e2 and al = a2 and the strong observable-(anti)correlation occurs exactly when one of the Eqs. (4a) or (4b) is satisfied. For strong correlation this now amounts to the condition:

E ij tr[Zi(Zj(T))] = E i2tr[ Zi(T)] ij

(14)

i

for all i and T. Therefore, if the premeasurement A/[ also fulfills the repeatability requirement (9) with respect to 7~,

tr [Zi (Zj (T) )] = tr [Zj (T)] hij

(15)

for each i,T, then one has p(E n, P~; V(T ® T,4)) - 1 for any state T (for which the correlation is defined). Clearly, if (15) holds true, then also (13) is satisfied. Condition (15) means that the final component states Ts(i, T) are eigenstates of Ei associated with their eigenvalue 1 (whenever p~T(Xi) ¢ 0),

EiTs(i, T) = Ts(i, T).

(16)

The eigenstate condition (16), or the equivalent repeatability condition (15), is thus a sufficient condition for strong observable-correlation. The converse implication holds if the reading scale 7~ is finite [Bus 96b]. THEOREM 4.2.1. Let Az[ be a premeasurement of an observable E, and let T~ be any reading sca/e. Then a) implies b), where:

a) E(Xi)Ts(i, T) = Ts(i,T) for all T e S(7-ls), Xi e n; E7¢

b) gr(pT~s(V(T®Ta))) # 0 and p(E n, P~; V(T ® TA)) = 1 for all T e S(7-ls) with a (pE~) ~ O. /f the reading scale 7~ is finite, then a) and b) are equivalent. If a premeasurement fulfills condition b) we shall call it a strong observable-correlation measurement (with respect to 7~). We emphasise once more that strong correlation is not sufficient for the eigenstate property a) if the reading scale is not finite. To close this section we demonstrate that also in the discrete case condition (13) is weaker than condition (15). Consider a simple POV measure E, defined on the two point set {Wl,W2}, with E({wi}) = Ei. Then O <_ Ei <_I and E2 - I - E l .

III.4. P r o b a b i l i s t i c C h a r a c t e r i s a t i o n s

55

Any E-compatible state transformer is generated by two operations (I)i (i -- 1, 2) with tr[~iT] = tr[TEi] for all T E S ( ~ s ) . Consider the Liiders operations:

+i(T) = E~/2TE 1/2

T E S(7-ls)

o

(17)

Since such operations are completely positive (cf. Sect. 2.2), there is a unitary premeasurement A/Iv of E which gives rise to such an state transformer. A direct computation shows that Eq. (10) is satisfied, that is, tr[(~l + ~2)(T)Ei] = tr[TEi] for each T and for both i = 1, 2. The conditions (da) and (db) for strong observable(anti)correlations reduce, respectively, to tr [TE 2] = tr [TEi] , t r [ T E 2] - t r [ T E i ] (2tr[TEi] - 1).

(lS ) (18b)

The first condition is satisfied for all T if and only if Ei is a projection operator, whereas the second holds true exactly when Ei is either O or I. The latter case leads to al = tr2 = 0, which was excluded from the consideration. Therefore, the premeasurement of this example is a strong observable-correlation measurement only when the measured observable is a (nontrivial) sharp one. III.4.3 Strong Correlations B e t w e e n Values The observable E n measured by the measurement scheme A4 n with the reading scale 7~ is discrete. One may ask to what degree the values of this observable and the pointer observable P ~ become correlated in the measurement. To answer this question requires studying the correlation p(Ei, Pi; V(T ® TA)) of the i-th values of these observables in the final object-apparatus state, that is, the correlation of properties Ei @ I and I ® Pi in the state V(T ® TA):

p(Ei, Pi; V(T ® TA)) = e12 - ele2

(19)

ff16r2

The respective numbers are easily computed: c12 = tr[Z2(T)],

(20a)

ex = tr[Z(I)(T)Ei],

(20b)

c2

(20c)

-

tr [TEi],

a 2 - tr[Z(I)(T)E 2] - tr[Z(I)(T)Ei] 2,

(20d)

a22 = tr [7~.4(V(T ® T,4))P 2] - tr[TEi] 2.

(20e)

Strong correlation is then equivalent to £12 - - £1¢~2 ~ 0"10"2

(21)

56

III. The Quantum Theory of Measurement

whenever the right-hand side is nonzero. Assume that the final component state Ts(i, T) is a 1-eigenstate of Ei (whenever pET(Xi) ~ 0); then one obtains e12 - el - e2 for all T. It follows that e l 2 - ele2 = a 2 <_ ala2 and thus al _ a2. On the other hand, the relation el = e2 = el2 together with a 2 _ e 2 - e2 = q 2 - el e2 = a 2 implies a2 _< hi. Therefore the correlation p(Ei, Pi; V(T ® T.a)) - 1 whenever 0 ~ pET(Xi) ~ 1. Another interesting implication of the eigenstate condition (16) and the ensuing equality a2 = e2 -%2 is the fact that the state TA (i, T) is a 1-eigenstate of P~. With these considerations we have established the following result. THEOREM 4.3.1. Let M be a premeasurement of an observable E and let T~ be any reading scale. Then for any state T of S, a) implies b) and c):

a) EiTs(i, T) = Ts(i, T)

t'or each i;

b) a(Si ® I; V(T ® T.a)) ~ 0 and p(Ei, Pi; V(T ® T.a)) = 1 i

ith o #

P,T (i, T) = T (i, T)

# 1;

i with p r(X,) ¢ O.

If b) is satisfied for all states T we say that the measurement is a strong valuecorrelation measurement (with respect to the given reading scale). This result then entails that any repeatable measurement is a strong value-correlation measurement. Moreover, a necessary condition for :M n to be a repeatable premeasurement is that the final component state T~(i, T) of ,4 is a 1-eigenstate of the pointer observable, that is, Ad must fulfil the pointer value-definiteness condition with respect to T~. We recall that this last property and in addition the pointer mixture property arise already as consequences of the orthogonality of the component states Ts(i,T), Ts(j, T) of S for i ~ j (Theorem 2.4.1). The notion of a correlation between values suggests that the observables in question do have definite values; yet it turns out that strong value-correlation does not require pointer value-definiteness, nor repeatability. Even the combination of b) and c) does not require the property a) to hold. To demonstrate this fact, consider again the standard measurement model of the unsharp position of Sect. 2.6 with respect to a reading scale ~ . Since the pointer observable, the scaled position of the apparatus, is sharp, condition c) of the above theorem is trivially fulfilled. Due to the special structure of the state transformer (2.43), that is, the commutativity of the operators Kq with the effects Ei, one has e12 - tr[2"~i(T) ] - tr[Zi(T)Ei] tr[TE2]. Since the measurement is of the first kind, that is, since (13) is fulfilled, and since the pointer is sharp, one also has el -- e2 and al - a 2 . Consequently, the equality (21) holds true and condition b) of Theorem 4.3.1 is satisfied. A direct computation shows that the eigenstate condition Xx, * e(Q) fx, KqTKqdq fx, KqTKqdq cannot be fulfilled for all T and i. This example shows that strong

III.4. Probabilistic Characterisations

57

value-correlation is essentially weaker than strong observable-correlation and does not imply repeatability. However, if a premeasurement Ad fulfils the pointer valuedefiniteness condition and the measured observable E is sharp, then the eigenstate condition is also necessary for strong value-correlation [Bus 96b]. THEOREM 4.3.2. Let Ad be a premeasurement of a sharp observable E and T~ be

any reading scale. For any initial state T of S, a) is equivalent to b)&c): a)

E~Ts(i, T) = Ts(i, T)

t'or each i;

b) a(Ei @ I; V(T @ TA)) ¢ 0 and p(Ei, Pi; V(T ® TA)) = 1 for each i with 0 ¢ pET(Xi) ¢ 1; c)

PiT.4(i, T) = T.4(i, T)

for each i with pE(x,) ¢ O.

III.4.4 Strong Correlations B e t w e e n Final C o m p o n e n t States In the two preceding sections it was demonstrated in which way strong observable and value correlations serve as characterisations of repeatable measurements. The corresponding eigenstate condition EiTs(i, T) - Ts(i, T) entails, in particular, that the final component states of the object associated with different outcomes i, j are mutually orthogonal,

Ts (i, T) . Ts (j, T) = O.

(22)

In some cases this orthogonality can be characterised in terms of strong correlations between the final component states of S and A. We consider again a measurement Ad of an observable E, reduced to A/In with respect to a reading scale 7~. We say that 3d le is a strong state-(anti)correlation premeasurement if for each initial state T of S it correlates strongly the final component states Ts(i,T) and T•(i,T) of the object and the apparatus. This calls for the study of the correlation p(Ts(i, T), TA(i, T); V(T ® TA)) of the probability measure defined by the self-adjoint operators Ts(i, T) ® I and I ® TA(i, T) and the final object-apparatus vector state V(T®TA). The following result is then obtained [Bus 96b]: THEOREM 4.4.1. Let Ad be a premeasurement of an observable E and T~ any reading scale. For any initial state T of the object system for which the component states Ts(i, T) and T,4(i, T) are vector states, a) is equivalent to b)&c):

a)

Ts(i,T) . Ts(j,T) = 0

for i C j;

b) p(Ts(i, T), T (i, T); V(T ® c)

= 1

i

itb 0 #

# 1;

T~a(i,T) is a 1-eigenstate of Zi for each i with 0 7~ p~(Xi) 7~ 1.

For a minimal measurement Ad~ of a discrete sharp observable the above result says that if the unitary coupling is generated by the family of unit vectors {¢ij }

58

III. The Quantum Theory of Measurement

(Sect. 2.5), then Az[~ is a strong state-correlation premeasurement if and only if

{¢ij) is an orthonormal system [Belt 90]. Finally, one may ask whether the requirement of strong correlation between the final S and ,4 states 7~s(V(T ® TA)) and T ~ ( V ( T ® TA)) imposes any constraint on the measurement scheme under consideration. That this cannot be expected in general can be seen in the case of a unitary premeasurement A~Iu. Note first that the reduced states of P[U(~ ® ¢)] have the same spectra, including multiplicities. The spectral decompositions can be given in terms of orthonormal systems ( ~ i ) , ( ¢ i ) defined by the biorthogonal decomposition U(~ ® ¢ ) - ~ i ci~i ® ¢i (ci > 0), and a straightforward calculation shows that

p(7~s(P[U(~ ® ¢)]), ~ A (P[U(~o ® ¢)]); P[U(~ ® ¢)]) = 1.

(23)

Hence these states are always strongly correlated. III.4.5 First K i n d M e a s u r e m e n t s The notion of a first kind measurement was discussed by Pauli [Pauli 1933]. He gave two definitions which he considered equivalent. The first definition says that a measurement is of the first kind if it leads to the same result upon repetition. The second definition says that a measurement is of the first kind if the probability of obtaining a particular result is the same both before and after the measurement. In the general context of measurement theory these two definitions are not equivalent. We shall adopt the first definition as the definition of repeatability (Sect. 4.6), and the second as the definition of first kind measurements. A premeasurement M of an observable E and its associated state transformer is said to be of t h e / i r s t kind if

= P'Rs(V(T®T.a))(X)

(2a)

for all X E ~" and T E S(7-/s). In Sect. 4.2 we pointed out necessary and sufficient conditions under which a first kind measurement is a strong observable-correlation measurement We also observed that for unsharp observables the first-kind property is not sufficient to entail strong observable-correlations. In order to further illustrate the connections between first kind and strong value-correlation measurements, let us consider once more a Liiders measurement of the two-valued observable wi ~ Ei, with the ensuing state transformer being given by the operations ~i(T) = E~/2TE~/2, T e S(7-ls). A direct computation gives

p(Ei,Pi; V(T ® TA)) 2 = tr[TE2] - tr[TEi]2 tr[TEi] - tr[TEi] 2 "

(25)

III.4. Probabilistic Characterisations

59

Hence the strong value-correlation occurs only if t r [ T E 2] = tr[TEi] for all T (for which 0 ~ tr[TEi] # 1), that is, Ei is a projection operator. In that case the final component state Ts(i,T) - ~i(T)/tr[TEi] is a 1-eigenstate of Ei whenever 0. A large class of first kind measurements are given by the standard model of Sect. 2.6. Indeed the coupling U - e i X A ® s defines always a first kind measurement scheme.

III.4.6 Repeatable Measurements Repeatable measurements are an important class of measurements. In fact repeatability was already pointed out to be fundamental to the value objectification. Intuitively a measurement of an observable is repeatable if its repeated application yields the same results. This idea can be formalised systematically within the theory of sequential measurements (see, for example, [Bus 90b, Dav 70]). A premeasurement A4 is repeatable if its repetition does not lead - from the probabilistic point of view - to a new result, that is, if

tr[Z~(Y)(ZM(X)(T))]

= tr[Z~(Y NX)T]

(26)

for all X, Y E ~" and for all T E 8(7-/s). This condition may be written in the following equivalent ways:

tr[Ts(X,T)E(X)] - 1

(whenever tr[TE(X)] ~ 0),

E ( X ) T s ( X , T) = Ts(X, T),

(27 ) (27b)

which are to hold for all X E 7" and all T E 8(7-ls). As an immediate observation we note once more that a repeatable premeasurement is always of the first kind. We say that an observable E admits a repeatable measurement if there is a premeasurement of E which is repeatable. It is an old issue in the quantum theory of measurement, dating from von Neumann's work [yon Neumann 1931], as to whether observables which admit repeatable measurements are necessarily discrete. It was always anticipated that this must be the case. On the basis of important contributions by Stinespring [Sti 55], Davies [Davies 1976] and others the problem was finally solved by the results of Ozawa [Oza 84] and Luczak [Lucz 86]: THEOREM 4.6.1. If an observable E admits a repeatable premeasurement 2~4, then

E is discrete. We note an immediate corollary to the correlation theorems. COROLLARY 4.6.2. Let M be a premeasurement of an observable E and 7~ be any reading scale. The following statements hold true. a) If All n is repeatable then it produces strong observable-correlations and strong

value-correlations.

60

III. The Quantum Theory of Measurement

b) Let T£ be finite. M ~ is repeatable whenever it is a strong observable-correlation premeasurement. c) Let ¢~4~ satisfy the pointer value-definiteness condition and let E be a a sharp observable. 2~4~ is repeatable whenever it is a strong value-correlation premeasurement. An observable need not be a sharp one in order to admit a repeatable unitary premeasurement. As an illustration, any collection of effects Ei, i E I, which all have eigenvalue 1 and which sum up to the unit operator I, constitute an observable admitting a repeatable unitary premeasurement. Indeed, choosing a 1-eigenstate Ti for each i, EiTi = Ti, the state transformer, with the operations ¢ i ( T ) : = tr[TEi]Ti, T E S(7-ls), is repeatable and completely positive (cf. Sect. 2.2), although the Ei need not be projection operators. We noted already that repeatable measurements are of the first kind, but a first kind measurement need not be repeatable. The standard measurement of an unsharp position (Sect. 2.6) or the Liiders measurement of a simple observable E, introduced in Equation (17), are of the first kind but in general not repeatable; the latter being repeatable if and only if El, and thus also E2, is a projection operator. Besides the first kindness, there is another probabilistic weakening of the notion of repeatability. We say that a measurement A~i of E is value reproducible if for any X E 9r and T E S(7-ls) the following implication holds true: if p E ( X ) = 1,

then

p ET s ( n , T ) ( X ) - 1.

(28)

Clearly any first kind measurement is also value reproducible, but the converse inclusion does not hold. In the case of sharp observables the notions of first kind, value reproducible and repeatable measurement coincide [Bus 95a]. THEOREM 4.6.3. Let E be a sharp observable and Z any of its associated state

transformers. Z is repeatable ff and only ff it is of the first kind, if and only if it is value reproducible. The von Neumann measurements of a discrete sharp observable A are repeatable, but not every repeatable measurement of A is a von Neumann measurement. Indeed the A-compatible state transformer Z ( { a i } ) ( T ) = tr[TEA({ai})]Ti is repeatable whenever tr[TiEA({ai})] - 1 for each i - 1 , 2 , . . . ,N. Yet it is not a von Neumann state transformer if some of the eigenvalues ai of A are degenerate.

III.4.7 Ideal M e a s u r e m e n t s No measurement that is capable of providing some probabilistic information can leave unchanged all the states of the measured system. Hence it is important to investigate to what extent state changes are necessary in a measurement. The value reproducibility and first kind properties introduced in Sects. 4.5 and 4.6 are probabilistic nondisturbance features which quantum mechanical meas-

III.4. Probabilistic Characterisations

61

urements may or may not possess. A more stringent nondisturbance property representing a kind of minimal disturbance is that of ideality [Beltrametti and Cassinelli 1981]. A measurement is ideal if it alters the measured system only to the extent that is necessary for obtaining a measurement result: all the properties which are real in the initial state of the object system and which are coexistent with the measured observable remain real also in the final state of the system. In the case of sharp observables A this intuitive conception of ideality leads to the following probabilistic formulation, called here p-ideality. An A-measurement A4 and its induced state transformer Z ~ are called p-ideal if for any state T E S(7~s) and for any sharp property P E C(7-/s), which is coexistent with A, the following implication holds true: if tr[TP] = 1, then tr [Z~ (R)(T)P] = 1.

(29)

When applied to the measured observable A itself, the p-ideality of A,[ implies that for any state T and for all value sets X the following implication holds true: if

pAT(X) =

1, then

pA~(R)(T)(X ) -- 1.

(30)

For sharp observables this condition is equivalent to the repeatability of Az[. Thus we have the following result [Lah 91]: THEOREM 4.7.1.

A p-ideal premeasurement of a sharp observable is repeatable.

The yon Neumann measurements of a sharp discrete observable are always repeatable and satisfy (28). However these measurements are not p-ideal except when they are Liiders measurements. As a corollary to the above theorem we note that a sharp observable A is discrete whenever it admits a p-ideal measurement (Theorem 4.6.1). In the case of a discrete sharp observable the p-ideality of an A-measurement AJ implies the following condition: for all i - 1, 2 , . . . and all T, if

tr[TEA({ai})]

= 1,

then

Z~({ai})(T)=T.

(31)

Indeed if ~ is a vector state for which (~lEA({ai})~) = 1, then, by p-ideality, tr[ZA4(R)(P[~])(P[~])] : 1. But in this case the support projection of the s t a t e / : ~ ( R ) ( P [ ~ ] ) is contained in P[~], which is possible only if = P[~o]. Since P[~] is a vector state with tr[ZA4({ai})(P[~])] - 1, we finally have ZA4({ai}) (P[~o]) : P[~]. By linearity the argument extends to arbitrary states T for which tr[TEA({ai})] = 1. We shall see below that condition (31) is in fact equivalent to the p-ideality of Z ~ . Condition (31) admits an immediate generalisation to arbitrary discrete observables E. Since this generalisation will turn out to be important for the objectification problem we formulate it as the definition of the d-ideality of a measurement, d referring to the discreteness-assumption.

:r~(R)(P[~])

62

III. The Quantum Theory of Measurement

Let E be a discrete observable with the generating (nonzero) effects E ({wit), i - 1, 2,.... An E-measurement A4, or an E-compatible state transformer Z, is d-ideal if it does not change the state of S whenever a particular result is certain from the outset: if tr[TE((wi))] : 1,

then

Z~((wi))(T):

T

(32)

for all i = 1, 2 , . . . and for any T E S(7-/s). For general discrete observables a d-ideal measurement need not be repeatable nor first kind. Indeed the Liiders state transformer T ~ E~/2TE~/2 of a discrete observable E (with the generating effects E1 and E2) is d-ideal but never repeatable unless the Ei are projection operators. Suppose the Ei have eigenvalue 1 with associated spectral projections E~, and let U " 7-/s ~ 7-/s be a unitary mapping which acts as an identity on the eigenspaces E~(Hs). Then the E-compatible state transformer T ~ UE~/2TE:/2U-1 is still d-ideal but not first kind, unless U commutes with the Ei. The question of the structure of ideal, repeatable measurements has been a major issue since von Neumann's work [yon Neumann 1932]. These properties are crucial for the realistic interpretation of quantum mechanics insofar as the existence of measurements with these properties ensure the interpretation of possible measurement results as potential properties of the system. The repeatability of a premeasurement A4 requires the measured observable E to be discrete (Theorem 4.6.1). The d-ideality or the repeatability of a measurement A4 of a discrete observable E do not imply that E is an sharp observable. In order to afford this conclusion, one needs to postulate an additional property of the measurement, its nondegeneracy. A measurement A4 of E is nondegenerate if the set of all possible final component states ( T s ( X , T ) " X e 3c, T e S(7-ls)) separates the set of effects; that is, for any B E £(T/s), if tr[Ts(X, T)B] = 0 for all X e 3~, T e S(7-/), then S = O.

(33)

The following theorem then holds true [Davies 1976]: THEOREM 4.7.2. A discrete observable E admits a repeatable, d-ideal, nondegencrate measurement/f and only if E is a sharp observable. In that case the premeas-

urement f14 is equivalent to a Liiders measurement of E; that is, the induced state transformer Zj~ is of the form Z~(X)(T)

-

Z

E((wi))TE((wi))

(34)

w~EX

for all T E S(7-ls) and for all X E 3c. It can be shown that a d-ideal measurement of a discrete sharp observable is nondegenerate [Bus 90b]. Furthermore, a result similar to Theorem 4.7.1 shows that also d-ideality implies repeatability [Lah 91]. Hence the considerations of this section give rise to the following statement.

III.4. Probabilistic Characterisations

63

4.7.3. The d-ideal premeasurements of a sharp discrete observable are exactly the Lfiders measurements. COROLLARY

A Liiders measurement of a sharp observable is p-ideal. Thus the concepts of p-ideality and d-ideality are equivalent in the case of sharp discrete observables. From now on we shall only consider the d-ideality as given by (32), and we refer to it as the ideality of a premeasurement. III.4.8 R@sum@- A Classification of P r e m e a s u r e m e n t s We summarise the main relationships between the various probabilistic characterisations of premeasurement discussed so far. While all of the notions to be listed can be formulated solely in terms of premeasurements, their consideration is actually motivated by the idea of measurements. We therefore drop the prefix 'pre' in the sequel. In the following summary we are using obvious abbreviations for the terms ideality (ID), repeatability (REP), first kind property (FK), value reproducibility (VR), strong observable-correlation (SOC), strong value-correlation (SVC), and strong state-correlations (SSC). It is convenient to consider four successive classes of measurements, each containing the subsequent one. A) A r b i t r a r y observables. There are measurements which are value reproducible but not of the first kind, and there are first kind measuremets which are not repeatable. But repeatable measurements are always of the first kind, and first kind measurements are always value reproducible. Repeatable measurements yield strong correlations, but there exist first kind measurements which produce neither strong observable nor strong value correlations. Hence we have the implications:

(REP) ~

(FK)==~ (VR);

(REP) ~

(SOC);

(REP) ~

(SVC).

If the reading scale is finite then one has: (REP) ~

(SOC).

An observable admitting a repeatable measurement is discrete. We recall also that the ideality and repeatability of a measurement of some discrete observable do not yet require this observable to be a sharp one. B) S h a r p o b s e r v a b l e s In this case the important notions of repeatability, first kind property, and value reproducibility are the same: (REP) ~

(FK) ~

(VR).

64

III. The Quantum Theory of Measurement

Furthermore, if the measurement fulfils the pointer value-definiteness condition (which is the case if the pointer observable is sharp), then (REP) ~

(SVC)

with respect to any reading scale. C) D i s c r e t e s h a r p observables (ID) ~

(REP)

In the case of minimal unitary measurements ~4~ one has, in addition: (REP) ==~ (SSC). As mentioned above, both the ideality and the repeatability requirements imply the discreteness of a sharp observable. For an arbitrary observable E, besides ideality and repeatability, yet another property, the nondegeneracy, of state transformers is needed in order to ensure that E is an sharp observable. Finally we recall that in general a von Neumann measurement is not ideal (though repeatable). In fact the ideal measurements of discrete sharp observables are precisely the Liiders measurements (LU)" oo

(ID) ~

(LU).

D) Nondegenerate, discrete sharp observable Here we note the additional fact: (ID) ~

(REP).

We emphasise once more that the notions of repeatability and ideality are crucial for the realistic interpretation of quantum mechanics. We have seen that quantum mechanics does allow for such measurements. But it is also important to realise that to any observable E belongs a large (infinite) class of E-measurements. This fact opens up the possibility of accounting for real (imperfect) measurements.

III.5. Information Theoretical Aspects of Measurements The incorporation of the concepts of entropy and information into quantum probability theory runs up against severe difficulties, which are mainly due to the existence of noncommuting observables. That this problem is of considerable interest in various branches of quantum physics is demonstrated, for instance, in the monographs of Helstrom [Helstrom 1976], Lindblad [Lindblad 1983] and Whirring [Thirring 1980]. It took a long time until a rigorous and comprehensive framework has been set up. An account of the present state of art is presented in [Ohya and Petz 1993]. Important landmarks of the earlier developments with some bearings

III.5. Information Theoretical Aspects

65

on quantum measurement theory are given, for example, by the works of von Neumann [von Neumann 1932], Everett [Eve 57], Ingarden [Ing 76], or Lindblad [Lin 73]. The application of information theoretical concepts within the quantum theory of measurement has recently attracted increasing attention, as documented in [Santa Fe 1989]. Information theoretical characterisations of measurements will be the subject of the present section. Any premeasurement A/[ of an observable E induces the following transformations

T ~ Ts (~t, T), E

(1)

(2)

where T is the initial state of the object system S. Moreover, any reading scale 7~ defines a natural decomposition of Ts(~, T):

Ts(f~, T) = ~pi Ts(i, T),

(3)

where Pi - pE(xi). Properties of the transformations (1) and (2) reflect the properties of A/I, and the decomposition (3) is the one for which one might tend to apply the ignorance interpretation. Various probabilistic features of the transformations (1) and (2) were studied in the preceding section. We next come to apply the concepts of entropy and information in order to further characterise these transformations. In particular, we are able to characterise those premeasurements which lead to an optimal separation of the final component states Ts(i, T) of the object system S (Sect. 5.1). We also specify the conditions under which the deficiency of information (implicit in the probability measure pTE) for predicting a certain measurement outcome can be identified with a potential information gain (associated with P~s(~,T)) obtained by reading the actual measurement result (Sect. 5.2). Some information theoretical aspects of von Neumann and Liiders measurements will be pointed out as illustrations of the general relations obtained. Finally an information theoretical characterisation of the commutativity of discrete sharp observables will be presented (Sect. 5.3). III.5.1 T h e C o n c e p t of E n t r o p y Following [von Neumann 1932], the entropy of a state T is defined as the (nonnegative) number S ( T ) " - -tr[Tln(T)]. Using the spectral decomposition T tiE T ({ti }) of T, one obtains the expression

S(T) = - ~ t i l n ( t i ) tr[E T({ti})].

(4)

Since all the spectral projections E T ({ti)) (with ti ¢ 0) are finite dimensional, we may also write S(T) = - ~ t i l n ( t i ) (5)

66

III. The Quantum Theory of Measurement

whenever T is nondegenerate, or if we allow each term tiln(ti) to appear in the series as many times as indicated by the degeneracy of the eigenvalue ti. S(T) is finite if Tln(T) is a trace class operator. In particular, if the dimension of the range of T is finite, say n, then 0 _ S(T) < ln(n). The case S(T) - 0 occurs exactly when T is a vector state, whereas S(T) - ln(n) holds if and only if T is totally degenerate (maximally mixed)[Feinstein 1958]. Consider any decomposition of a state T into some other states Ti, T - ~ wiTi, with 0 < wi <_ 1, ~ wi = 1, as in Eq. (3). The concavity and the subadditivity of the entropy functional give rise to the following inequalities.

wiS(Ti) <_ S( ~ wiTi) = S(T) <_ ~ wiS(Ti) + S({wi})

(6)

Here we have defined S({wi}) := - E wiln(wi). The lower bound of S(T), S(T) = wiS(Ti), is obtained if and only if all the component states Ti are the same, that is, Ti - Tj for all i,j. The upper bound, S(T) - ~, wiS(Ti) + S({wi}), is assumed exactly when the component states Ti are mutually orthogonal, i.e., TiTj - 0 for all i ¢ j [Lin 73]. In particular, if T = ~ w~T~ is an orthogonal decomposition of T into vector states Ti = P[~i], then S ( T ) = S({wi}), as in Eq. (5). In an intuitive sense the entropy of a state T is a measure of the degree of 'mixedness' of T. In the case of an orthogonal decomposition of T into vector states, T - ~ wiP[~/i], one might ask whether the number S(T) - S({wi}) could be interpreted epistemically as the (average) deficiency of information on the actual vector state "yi of the system if the system is known to be in the mixed state T. However, such an interpretation seems to require the Gemenge interpretation of T with respect to its decomposition ~ wiP[~/~]. The nonobjectivity arguments of Sect. II.2.6 show that such an interpretation cannot be justified in general. However, we tentatively assume that in a measurement context the ignorance interpretation can indeed be applied to the final state Ts(f~,T) of the object system with respect to its natural decomposition (3). This assumption is necessary for the physical interpretation of the subsequent considerations. The validity of the corresponding formal results does not, of course, depend on interpretational issues. Consider a measurement M of an observable E. The state change T Ts(~, T) induced by A4 [Eq. (1)] is accompanied with a change of entropy, S(T) S(Ts(f~,T)). Fix a reading scale n . Then (3) is the decomposition of Ts(f~,T) with respect to 7~, and the inequalities (6) give bounds to the entropy S(Ts(~, T)) for this decomposition. We call A4 a maxima/state-entropy measurement with respect to 7~ if

S(Ts(f~,T)) - S ( ~ p i T s ( i , T ) )

= ~ , p i S ( T s ( i , T ) ) + S({pi})

(7)

for any initial state T of S. On account of (6) this is the case if and only if the component state Ts(i, T) are mutually orthogonal for any T. In such a case the measurement leads to an optimal separation of the final component states.

III.5. Information Theoretical Aspects

67

Let 3/l be a repeatable measurement of E (Sect. 4.6) and 7~ be any reading scale. For any state T and any value set Xi (with pE(Xi) ¢ 0) the state Ts(i, T) is a 1-eigenstate of E(Xi). The effects E(Xi) and E(Xj) associated with disjoint sets X~, Xj satisfy E ( X i ) + E(Xj) <_ I, which implies that the associated 1-eigenstates are orthogonal. We thus obtain the following result. THEOREM 5.1.1. A repeatable premeasurement A4 of an observable E is a maximal state-entropy measurement with respect to any reading scale T~. As a further illustration of the maximal state-entropy measurements we shall now consider unitary premeasurements Ad~ of a sharp discrete observable A. Assuming that the eigenvalues of the pointer observable form a closed set there is a natural finest reading scale consisting of the pointer eigenvalues. We shall assume throughout the present subsection that this reading scale is chosen, so that the letter 7£ may occasionally be dropped. If qo is an initial vector state of S then U(~o ® ¢) = ~ x/~'Yi ® ¢i is the final state of 8 + .4 (Sect. 2.4). It follows immediately that jt4~ is a maximal state entropy measurement if and only if the vectors "yi are mutually orthogonal for each ~o. According to Theorem 4.4.1 this is the case exactly when Ad~ is a strong state-correlation measurement. Note that this does not require Ad~ to be a repeatable or a strong value-correlation measurement (see Theorems 4.3.2 and 4.6.2). Hence we have the following result. THEOREM 5.1.2. Let A4'~ be a premeasurement of a sharp discrete observable A with V(gPij @ ¢) ¢ij ® ¢i" ~ is a maxima/state-entropy measurement ff and only if the set {¢ij } is orthonormal. -

-

The above statements apply to the von Neumann and Liiders measurements of A, which are repeatable and therefore also maximal state entropy measurements. With a notation similar to that of Sect. 2.5, if ~o is an initial vector state of S, then the final states of 8 after a Lfiders measurement and a v o n Neumann measurement of A are

TL (A, ~o) := 7Zs (P[UL (~o ® ¢)1) = Y~.PiP[qoa,] TvN(A, ~o) := n A (P[UvN(~O ® ¢)]) = Y].Pi T,N(i, ~o) .

(8) (9)

Here TvN(i, ~o) := Ts(i, ~o) = p7~ 1 ~-~j P[qoijlP[qolP[qoij] if Pi 7~ O, and Tvg(i, ~o) = 0 otherwise. We consider only those von Neumann measurements of A which are induced by a maximal refinement of A. Application of Eq. (7) gives

S(T (A,

= S({p,}),

(10) (11)

If the degeneracy of an eigenvalue ai is finite and equal to n(i), say, then one has S(TvN(i, qo)) _< ln(n(i)).

68

III. The Quantum Theory of Measurement In a measurement A/I, with a reading scale :R, the change of state entropy

S(T) ~-~ S(Ts(~,T)) can be used in various ways to characterise jk4. Above we studied only the entropy of the final state Ts(~, T) with respect to a given reading scale 7~. In addition, one may ask under what conditions the difference

S(Ts(fl, T)) - S(T)

(12)

would be nonnegative. Clearly this is the case whenever the initial state is a vector state. For a Liiders measurement of A one has S(TL(A,T)) - S(T) >_ 0 for any T [Lin 73]. Another result is the Groenewold-Lindblad-Ozawa inequality [Gro 71, Lin 72, Oza 86] which shows under what conditions the quantity

S (T)- Epi S (Ts (i, T) ) = [S(Ts(fl, T)) - ~-~piS(Ts(i,T))] - [S(Ts(fl, T)) - S(T)] is nonnegative. This quantity was considered by Groenewold [Gro 71] as the difference between the potential information gain upon reading the result and the increase in the deficiency of information about the actual state of the system. However, in the general context of measurement theory such an interpretation of this quantity is not possible. It presupposes repeatable measurements (cf. the next subsection). The domain of validity of the Groenewold-Lindblad-Ozawa inequality can be easily illustrated by the unitary premeasurements M ~ of discrete sharp observables A. Such measurements share the property that any final component state is a vector state (P[~/i]) whenever the initial state T is such (T - P[7~]). This property is indeed characteristic of the nonnegativity of the quantity (13) [Oza 86]. Hence in the case of a measurement M ~ of A we have, in fact, the trivial result" S ( T ) - ~-]~piS(Ts(i,T)) - 0 - 0 - 0 whenever T - P[~]. A further option of characterising M via the entropy change is to study the relative state entropy

S(T[Ts(fl, T)) := tr [Tln(T) - Tln(Ts(fl, T ) ) ] .

(14)

This quantity is finite only if the range of T is contained in the range of Ts(fl, T) [Lin 73]. Thus, when applied to a unitary measurement AJ~ of a discrete sharp observable A, the relative state entropy

S(P[ ollTs( , T))

=

-( o]ln(Ep, P['rd) o>

(15)

is finite for each ~o if and only if M ~ is a Liiders measurement. In this case one

has S(P[7~][TL(A, 7~)) = S({pi}).

III.5. Information Theoretical Aspects

69

III.5.2 The Concept of Information The notion of average information of a probability measure, as introduced by Shannon, can be employed to yield characterisations of measurements. Consider a measurement A4 of an observable E and fix a reading scale T~. Then for any state T the average deticiency of information of the discrete probability measure Xi ~, pE (Xi) = Pi is given by the quantity

H(E,T; Tt) := S((pi)).

(16)

In accordance with the minimal interpretation, H(E,T; T~) can be interpreted as the (average) deticiency of information for predicting a measurement outcome Xi in the initial state T of the system S. In particular, the case H(E, T; :R) - 0 of vanishing deficiency of information occurs exactly when T is an eigenstate of E, that is, if E(Xi)T = T holds for some Xi E T~. Similarly, the state Ts(~t, T) of S after the measurement gives the probability E (Xi) = qi, with the corresponding deficiency of information measure Xi ~ PTs(f~,T)

H(E, Ts(f~,T); T~) = S({qi}).

(17)

Assuming that value objectification takes place in the E-measurement A/I, then this number could be interpreted as the potential in[ormation gain upon reading the result Xi E T~. On this assumption it becomes interesting to compare the initial deficiency of information, given by H(E, T; T~), with the potential information gain, H(E, Ts(fl, T); 7~) and to ask under what conditions they are the same. E Noting that Ts(~, T) = ~ piTs(i, T) and hence PTs(n,T) = ~ pipETs(i,T), one has the basic inequalities [Lin 73]

~piH(E, Ts(i,T); n) <_ H(E, Ts(~,T); n) <_ ~'~piH(E, Ts(i, T); n) + H(E, T; T~)

(18)

which allow one to compare the quantities H(E,T; T~) and H(E, Ts(~,T); T~). If E A/I is of the first kind, that is, p~ = PTs(f~,T), one then has

H(E, T; n) = H (E, Ts(~, T); :R)

(19)

for any state T of S and any reading scale n . The quantity ~p~H(E, Ts(i,T); n), which appears in the inequalities (18), describes the average deficiency of information on the value of E when the system is known to be in one of the states Ts(i, T) after the measurement. Equality in the second line of (18) does not require the vanishing of this quantity, as exemplified by the measurement of an unsharp simple observable defined via Eq. (4.17). However, for repeatable measurements this quantity is always zero: H (E, Ts(i, T); 7~) = 0 (20)

70

III. The Quantum Theory of Measurement

for all i = 1, 2,..., for any reading scale T~. On the other hand, (20) can hold only if E(Xj)Ts(i, T) - Ts(i, T) for some Xj e T~. Hence if we require that (20) holds for each initial state T and for any reading scale T~, then j~4 necessarily leaves the object system in a mixture of eigenstates of the measured observable. This does not imply the repeatability of ~t. For example, a unitarily modified Liiders measurement, with Z({wi})(T) - UEiTEiU -1, of a discrete nondegenerate sharp observable E, with an U permuting the eigenvectors of E, satisfies conditions (19) and (20), though it need not be repeatable. Although repeatable measurements are not uniquely singled out by these requirements, they nevertheless are optimal in the sense that for any initial state of the object system and for any given reading scale the deficiency of information for predicting a measurement outcome always equals the potential information gain upon reading the result, and there is no remaining deficiency of information on the value of the measured observable in the final component states of the object system. We summarise this result in the following theorem. THEOREM 5.2.1. In a repeatable measurement of any (discrete) observable the deficiency of information for predicting a measurement outcome always equals the

potential information gain upon reading the measurement result. Repeatable measurements are known to possess the following important properties: they are maximal state entropy measurements, and for them the deficiency of information always equals the potential information gain. These two aspects allow one to establish a quantitative connection between the entropy of the final state (3) and the average informations (16) and (17). Indeed we have

S(Ts(~, T)) = ~']~piS(Ts(i, T)) ÷ H(E, T; T~)

(21)

= ~-~p~S(Ts(i,T)) ÷ H(E, Ts(~,T);T~). In particular, this yields the following inequality

S(Ts(~,T)) - H(E,T; n) = S(Ts(~,T)) - H(E, Ts(~,T); n)

(22)

= Ep~S(Ts(i,T)) >_O. Hence for repeatable measurements the deficiency of information S(Ts(~2, T)) about the actual final state of the system is never less than the potential information gain H(E, Ts(~, T); T~) upon reading the actual result. The quantities coincide exactly when all component states Ts(i,T) are vector states. For arbitrary repeatable measurements this need not be so. We can illustrate the above results as characterisations of the von Neumann and Liiders measurements of a discrete sharp observable A. Since these measurements are repeatable, the relations (21) may be applied. For a Liiders measurement of A one obtains S(TL(A, 7~)) = H(A,P[7~]) = H(A, TL(A, 7~)) (23)

III.5. Information Theoretical Aspects

71

for any initial vector states ~. On the other hand, for a (maximal) von Neumann measurement of A one has only

S(T N(A.

- H(A. P[(p]) = S(T.N(A. cp)) - H(A. TvN(A. = Ep,

(2a)

>_ O.

so that the deficiency of information on the actual state of the system after the measurement is not less than the potential information gain upon reading the result. As in the case of state entropy, there are several alternative information theoretical characterisations of the probability changes pE ~-~ P TEs ( ~ , T ) associated with an E-measurement j~4. In addition to the above considerations we may mention the concept of average relative information, which can be used to compare the initial and final probability measuresp E and PTs(~,T)'E Assume that the final probability measE E ure PTs(~,T) is absolutely continuous with respect to pE, that is, pTs(~2,T)(X) -- 0 E whenever p E ( x ) -- O. Then the average relative information of PTs(~,T) with re-

spect to pE is defined as the nonnegative number [Jan 72]" dpETs(n,T) dpET In

dp E

dp E.

(25)

Consider an E-measurement Az[ with a given reading scale TO. Assume that the final probability measure xi ~ p TEs ( g t , T ) ( X i ) -- qi is absolutely continuous with respect to the initial one, Xi ~ pE(Xi) = Pi, that is, qi - 0 whenever Pi = O. Then one obtains

H ({qi}I(pi})= ~ - ~ q i l n ( q ~ p i " - ' \ ] P iPi

(26)

= - Z q i l n ( p i ) - H(E, Ts(~, T); 7~). This relation can be used, in particular, for a characterisation of repeatable unitary measurements M ~ of A. Assuming that the final probability measure is absolutely continuous with respect to the initial probability measure for any initial (vector) state of S, it follows that the A4~-generating set {¢ij } (cf. Sect. 2.5) is a complete set of eigenvectors of A. Hence A/Iff is repeatable so that one has H(pAs(n,T) IpA) -0 for any state T. The notion of average information (16) of a discrete probability measure is not the only one ever proposed in the context of quantum mechanics. Let A be a discrete sharp observable such that the degree of degeneracy n(i) of any eigenvalue ai is finite. Then the formula

H*(A,T) "- - - Z p A ( a i ) ln (pA!?:) ~ \ n(.t) /

(27)

72

III. The Quantum Theory of Measurement

defines an information functional considered by Everett [Eve 57]. Obviously we have H* (A, T) = H(A, T) + ~pAT(ai ) ln(n(i)) _ H(A, T). (28) In order to give a first intuitive comparison of the two concepts of information H and H*, we shall consider a situation in which the outcome ak is known before the measurement, that is, pA(a~) = 5ik. It follows that H(A,T) = 0 so that there is no ignorance left about the measuring result. On the other hand, we find H*(A,T) = ln(n(k)), that is, there is still some deficiency of information. This result can be explained in the following way. Even if the result ak is known, nothing is known about the pure state 7~kj of S which lies in the subspace with dimension n(k). There are n(k) orthogonal states of this kind with probability w~j -- n (1k ) , that 1 is, the system with eigenvalue ak is described by the state Tk = ~ ( k 1) n(k) P[7~kJ]• In order to determine the pure state ~kj of S, one could measure some nondegenerate observable Sk = ~(=k) bjP[7~kj]. The deficiency of information about the value and thus about the state 7~kj is then given by

H(Bk, Tk) = --~~jwkjln(wkj) = ln(n(k))

b

(29)

in accordance with the formula for H*(A, T). Hence H*(A, T) describes not only the deficiency of information about the value ak but also about the pure state of the system S. This consideration leads to the following more general remark. Let A, B be two sharp discrete observables such that B is a refinement of A, so that A - f(B) for some function f. It follows that the potential information gain with respect to B is larger than that for A:

H(B, T) >_H (f(B), T).

(30)

For a given state T, there exists a maximal (nondegenerate) refinement B -- A0 ~_,~j aijP[cpij] of A such that the probabilities pAT°(aij) are independent of the degeneracy index j , that is, leads to

pAT°(aij) -- ~ 1 pAT(ai) • This T-dependent choice of A0 H(Ao,T) = H*(A,T)

which again illustrates the meaning of the entity equality (28) is seen to be a special case of (30). III.5.3

Information

and

(31)

H*(A, T). Furthermore, the in-

Commutativity

The commutativity of two bounded sharp discrete observables A and B can be given an operational meaning by means of Liiders measurements. If T is an initial state of S, then TL (A, T) is the state of S after a Liiders measurement of A. For a measurement of another discrete sharp observable B we may (c~) measure B directly

III.6. Objectification

73

on the system S in the state T, or (fl) first perform a Liiders premeasurement of A on the system in the state T and then perform a B-measurement on ,~ in the state TL(A, T). Even if the operators A and B commute, the outcome of a single B-measurement will in general be different in the two cases (a) and (/3) since in the case (/3) the A-measurement produces changes of the state of the system, which may influence the result of the succeeding B-measurement. However, in the statistical average these differences disappear. This is the content of the following theorem due to Liiders [Liid 51]. THEOREM 5.3.1. Bounded discrete sharp observables A and B commute//and only if t r [ T S ] - tr[TL(A, T)B] for all states T. This theorem provides an operational meaning of the concept of commutativity: if for any preparation T a Liiders measurement (without reading) of A has no influence on the expectation value of B, then it follows that A and B commute, and vice versa. It is essential for the theorem that only Liiders measurements of the discrete observable A are considered. However, it is straightforward to generalise the theorem for continuous observables in the sense of referring to all possible discretised versions of them. The intimate connection between Liiders measurements and the concept of information makes it possible to characterise the commutativity also in terms of information theory. The deficiencies of information about B which correspond to the measuring procedures (a) and (~) are H(B,T) and H(B, TL(A,T)) respectively. An information theoretical characterisation of the commutativity can then be formulated in the following way ITS1 87]: THEOREM 5.3.2. Discrete sharp observables A and B commute ff and only ff H(B, P[~]) - H(B, TL(A, ~)) for all vector states qo. This means that A and B commute if and only if for any state P[~] the deficiency of information about B does not depend on whether a Liiders measurement (without reading) of A has been performed or not. A Liiders measurement (without reading) of an observable A which does not commute with B may thus change the deficiency of information about B. It is remarkable that the initial ignorance H(B, ~) about B can be either increased or decreased in this way ITS1 87].

III.6. Objectification III.6.1 The Objectification Problem A premeasurement of an observable is a measurement if it satisfies the objectification requirement. This condition rests on the idea that a measurement leads to a definite result. Objectification has proved to be the key problem of the quantum theory of measurement. If a premeasurement j~4 of an observable E is performed on the object system S in the state T then it leads to the final state V(T ® TA) of S +.A. The final states

74

Ill. The Quantum Theory of Measurement

of S and ,4 are the reduced states

Ts(f~,T)

:=

T,a(f~, T) :=

Tis(V(T ® T~4)), 7~,4(V(T ® T.4)).

(1) (2)

In order to formulate the idea of definite measurement results we refer to the notion of the objectivity of an observable (Sects. II.2.4 and III.2.4). In fact pointer objectification will have been achieved exactly when the pointer observable PA is objective in the final apparatus state (2). If in addition the measured observable E is objective in the final object state (1), then also value objectification will have been reached. Since from the outset the pointer observable need not be a discrete sharp observable and the pointer function f need not be an identity function, some care is required in the elaboration of these conditions. Let 7~ be the reading scale with respect to which the registration and reading of a result will be carried out and let T~ denote the final apparatus state conditioned by this reading scale,

T~'=

EpE(Xi)TA(i,T).

(3)

The objectivity of the coarse-grained pointer observable P ~ in the state (2) requires the equivalence of this state with the state (3) and the validity of the pointer value-definiteness condition (2.11). We collect the implications of previous theorems on the structure of the premeasurement M if these objectivity conditions are to be fulfilled. According to Theorem 2.4.1, a sufficient condition for both the pointer mixture condition [TA(f~, T) = T~] and the pointer value-definiteness condition [tr[TA(i,T)Pi] = 1 for all i e I with pET(Xi) ~ 0] is furnished by the mutual orthogonality of the object states Ts(i,T), i E I. Moreover for unitary measurements the condition Ts(i, T). Ts(j, T) = O, i ~ j, to hold for for all vector state preparations T - P[7~] of S, is also necessary for the equality of (2) and (3). (In such measurements the pointer value-definiteness condition is automatically fulfilled since the pointer is a sharp observable). In case the final component states Ts(i, T) and T~4(i, T), i e I, are vector states, the equality T,4(~, T) = T~ is given exactly when M is a strong state-correlation measurement (Theorem 4.4.1). The objectivity of P ~ in the state T.a(f~, T) is achieved if and only if i) the states (2) and (3) are equivalent, ii) Pi is real in T,4(i,T), and iii) the ignorance interpretation can be applied to the particular decomposition given in (3). If this is the case, then the 'definite' value f - l ( x k ) , say, of PA can be determined without changing the apparatus state in any way. Due to the reality of Pi in TA(i, T) the decomposition (3) of T~ is an orthogonal decomposition. The applicability of the ignorance interpretation to the state (3) also requires that the states V(T ® T~4) and ~ I ® P:/2V(T ® T.4)I ® p:/2 are equivalent. This corresponds to the fact that the objectivity of P ~ entails that of I ® P ~ in the

III.6. Objectification

75

present situation (cf. Sect. II.2.6). We conclude therefore that the equivalence

V(T ®

I®

V(T ® TA) I ®

(4)

is a necessary condition for the objectivity of P ~ in the final apparatus state TA(~, T) and also for the objectivity of I ® P ~ in the final state V(T ® Tx) of S + A. Obviously, condition (4) is not satisfied in general. A sufficient condition for (4) is that P ~ is a classical observable of A. In the case of a minimal unitary measurement A/[~ of a discrete sharp observable the classical nature of the pointer observable will be seen to be even necessaryfor (4) (provided that the object system is a proper quantum system). If the pointer objectification with respect to 7~ had been obtained, one would still face the question of whether the registered value f-l(Xk) of PA indicates a real property of the object system S. One could consider the requirement that the premeasurement A/t produces strong correlations between the final component states Ts(i,T) and Tx(i,T) of S and ,4. But even if the state-correlation conditions p(Ts(i,T),TA(i,T); V(T @ TA)) - 1 are realised, it could still happen that tr[Ts(i,T)Ei] ~ 1, so that E~ is not real in the state Ts(i,T). Thus, to allow for the objectivity of E n, one could stipulate further that the premeasurement A/In would produce also strong value-correlations: p(Ei, Pi; V(T @ Tx)) - 1 for all i e I and all T e S ( ~ s ) (with 0 ~ pE (X i) ~ 1). For a sharp observable E 7¢ strong value-correlation is achieved only if the measurement is repeatable (Theorem 4.3.1). Then the relation tr[Ts(i,T)Ei] = 1 holds for each i e I and for all T e S(7-/s) such that tr[TEi] ~ 0. Under these conditions the measurement A4 leads to value objectification with respect to the reading scale T~, since the objectivity of P ~ is then inherited by E n (Sect. II.2.6). Similarly, in the more general case of a strong state-correlation measurement, the objectivity of the pointer is transferred to that of those properties which are real in some of the (mutually orthogonal) states Ts(i, T).

III.6.2 Insolubility of the Objectification Problem The objectification requirement becomes problematic if the measurement coupling is taken to be unitary and both S and ,4 are proper quantum systems. In fact for a unitary measurement A/Iv of a sharp discrete observable A, the state U(~ ® ¢) is a superposition of eigenstates of the pointer observable I ® AA whenever ~ is a superposition of A-eigenstates. Thus, as long as A is not real prior to measurement, the pointer cannot be objective after the interaction. This simple model presentation of the objectitication problem rests on the idealising assumptions that the measured observable is a (discrete) sharp observable, the apparatus is initially in a pure state, and the pointer is a sharp observable. Neither of these assumptions is, however, necessary for the conclusion reached. Thus the potential objection that the measurement problem might result only from some

76

III. The Quantum Theory of Measurement

excessive idealisations, and that it would disappear if a more realistic account were given, is not tenable. The following result, the most general one to be conceived in the present formulation of the measurement theory, is obtained as the finishing touch [Bus 96c,96d] to a long development, marked by contributions of von Neumann [von Neumann 1932], Wigner [Wig 63], and many others, eg., [Fine 70, Shi 74, Bro 86]. THEOREM 6.2.1. Let E be a (nontrivial) observable of a proper quantum system 8. There is no premeasurement (7-/A, P.a,T~4, U, f} of E, with ,4 being a proper quantum system and U a unitary measurement coupling, that satisfies the pointer value-definiteness condition for ,4 and the pointer mixture condition for S + ,4,

U(T ® TA)U*

= E

I ® p:/2 U(T ® T.a)U* I ®

p1/2

(5)

for some (nontrivial) reading scale T~ and all initial states T of,9. The insolubility theorem shows the assumptions that are left open for challenge. a) Apparatus/s a proper quantum system. As mentioned in Sect. 6.1, assuming the pointer to be a classical observable would be sufficient to grant the objectification. In the next subsection it will be shown that some classical features of the apparatus are necessary for objectification. The difficulties arising with this demand are discussed in Sects. 7.2 and IV.4.1-2. b) Measurement coupling is unitary. New possibilities arise indeed if this assumption is given up. In fact, regarding (tacitly) both S and ,4 as proper quantum systems, Wigner [Wig 63] concluded that the linearity of the quantum mechanical dynamics cannot be maintained if objectification is to be achieved. This argument is taken up in various recent attempts to introduce modified quantum dynamics, thus giving room for spontaneous, autonomous processes leading to the objectivity of some macroscopic observables (cf. Sect. IV.4.3).

c) Measurement scheme provides pointer value-definiteness and pointer mixture property for 8 + ~4. Some approaches try tO dissolve the objectification problem by redefining the notion of objectivity in such a way that the pointer mixture property for S + ,4 need not be stipulated in order to be allowed to attribute definite values to the pointer. This poses the task to formulate new interpretations of quantum mechanics (cf. Sect. IV.5.1-2). Alternatively, it may turn out that pointers, being macroscopic quantities (Sect. IV.4.1), are genuinely unsharp observables; that is, the pointer value-definiteness may only be approximately realisable such that the probability pP~(x,,T)(f-I(xi)) is never equal to unity but rather equal to l - e for some perhaps small but nonzero number e. Hence there may be room for a resolution of the quantum measurement problem by relaxing the requirement of objectification into unsharp objectification. This idea will be explored in Sect. IV.5.5.

III.6. Objectification

77

III.6.3 Classical P o i n t e r O b s e r v a b l e If the assumption is given up that the apparatus is a pure quantum system, it becomes possible to fulfill the necessary objectification condition (4). This will then imply that the apparatus has some classical properties. We illustrate this consideration for the case of a minimal unitary premeasurement M ~ of a discrete sharp observable A. If ~ is an initial (vector) state of S, then the final states of S + A, S, and A are

A T~s(P[U(~ ® ¢)1) - Zp~(ai)P[Til 7~A(P[U(T ® ¢)]) = Z

~/pA(ai)pA(ak)(TilTk)lCk)(¢i[

(6b) (6c)

respectively. A natural reading scale is 7~ = {ai}ieI, so that An - A~, and the reading scale conditioned final apparatus state is A

T~ = Z Pv (ai)P[¢i].

(7)

It will be immediately observed that the states (6c) and (7) are the same for all if and only if the vectors 7i are pairwisely orthogonal (for each ~), that is, the U-defining set {¢ij } is orthonormal. In general this is not the case. The necessary condition (4) for the pointer objectification now reads

P[U(~ ® ¢)] ~ Z

(I ® P[¢i]) P[U(~ @ ¢)] (I ® P[¢i]).

(8)

Let R E/~(7-/A). If (8) is to hold for S being a proper quantum system, that is, (U(~o ® ¢)[(P ® R ) U ( ~ ® ¢)) = Z

(U(~ ® ¢)[(P ® P[¢i]RP[¢i])U(~o®¢)) (9)

for all P E P(7-ls) and for all ~o E 7"/s, then it follows that R must commute with all P[¢i]. Obviously the commutativity of R with An is also sufficient for (8). This shows that, in the present situation, the commutativity of the pointer observable with any other observable of the apparatus is a necessary and sufficient condition for the pointer objectification. Since classical observables are always objective, we can formulate this result as follows [Belt 90]. THEOREM 6.3.1. Let Az['~ be a minimal unitary premeasurement of an observable A performed on a proper quantum system S. The pointer objectification is obtained

if and only if the pointer observable AA is a classical observable. This theorem can be extended to more general measurement situations, as, for example, to an A-measurement induced by a measurement A4~ of a refinement B

78

III. The Q u a n t u m Theory of Measurement

of A = f ( B ) . The equivalence (4) still assumes the form (8), but now it only follows that the degenerate pointer observable f ( A A ) is a classical observable of ,4. In the general case of a unitary measurement Jk4v, the objectification requirement leads to the conclusion that the apparatus must have some classical properties, excluding, in particular, superpositions of 1-eigenstates of the pointer observable corresponding to different readings [Bus 90c]. The proof given in [Bus 90c] can be easily extended to the case where the pointer observable is not sharp but the measurement scheme provides pointer value-definiteness. We assume now that the pointer observable Ax of the premeasurement 2~4~ of A is indeed classical. A maximal pointer observable Ax being classical implies that any other observable of ,4 is a function of Ax and hence that that ,4 is a classical system. In that case the only pure states of the measuring apparatus ,4 are the eigenstates P[¢k] of the pointer observable. The pointer objectification condition (8) is fulfilled. In particular, the final apparatus state (6c) is equivalent to (7). The decomposition (7) is the only decomposition of T~ into pure states A of ,4. This means, in particular, that the pointer probabilities p~(ai) as well as the final state of ,4 allow an ignorance interpretation: when the apparatus ,4 is in the state T~, then it is actually in one of the pure states P[¢k], the coefficients p~A (a~) describing our knowledge about the actual state of A. The actual value of the pointer observable AA can be read without changing the actual state of ,4. The final state of the object system ,S is not directly affected by the assumption that Ax is classical. However, if the measurement A4~ produces strong correlations between the component states P['Yi] and P[¢i] (in which case the "Yi are pairwise orthogonal), then the ignorance interpretation can be applied to the final object state as well. Assuming that this is the case, if P[¢k] is the actual final state of ,4, then P['Yk] is the actual final state of S. Still, the probability (~/klEA({ak})'yk) need not be 1. If, in addition, the measurement M ~ is a strong value-correlation (or repeatable) measurement, then the U-generating vectors ¢ij, and thus ~i, are eigenvectors of A, and A is objective in the actual final state of 8. The value objectification is thereby achieved. In the present approach the classical nature of the pointer observable in a measurement M ~ of a discrete sharp observable is a sufficient (as in the general case) but also a necessary condition for the pointer objectification. The difficult problem of how to realise classical (pointer) observables within quantum mechanics cannot be tackled here. Nevertheless the above result gives an illustration of what is needed for ensuring the consistency of the minimal interpretation with the assumption that quantum mechanics is a complete theory of individual objects. In addition to the question of how to explain the existence of a classical pointer observable, this solution of the objectification problem bears with itself some further problems. First, the assumption that the unitary measurement coupling U represents an observable H of S + ,4 via the relation U = e iH cannot be reconciled with

III.6. Objectification

79

the classical nature of A,4. In other words, if H commutes with I ® AA, then there is no measurement, that is, the measured observable is trivial: E(X)= p ~ ( X ) I . Since on the other hand the classical nature of AA is inevitable for the pointer objectification and thus for the measurement, one arrives at the surprising conclusion that the unitary operator U represents a measuring coupling only if H is not an observable (Sect. 7.2). Second, if AA is a discrete maximal observable, then its classical nature forces the apparatus jt to be a discrete classical system. But such a system cannot be a carrier of the Galilei covariant canonical position and momentum observables [Mackey 1989, Piron 1976, Varadarajan 1985]. Thus a measuring apparatus cannot be a Galilei invariant quantum mechanical system and at the same time have a classical pointer. Obviously this conclusion invalidates some of the presuppositions of a universally valid quantum mechanics (Sects. 1.3, III.l.1). With these problems arising from the fulfilment of the objectification requirement, we are facing the following important conclusions, which were anticipated in Table 1 (Sect. 1.3). If it is possible to explain the origin (and to ensure the existence) of classical (pointer) observables within quantum mechanics, then the incorporation of fundamental symmetries (Galilei covariance) still requires a framework more general than that furnished by a single separable Hilbert space. Hence quantum mechanics cannot be a universal theory. If, on the other hand, quantum mechanics is unable to account for classical (pointer) observables, then there are two possible conclusions. Either one believes that (continuous) superselection rules do exist in a strict sense, which in turn means that one has to give up the universal validity of quantum mechanics. Or one accepts that quantum mechanics properly accounts for the fact that superselection rules and classical observables are only approximately realised in nature. Then quantum mechanics may be regarded as a universal theory, but at the price of a fundamentally weakened conception of reality, perhaps even an intrinsically unsharp reality.

III.6.4 Registration and Reading Registration and reading are the final steps in a measuring process. First, after the measurement interaction, the apparatus reaches a stage at which it records some outcome; that is, the pointer assumes some value on the reading scale. In this sense the process of registration is nothing but the pointer objectification, so that the respective apparatus state TA(f~, T) ~- T~ admits an ignorance interpretation with respect to its components TA(i, T). The remaining step, the reading, is performed by the observer, who in this way eliminates his ignorance and changes the description of the apparatus state according to the registered outcome. In the preceding considerations the reading scale was considered discrete. In fact it was defined as a partition of the value space of the pointer observable. There exist various types of arguments indicating that this discreteness is mandatory. a) Pragmatic need. Any physical experiment is designed to yield definite outcomes out of a collection of alternatives. These outcomes must be described by

80

III. The Quantum Theory of Measurement

essentially finite means, either by digital recordings, or by estimating a pointer position in terms of a rational number on an apparently continuous scale. b) Statistics requirement. The statistical evaluation of experimental outcomes is based on counting frequencies of mutually exclusive events out of a countable collection. This again requires the fixing of a partition of the value space of the pointer observable. c) Pointer objectitication. The fact that the pointer ultimately assumes a definite position is to be interpreted as a repeatable measurement of the pointer observable. Hence either this observable itself or one of its (actually measured) coarse-grained versions must be discrete. These arguments for the need for discrete reading scales can be substantiated in formal terms. First, the pragmatic argument a) is well illustrated by means of the information concept. Let E be a continuous sharp observable on B(R), meaning that for all X E B(R) there exists Y E B(R) such that Y C X and 0 ~ E(Y) < E(X). Further, let T~a, 7~2, ... be a sequence of increasingly finer reading scales on B(R) such that T~n+l consists of partitions of the elements of T£n. As shown in Sect. 5, we have

H(E,T;nn+x) >_H(E,T;TCn)

(10)

for all states T and for all n. Assume that the partitions approach points so that the maximum size of the partition intervals X} n), i - 1, 2, ..., of 7~n tends to zero in the following sense: for each state T, the sequence sup{tr[TE(X}n))] • i - 1, 2, ...} converges to zero. Then it follows that the sequence of numbers H(E,T; T~n) increases indefinitely for all T. Hence if there were a continuous reading scale then an E-measurement would have to lead to an infinite increase in information. Argument b) refers to the intended empirical content of the minimal interpretation of the probability measures pS. if a measurement of E in a state T had been repeated n times, and the result X had occurred v times, then limn-.oo v/n - p~ (X). A formal justification of such a statistical interpretation was reviewed in Sect. III.3. Here we recall only that the measurement statistics interpretation of the probabilities BE(x) , X E ~', T E 8(7-/s), with respect to a given E-measurement 2~4, induces a family of discretised pointer observables P ~ , 7~ a reading scale, from which the relevant probabilities

T~A(V(T®TA))

(11)

---- P T C ~ ( V ( T ® T ~ ) )

for each 7~ and any Xi (in T~) are obtained as relative frequencies; that is, for each T and each T~ there is a sequence FT,T¢ of P~-outcomes such that for all i,

relf(X~T¢) FT,~Z)="P~ ({i}) ~"R.A(V(T®T.4)) '

"

(12)

III.7. Measurement Dynamics

81

The third argument c) referring to repeatability is based on Theorem 4.6.1, applied here to the apparatus system. This argument can be carried further if also the value objectification is taken into account, which also requires repeatability (Sect. 6.1). Therefore it can be achieved only if the measured observable is discrete. The discrete nature of the reading scale entails that a given measuring apparatus allows only a measurement of a discrete version of the observable under consideration. With this we do not, however, deny the operational relevance of continuous observables. On the contrary, their usefulness as idealisations shows itself in the fact that they represent the possibility of indefinitely increasing the accuracy of measurements by choosing increasingly refined reading scales.

III.7. Measurement Dynamics III.7.1 T h e P r o b l e m An important problem of the quantum theory of measurement related both to the possibility of premeasurements as well as to the objectification requirement is the question of the physical realisability of the appropriate measurement couplings, that is, the state transformations V in the measurement schemes (7-/x, PA, TA, V, f}. Within the conventional description of the dynamics of isolated quantum systems, there are logically two possibilities: (i) the system S + jt consisting of object and apparatus can be considered as an isolated system; or (ii) the influence of the environment E on S + ,4 cannot be neglected. In the case (i) the usual description of dynamics applies, and V should be in the range of the mapping t ~ Ht, t E R, the dynamical group of S + ,4 (cf. Sect. II.1.3.). More explicitly, S and A should be dynamically independent before and after the measurement, that is, before a time t = 0 and after some time t = T > 0. This implies that the Hamiltonian H, which generates the dynamics lit, t E R, coincides with the free Hamiltonian Hs + HA before and after the measurement, while in the time interval 0 < t < r the measurement interaction comes into play:

H = Hs + HA + Hint.

(1)

The mapping V should be identified with Hr. But then the unitary dynamics t Ut = e x p ( - ~i Ht) is either discontinuous, or a time-dependent interaction Hint(t) is to be introduced. Both possibilities are, however, excluded by the continuity and the group properties of the dynamics t ~/act. This problem of incorporating V into the dynamics t ~ Ht of S + ,4 seems to allow only solutions in the sense of some approximations. The interaction part Hint of the total Hamiltonian H should be negligible before and after the measuring process. If the actual duration of the interaction is of no concern, then the canonical approach is that of describing measuring processes as scattering processes, so that V is identified with the respective scattering operator [Ludwig 1987]. However,

82

III. The Quantum Theory of Measurement

it may be desirable to account explicitly for the finite times of preparation and registration. In this case the finite duration of the interaction, that is, the apparent time dependence of Hint, needs to be explained. This can be achieved by regarding the relative motion of S and ,4 as the relevant 'clock' determining approximately the times of turning on and off the interaction. In [Aha 61] it is indicated by means of a simple model how in this way an effectively time-dependent Hamiltonian is obtained from a unitary dynamical group if the system ,4 is 'large' in some suitable sense. Another aspect of the problem of measurement dynamics is the limited number of interactions available. This contingent fact leads to a 'natural' restriction of the set of operators which correspond to actually observable quantities. In particular, the semiboundedness of the Hamiltonian entails that the probability reproducibility may be achieved, in general, only in an approximative way [Grab 90]. Turning to the second option, case (ii) above, the measurement coupling V should result from a family Pt, t E R, of linear state transformations representing the reduced dynamics of S + ,4 derived from the unitary dynamics t ~ Lit of the isolated system S + ,4 + g. But treating S + ,4 + g as an isolated system entails exactly the same problems as those encountered in case (i).

III.7.2 An Inconsistency Besides the question of realising the measurement coupling V as a part of the dynamics t ~ Pt, the objectification requirement poses additional constraints on the measurement interactions. Apart from the fact that the measurement coupling V should give rise to suitable correlations in the final state V(T ® T,4) of S + ,4, the apparatus should assume a definite pointer value at the end of the measurement. As shown in Sect. 6, this can be achieved by assuming that the pointer observable is a classical observable. This state of affairs forces one to consider a modification of quantum mechanics; in particular, it implies constraints on the dynamics of ,4 as well as of S q-,4 [Beltrametti and Cassinelli 1981]. The assumption that the pointer observable is classical implies a further puzzling feature, namely the measurement coupling cannot represent an observable of S-b A [Belt 90]. To discuss this problem, consider a unitary measurement A4u of a discrete sharp observable A, and assume that the pointer observable AA is in fact classical. The pointer observable Ax can be interpreted as an observable I ® AA of the compound system S + A. As the von Neumann algebra L:(7-/s ® 7-/,4) of bounded operators on 7-/s @ ?/,4 is generated by the operators of the product form B ® C, B E £(7~s), C E L:(7~x), one recognises that I ® AA is a classical observable of S + ,4 as well [Takesaki 1979]. This fact has an important consequence. Indeed, writing the unitary measurement coupling as U - exp(iH), and assuming that H commutes with A,4, one then has =

P~®¢

=

Pu(~®¢)

=

P

(P[u(~®¢)])

=

P~

(2)

III.8. Limitations on Measurability

83

for any unit vector ~o E ~ , since I ® Act commutes with U. Since p ~ does not depend on the initial state of the object system, Equation (2) is incompatible with the probability reproducibility condition unless A is a trivial (constant) observable. Hence we have established the following result. THEOREM 7.2.1. Let a measurement scheme (7-/ct, Act, ¢, U) be a candidate for a unitary premeasurement M u of a discrete sharp observable A. If Act is a classical observable and if the coupling U is generated by an observable o[ ,~ + A, then

(7-lct, Act, ¢, U) cannot ful~l the probability reproducibility condition. We conclude that if Act is a classical pointer observable associated to a premeasurement M u of A, then H cannot be an observable of S + A, and vice versa. As shown in Sect. 6.3, the classical nature of the pointer observable Act is a consequence of the objectification requirement in the case of a minimal unitary premeasurement M ~ . Therefore such a measurement cannot be realised by means of a unitary dynamical group/4t, the generator of which being an observable of S + ,4. If, on the other hand, only dynamical groups of this kind are available - as it is usually assumed - then no premeasurement j~4~ can serve as a measurement, since the objectification is impossible. Finally, insisting on b o t h - classical pointer Act and unitary measurement dynamics b/t - forces one into the strange conclusion that the generator H, the Hamiltonian of ,S + ,4, is no observable. The problem of incorporating the measurement coupling between 8 and j[ as a part of the dynamics of S + ,4, or $ + ,4 + g, and the inconsistency between the classical nature of the pointer observable and the physical realisability of a unitary measurement coupling lead to the consideration of modified descriptions of dynamics. It has been shown that a suitable additional term in the von Neumann-Liouville equation leads to a spontaneous pointer localisation and thus to the effective classical nature of pointer observables [Ghi 86]. Still the nonunique decomposability of mixtures would require an explicit description of dynamics as a stochastic process on the level of the Gemenge representation of states. This, in its turn, suggests considering stochastic, nonlinear modifications of the SchrSdinger equation [Gis 84] (Sect. IV. 4.3). III.8. L i m i t a t i o n s o n M e a s u r a b i l i t y

In investigating the measurement possibilities of quantum mechanics, the quantum theory of measurement also reveals the limitations on measurability. There are two types of such limitations, those implied by the theory itself, and those which may arise when the theory is supplemented by some further assumptions. The question of practical limitations in the sense of what actually can be measured in a laboratory is outside the scope of the present account. In the first group there are limitations such as 'only discrete observables admit repeatable measurements' and 'complementary observables cannot be measured

84

III. The Quantum Theory of Measurement

together'. Thus, for example, the usual position observable does not admit a repeatable measurement. Such a result is connected with our very possibilities of constituting physical objects. The fact that, say, position and momentum cannot be measured together is a basic and well-understood feature of quantum mechanics. However, the idea that the position-momentum uncertainty relations open a way to circumvent this limitation has been more controversial and has been carried out not until rather recently (for an overview, see [Busch, Grabowski, Lahti 1995]). Among the second type of limitations there is the fact that only observables which commute with all conserved observables can be measured at all. For instance, if momentum conservation is a universal conservation law, then position cannot be measured at all. However, conclusions of this type presuppose that the dynamical problem of the previous section has been solved. There are also the fundamental limitations which concern the very nature of the measuring apparatus and the measurement coupling. As became evident in Sect. 6, no proper quantum mechanical object can serve as a measuring apparatus. Moreover, if the pointer observable is to be classical, then the unitary measurement coupling cannot represent an observable in the sense of an interaction Hamiltonian of the object-apparatus system. Or conversely, if a unitary measurement coupling is generated by an observable interaction Hamiltonian, then the pointer observable cannot be classical. These fundamental limitations indicate the directions for searching the possible resolutions of the objectification problem. Finally there is also a limitation on the determination of the past and the future of an object system: complete (statistical) determination of the state (before the measurement) cannot be achieved by means of repeatable measurements. This phenomenon will be explained in some detail in Sect. 9. In the present section we shall discuss limitations on measurability implied by quantum measurement theory. Before entering into this subject, we ought to remind ourselves of some basic positive results of the measurement theory. First of all, for each observable of the object system there do exist premeasurements. Leaving aside the objectification problem, this fact confirms the idea that physical quantities are, indeed, observables, that is, they can be measured. Furthermore, for single sharp observables no a priori limitations on their measurement accuracies arise.

III.8.1 Repeatable Measurements and Continuous Observables If an observable E of S admits a measurement A4 which is repeatable, then E is discrete (Theorem 4.6.1). This fact has an important (though obvious) corollary. COROLLARY 8.1.1. No continuous observable admits a repeatable measurement. This result causes difficulties in our understanding of the operational definition of continuous observables, among them position, momentum, and energy - observables which are most important for the concept of a particle in quantum physics. We

III.8. Limitations on Measurability

85

shall illustrate these difficulties by considering the localisation observable of an object residing in three dimensional Euclidean space R 3. In fact the localisation observable of such an object can be decomposed into three similar parts referring to the three component spaces R. If the object has a nonzero rest mass, then its localisation observable in R is simply the spectral measure E Q of the position Q:

(Q~o)(x) = x~o(x) for a.e. x 6 R, ~0 6 dom(Q) c 7-/= £2(R, dx)

(1)

E Q(X)~o = Xx ~0 for any X 6 B(R), where Xx is the characteristic function of the set X 6 B(R). The spectrum of Q is the whole real line R. Q is continuous so that it does not admit a repeatable measurement. This raises the question how to define Q operationally. From the results of Sect. 2 we know that Q admits, in particular, unitary measurements, but none of them can be repeatable. Being unable to provide a canonical answer to the question posed, we shall content ourselves with demonstrating that the most obvious way of defining Q operationally is in fact ruled out. To begin with, we recall that the Borel a-algebra B(R) of R is generated by the closed intervals I of R. Hence also the range of E Q, E Q (B(R)) - {EQ(X) • X 6 B(R)}, is generated by the projections EQ(I) associated with such intervals. This means that the mapping I ~ E Q (I) (on the closed intervals) extends to the spectral measure X ~ E Q ( x ) (on B(R)) [Varadarajan 1985]. Thus to define Q it suffices to define the localisations E Q (I) associated with the intervals I. Each E Q (I) can be defined, for instance, via the state transformation T ~ E Q ( I ) T E Q (I) which corresponds to the yes-outcome of the Liiders measurement of the simple observable X, (Q). A diaphragm with a slit I is a prototype of an experimental arrangement leading to this state transformation. The natural question then is whether the mapping I H Lr(I), with I ( I ) ( T ) = E Q ( I ) T E Q (I), extends to a state transformer of Q, that is, whether by varying slit I in the diaphragm one can define Q. The answer to this question is negative. Indeed, if there were a state transformer I Q of Q such that :[Q(I) = ~(I) for all closed intervals I, then due to the additivity of the state transformer

EQ(I)TEQ(I) = EQ(I1)TEQ(I1) + EQ(I2)TEQ(I2)

(2)

for all states T and for any partition of I into disjoint subintervals 11 and/2. But (2) cannot hold true, for example, for the vector states ~ for which (~IE Q (11)~) # 0 # <~[EQ (12)~>. In fact, one can show that there is no state transformer iTQ associated with Q for which I Q ( x ) = E Q (X)TEQ (X), T 6 S(7"l), for some X 6 B(R). This result holds true for all continuous observables [Bus 90b]. These considerations show that an operational definition of the localisation observable in terms of ideal or repeatable measurements is impossible. The usual way out of this difficulty consists of restricting oneself to discrete versions of Q

86

III. The Quantum Theory of Measurement

[yon Neumann 1932] (see also argument b) in Sect. 6.4). A disadvantage of this approach is that it destroys the translation covariance characteristic of the localisation concept. Another approach is to relax strict repeatability into 8-repeatability [Dav 70]. A Q-compatible state transformer Z is 8-repeatable if for all states T and all X 6 B(R) one has tr[Z(X6)Z(X)(T)]

= tr[Z(X)(T)]

(3)

Here X~ = {x 6 l~ : I x - x ' I < 8 for some x' 6 X} denotes the closed 6neighbourhood of X. There do exist 6-repeatable, covariant, completely positive Q-compatible state transformers; but still the known examples are of a somewhat artificial nature. Therefore an even more general concept of approximate repeatability has been proposed admitting more natural realisations. A Q-compatible state transformer is (e, 8)-repeatable if it satisfies the following for all states T and all Borel sets X: tr[~(X~)Z(X)(T)] >_ (1 - e) tr[Z(X)(T)] (4) Here e and ~ are some fixed ('small') numbers. This notion of approximate repeatability may even be applied to the operational definition of general continuous observables [Busch, Grabowski, Lahti 1995].

III.8.2 Complementary Observables Consider two sharp observables A and B of an object system S. Any of their measurements can be combined into the sequential AB- and BA-measurements (cf. Sect. 4.6). The induced state transformers are the composite state transformers ~-S o ~-A and ~-A o ~-B, respectively. It may happen that for some measurements of A and B the sequential AB- and BA-measurements are equivalent, that is, ~'S o ~-A _. ~-A o ~-B. In that case we say that the sequential measurements are order independent. The existence of order independent sequential measurements for a given pair of observables A and B implies their commutativity, or, in general, their coexistence (when arbitrary observables are considered) [Lah 85]. For sharp discrete observables also the converse holds true, that is, commutativity entails the existence of order independent sequential measurements. This fact is closely related to the coexistence of A and B in the sense discussed in Sect. 5.3. Complementary observables represent an important extreme case of noncoexistent observables. Any two observables are complementary if the experimental arrangements which permit their unambiguous (operational) definitions are mutually exclusive [Pauli 1933]. This conception of complementarity of two observables A and B lends itself readily to a formal representation as a relation between the A- and B-compatible state transformers [Busch, Grabowski, Lahti 1995]. For the present purposes it is unnecessary to go into formal details. Instead we state the obvious result:

III.8. Limitations on Measurability

87

COROLLARY 8.2.1. Complementary pairs of observables have no order independent sequential measurements.

As another related no-go-theorem for the measurability of complementary observables we state the one resulting from the strong noncoexistence (total noncommutativity) of such observables. COROLLARY 8.2.2. Complementary pairs of observables do not admit any joint measurements.

Canonically conjugate position and momentum observables are complementary. The same holds true, for example, for any two (different) spin components of a spin -1 object. Finally we recall that according to the measurement inaccuracy interpretation of the uncertainty relations complementary observables can be measured together if the involved measurement inaccuracy is sufficiently large. For a systematic measurement theoretical justification of this interpretation, see [Busch, Grabowski, Lahti 1995].

III.8.3 Measurability and Conservation Laws We consider next limitations on measurability implied by the existence of universal conservation laws. Such limitations were discovered by Wigner [Wig 52]. To begin with, we restate Wigner's result as elaborated by Araki and Yanase [Ara 60], using our notations. THEOREM 8.3.1. Let (7-lA, AA, ¢, UL>be a Liiders measurement of a discrete sharp observable A. Let K = K ® 1,4 + I ® K,4 be a bounded self-adjoint operator on 7-ls ® TIA. Assume that [( is a constant of motion of S + ,4 with respect to UL) that is, [K, UL] = O. Then also [K, A] = O. This theorem suggests two divergent interpretations: INTERPRETATION I. Any discrete sharp observable A admits a Liiders measurement with a measurement coupling UL. If [( is any additive (bounded) observable of S + ,4 which is a constant of motion with respect to UL, then K commutes with A. If [K, A] ~ O, then ~[ is not a constant of motion with respect to UL.

INTERPRETATION 2. Assume that [~ represents a universal conservation law, that is, it is a constant of motion with respect to all physicaJly admissible evolutions of S + ,4. If [K,A] ~ O, then the Liiders measurement of A, with UL, is not a physically realisable measurement, that is, UL is not (a part of) a physically admissible evolution of S + ,4.

Interpretation 2 follows Wigner who writes: 'Only quantities which commute with all additive conserved quantities are precisely measurable ([Wheeler and Zurek 1983], p. 298; note that here 'precisely' refers to a Liiders measurement). To accept

88

III. The Quantum Theory of Measurement

Wigner's interpretation of the above theorem as a limitation on the measurability of certain observables it is desirable to extend this theorem to a larger class of measurement couplings between S and ,4 than the Liiders measurements. The next theorem provides such a generalisation [Belt 90]. THEOREM 8.3.2. Let M'~ be a minimal unitary premeasurement of a discrete sharp observable A. Let K = K ® In + I ® K n be a bounded self-adjoint operator on ?is @ 7-ln. If K commutes with U, then K commutes with A

(a)

provided that one of the following conditions is satisfied: P(P['Ti], P[¢i], U(9~ ® ¢)) = 1

(b)

for any i = 1 , . . . , N, and for all ~ E 7-l, [[ ~ [[= 1 for which 0 ~ N 2 ~ 1; K n commutes with An.

(c)

The objectification requirement implies that the pointer observable An is classical. Hence condition (c) of this theorem is always satisfied. Condition (b) is the strong state-correlation condition of the measurement, which is fulfilled, for instance, in the case of a repeatable measurement. Theorem 2 extends the scope of Theorem 1 in several respect. However, the relevance of these theorems as limitations on measurability is somewhat open due to the difficulties of realising the measurement coupling U as part of a unitary evolution 1At of S q- ,4. In particular, the classical nature of the pointer observable An, which ensures condition (c) of Theorem 2, entails that the generator of the group/At, the Hamiltonian H of S + ,4, cannot be an observable of S + ,4. Hence this theorem pertains to a rather strange measurement situation which certainly does not belong to the domain of conventional quantum mechanics. Nevertheless these theorems suggest that conservation laws associated with fundamental symmetries may lead to limitations on the measurability of observables. Consequently it would be desirable to investigate further extensions of Theorem 1, both for general (POV) observables [Hell 71] as well as for generalised dynamics. In particular, it is an open question whether also unbounded operators/~ give rise to limitations. There are some results indicating that this is the case for the important position-momentum pair, that is, the conservation of momentum excludes (unitary) measurements of (discretised) position [Bus 85, Ste 71]. On the other hand, it turns out that the limitations stated in Theorem 1 can be circumvented by means of introducing an arbitrarily small inaccuracy [Ara 60, Ste 71, Wig 52]. It has been shown that such inaccurate measurements can be described in terms of POV measures representing unsharp versions of the observables to be measured [Bus 85,89a].

III.9. Preparation and Determination of States

89

III.9. Preparation and Determination of States III.9.1 State Preparation In the present treatise the possibility of preparing arbitrary states T E S(7-/) was taken for granted without further analysis. Nevertheless problems associated with the concept of state were encountered in several places. In Sects. II.2.4 and II.2.6 it was pointed out that the interpretation of mixed states depends on the method by which these states are prepared. The consideration of compound systems and the omnipresence of interactions (Sect. 1.1) provide evidence for the fact that objects can presumably be prepared at best in an approximate way. Finally the ignorance interpretation of mixed states turns out crucial for the objectification problem (Sect.

6). It seems difficult to conceive of a general theory of the preparation of systems in quantum mechanics. Yet the quantum theory of measurement allows for a modelling of the process of preparation in terms of filters. In fact, a repeatable measurement is preparatory in the sense of producing systems with definite real properties. A tilter is a repeatable measurement applied to a class of similar systems and combined with a selection of the systems having a particular value of the measured observable. Combining a sequence of filters associated to a complete set of commuting discrete (sharp) observables yields a filter preparing a pure state. Equivalently a pure state may be prepared by means of a filter using a Lfiders measurement of a maximal (nondegenerate) discrete observable. Clearly this all presupposes that the measurement outcomes can be objectified. A filter based on von Neumann measurements of a degenerate discrete observable prepares mixed states, the interpretation of which depends on the particular choice of pointer observables. To illustrate this point, consider an observable A = ~-~i aiEi, with the spectral projections Ei = ~'~j P[9~ij]. Let A o - ~ aijP[~ij] be a maximal refinement of A, so that A - f(Ao), with f(a~j) - a~ for each j and for all i. Let A/tumL be a Liiders measurement of A0. Then (7"/,4,A`4, ¢, UL, f) and (7-l`4, f(A`4), ¢, UL) are two equivalent von Neumann measurements of A, leaving the object system S in the component states

Ts(i, 7~) - p~A (ai) - 1 ~jP[7~ij]P[7~]P[7~ij].

(1)

The interpretation of this state depends, however, on the measurement applied. Indeed in the first case, the object-apparatus state after reading a value ak is ~ j (I ® P[¢kj]) U(T ® T`4)U -1 (I ® P[¢kj]),

(2)

whereas in the second case the corresponding state is (3)

90

III. The Quantum Theory of Measurement

Here T - P[7~] and T.4 - PIe]. In the first case the final state (2) is a mixture representing the Gemenge Fk = ( (l(7~kjlcpll2, P[7~kj] ® P[¢kj]) " j = 1,2,...,n(i)}; for it is assumed that the maximal pointer observable A~4 is objective though its actual value in the set f-1 ((ak}) is ignored. In the second case the entanglement between the object system and the apparatus is not completely destroyed after the reading. In fact state (3) is generally a pure correlated state. In both cases the reduced state of the object system S is Ts(k, 7~) as given in (1) since the two measurements are equivalent. Their interpretation is, however, different. In the first case the state Ts(k, 7~) represents the Gemenge Fk admitting thus an ignorance interpretation. In the second case the state Ts(k, ¢p) is the reduced state of a pure correlated state and, as such, does not admit a similar interpretation. These model considerations demonstrate the possibilities of preparing states - pure states, quantum mixtures as well as Gemenge states - by means of filters. The realisation of these possibilities requires that the measuring results can be objectified, and thus it depends on the solution of the objectification problem, which has not yet been achieved. III.9.2 State Determination

Versus State Preparation

Among the principal limitations on measurements is the following mentioned in Sect. 8. No measurement is capable of determining an arbitrary initial state T from a single measurement outcome. Even the statistics of a measurement generally do not !cad to a unique state determination. In fact for a given observable E there are in general many states T, T~, ... yielding one and the same probability distribution pT E = pE, = .. ". An observable is called in?ormationally complete if it separates the set of states, that is, if for any two states T, T ~ E S(7-/), pT E = pE, implies that

T = T'.

(4)

It is well known that no sharp observable is informationally complete, whereas there do exist informationally complete unsharp observables. This fact gives rise to a new mode of complementarity in quantum mechanics - the complementarity between the determinations of the past and the future of a system [Bus 89b,95b]. Indeed one can show that an informationally complete observable E does not admit a repeatable measurement. If one aims at an optimal determination of the future (preparation) of a system, then one would choose a repeatable measurement. But then the measured observable is not informationally complete. Conversely, an optimal determination of the past via measurement of an informationaUy complete observable excludes optimal state preparation. It turns out that these two complementary goals can be reconciled in an approximative way, using the notion of (e, ~)-repeatability (cf. Sect. 8).

IV. Objectification and Interpretations of Quantum Mechanics IV.1. Routes Towards Solving the Objectification Problem The heart of the measurement problem in quantum mechanics is the objectification requirement and its implications studied in Chap. III. The theorems established there allow one to give a systematic overview of all logical possibilities left open for searching a resolution. We collect the statements and assumptions on a measurement scheme A/f that are involved in the insolubility theorems of Sect. III.6: (PR)

A/f yields probability reproducibility for an observable [Sect. III.1].

(QS)

S is a proper quantum system [Sect. II.2.3].

(QA)

,4 is a proper quantum system [Sect. 11.2.3].

(U)

The measurement coupling is given by a unitary operator [Sect. III.2.2].

(PVD) A/t satisfies the pointer value-definiteness condition for jt [Eq. (III.2.11)]. (PM)

A/f satisfies the pointer mixture condition for 8 + ,4 [Eq. (III.6.5)].

(o)

M fulfills the objectification requirement (Sects. II.2.4, III.6.1).

The quantum mechanical objectification problem, stated in Theorem III.6.2.1, can then be summarised as follows.

(PR) & ( Q S ) & ( Q A ) & (U)&

(PVD)

',. -~(PM)

(OP)

It should be noted that (PM) is stronger than the pointer mixture condition for ,4 alone. Indeed all the properties (PR), (QS), (QA), (V), (PVD), and the pointer mixture condition for ,4 are fulfilled for instance in a v o n Neumann-Liiders premeasurement of a sharp discrete observable A as introduced in Sect. III.2.5. The sentence (OP), as it stands, expresses the logical incompatibility of six assumptions. The physical meaning and relevance of this formal result depends on the evaluation of the role of each of these assumptions and therefore on the kind of interpretation of quantum mechanics that one intends to pursue. This observation leads us back to the decision tree of Table 1, Chap. I. There it was shown how the various intentions that one might attach to the understanding and purpose of a physical theory lead to different decisions as to which interpretation one will have to be committed to. The first two premises are not likely to be open to dispute: (PR) defines the relation between the given measurement scheme and some observable as the measured one, while (QS) expresses the fact that one is interested in understanding measurements performed on a proper quantum system. We shall explicate next

92

IV. Objectification and Interpretations

the evaluation of the remaining statements (QA), (U), (PVD), and (PM) from the point of view of the interpretations indicated in the decision tree. We begin with the standpoint of the realistic intentions pursued in this book, which can be summarised as follows (cf. Table 1). (A) The referents of quantum mechanics are individual systems, their properties and their time evolution. (B) Quantum mechanics is a complete theory: its state description gives an exhaustive account of the properties of a system that can be real at a given time. (C) Quantum mechanics treats measurements as autonomous physical processes which lead to definite outcomes, that is, objectification. (D) Quantum mechanics is a universally valid theory; it applies to all physical systems, whether microscopic or macroscopic. Turning these intentions into an explicit and consistent interpretation of quantum mechanics requires, first of all, specifying the meaning of the terms 'real property', 'definite outcome', and 'objectification'. To begin with, we adopt the definitions given in Chap. II. A property is real in a given state if that state is an eigenstate associated with the eigenvalue 1 of the corresponding effect. To regard this condition as being sufficient and necessary amounts to one way of adopting (B). The objectification condition was based on the ensuing notion of objectivity (Sect. II.2.4) in such a way that one has the implication (O)

~

( P V D ) & (PM).

Now (C) requires us to insist in (O), that is: objectification is to take place in each single run of an experiment. Thus our realistic attitude forces us to read the implication (OP) as follows. (Here, as well as in the forthcoming implications, the premises are taken to be valid, so that the conclusions show the price to be paid.) (PR) & ( Q S ) &

(O)

;- - ~ ( U ) v - ~ ( Q A ) .

To solve the objectification problem, one must therefore abandon either (U) or (QA). One has to commit oneself to one of the following options. (PR) & ( Q S ) & (PR) & ( Q S ) &

(QA)&

(O)

==~ -~(U);

(U)&

(O)

==,

~(QA).

(MD) (CP)

We refer to these conclusions as the modified dynamics (MD) and classical pointer (CP) option, respectively. Both amount to accepting that quantum mechanics cannot fulfil a/1 of the intentions (A), (B), (C), (D) in the sense formalised here. In fact the unrestricted, universal validity (D) of quantum mechanics in its original form is

IV.1. Solving the Objectification Problem

93

called into question: either one takes into consideration some nonunitary dynamical law, or one starts looking for theories of macrosystems that differ from quantum mechanics at least to the extent that some classical observables (superselection rules) are taken into account. In the latter conclusion we have made use of the fact that (QA) is invalidated in the sense described in Theorem III.6.3.1. The two options outlined here require the objectification to take place in the strong sense formulated in Chaps. II and III. They give rise to a class of approaches towards solving the measurement problem which are ready to go beyond quantum mechanics, thereby giving up (D) while maintaining (A), (B) and (C) for the modified theory (Sect. 4). It may still be possible to preserve all four realistic intentions, including the universality (D). The only way then to cope with (OP) is to modify the notions of reality, objectivity and objectification, (PR) & ( Q S ) &

(QA)&

(U)

==,

--(O).

(MO)

This line of approach imposes on any interpretation the task of explaining why and in which sense objectification appears to take place 'in the eyes (or minds)' of conscious observers. More generally, one sets out to meet the challenge of understanding how quantum mechanics can account for the emergence of a classical world as it is perceived by human observers. This task has been pursued in the many-worlds interpretation and some more recent variations of it (Sect. 5). Most of these approaches are based on the conception of definite, sharp pointer positions, so that (PVD) is maintained. Also, the realisation of the pointer mixture condition for A is still being pursued, while the corresponding condition for ,S + ,4 (PM), and thereby (O), is explicitly given up. There are two opposing attitudes, an epistemologically optimistic and a skeptical one, which call into question the whole project of a realistic interpretation of quantum mechanics. The optimistic attitude entertains the hope that there is a more comprehensive, hidden variable theory which renders quantum mechanics incomplete; this amounts to denying (B). As a consequence one would view quantum mechanics as a mere ensemble theory, reflecting the statistical aspects of the underlying theory. In this way the formal result (OP) would become irrelevant: the phenomenon of nonobjectivity, which gives rise to (OP), would have to be reinterpreted in terms of some kind of 'cryptodeterminism' (Sect. 3). The skeptical attitude would reject the realistic point of view altogether and stick to the empiricist-instrumentalist wisdom that quantum mechanics, as any physical theory, is just a tool for computing and predicting measurement outcomes. In this view (A) is given up because one does not want to get involved with interpretational problems that may have no bearing on the empirical content of the theory. Instead one adheres to the minimal interpretation and nothing more. A similarly cautious attitude was adopted by the pioneers of quantum mechanics who announced some or another variant of Copenhagen interpretations (Sect. 2).

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With the last-mentioned option we have reached the top of the decision tree (Table 1) and exhausted the list of possible ways of dealing with (OP). There is yet one residual potential loophole to the derivation of (OP): the measurement theory developed in Chap. III makes extensive use of the assumption that initially the object system and apparatus are independent of each other, and that the apparatus can be prepared in one and the same initial state T~ for several runs of the measurement. Since the apparatus is a macroscopic system, this hypothesis of the reproducibility of Tx may be regarded as being of a counterfactual nature. There are various attempts to exploit the possible microscopic nonreproducibility of the apparatus preparation [Mach 80, Schu 91]. In one approach [Schu 91] it is argued that in each single run of an experiment the apparatus is in some 'special initial state' such that after the interaction a definite pointer state is reached; and that some 'pre-established randomness' in the object system's environment is capable of recognising the state of S and thus of conspiring with the apparatus to make it show the right statistics. It does not seem obvious to us how such a view should provide (a natural explanation) for the probability reproducibility condition (PR); at the least, without the assumption of the initial independence of S and ,4, the whole formulation of quantum measurement theory would need a fundamental revision. Therefore we do not pursue this particular option here. Each of the options listed here has been taken up in one or another approach to the interpretation of quantum mechanics, and is still the subject of intensive efforts. A commonly accepted resolution of the measurement problem does not seem to be in reach, and it may still be too early for any conclusive assessments. This seems true even in view of the current appearance of various comprehensive accounts of the different approaches. Therefore we feel free in our subsequent survey to just indicate the systematic position of the different approaches in the logical scheme developed here, with some comments on their merits and open questions. In order to explain the spirit of our modest critical remarks, we first describe our general views of the purposes that an interpretation of quantum mechanics should serve and of the criteria that the formulation of an interpretation should meet. To establish an interpretation of quantum mechanics (or of any physical theory) first of all means to formulate a set of rules that connects symbols and relations of the mathematical formalism with a certain domain of the 'physical world', that is, a domain of experience constituted by a collection of experimental procedures. Thus an interpretation is that part of a physical theory that determines the physical meaning of concepts and laws and thereby makes the formalism a mathematical picture of the given domain of physical experience. In general there is not a unique interpretation associated with a given formalism; and strictly speaking, each interpretation will fix another theory. Therefore it may sound paradoxical to speak of different interpretations of quantum mechanics. We resolve this puzzle by understanding 'quantum mechanics' or 'quantum theory' as that physical theory that is

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based on the usual Hilbert space formalism and equipped with the minimal interpretation introduced in Chap. II; any other 'interpretation of quantum mechanics' then augments this theory with further statements attaching meaning to some symbols and relations. It is important to realise that there is no complete freedom and arbitrariness in choosing an interpretation of quantum mechanics. There is always a tension between the desiderata that one might wish an interpretation to fulfil and the empirical and formal constraints that have to be taken into account. On the one hand, an interpretation should be sufficiently rich so as to be an instructive guide in the consideration of physical problems and the development of new ideas. We maintain that in the last instance it is only with a well-elaborated interpretation that has been checked in as many of its facets as possible that one can ascertain the correctness of the intuitive pictures that one always uses in the formulation of models and the conception of experiments. On the other hand, any interpretation must obey the constraints of empirical adequacy [i.e., there must not be a contradiction with experimental results] and self-consistency [i.e., the interpretation should not entail relations that are in logical conflict with the formal structures of the theory]. The empiricist skepticism points to the danger that too rich an interpretation may produce inconsistencies or paradoxa, which may in fact be quite subtle and hard to discover. The objectification problem itself is an example of the kind of problems that one may run into when it comes to formalising certain ideas: here it is the notion of reality or objectivity that one tries to implement into the quantum mechanical formalism. However, against this skepticism we hold that all the intuition and creativity invested into the development of physics always went far beyond what would be allowed by a purely empiricist or instrumentalist position. Therefore it would be the genuine task of a realistic interpretation, the intentions and expectations of which are expressed in statements such as (A), (B), (C), (D) above, to provide the conceptual tools allowing one to control and hopefully reconfirm the intuitive ideas about what picture of physical reality the given theory yields. The subsequent sections will illustrate further some of the formal problems that need to be addressed in the elaboration of any interpretation of quantum mechanics that sets out to solve the objectification problem. We start with a more or less historical review of the extreme epistemological positions mentioned above which try to evade the objectification problem in different ways.

IV.2. Historical Prelude - Copenhagen

Interpretations

The problem of measurement, that is, the clash between the phenomenon of nonobjectivity and the need for a classical description of the apparatus, was clearly envisaged by the pioneers of quantum mechanics. They tried to deal with it by emphasising the methodological necessity of placing a 'cut' between the object to be

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described by the theory and the means of observation. In order to evaluate the current attempts at solving the measurement problem in their proper historical context, we find it useful to review briefly some of the representative views of the Copenhagen school on the measurement problem. We shall start by presenting the intuitive ideas developed by Bohr and Heisenberg and then go on with the more systematic studies by von Neumann and Pauli. We also recall the related views of London and Bauer and of Wigner. In speaking of the Copenhagen interpretation of quantum mechanics one is usually referring to the interpretation of quantum mechanics which resulted from the discussions between Niels Bohr, Werner Heisenberg, and Wolfgang Pauli with contributions from Max Born and Johann von Neumann. In very broad terms one may say that these discussions were the first attempts to solve the interpretational problems of quantum mechanics considered as a fundamental theory of individual atomic objects. The classical papers by Born [Born 26], Heisenberg [Heis 27] and Bohr [Bohr 28] are important landmarks in this early development, which led to the systematic treatises by von Neumann [yon Neumann 1932] and Panli [Faun 1933] on the mathematical and conceptual foundations of quantum mechanics. The acceptance of the probability interpretation for the Schrbdinger wave function and the acknowledgement of some fundamental limitations on the applicability of the concepts of classical physics in the description of atomic phenomena were important elements in the Copenhagen interpretation, which, however, never developed into a coherent systematic interpretation of quantum mechanics. Still Bohr's Como lecture may be considered to some extent as a codification of the Copenhagen viewpoint. In fact, as is now well known, the views of Bohr, Heisenberg and Pauli on the interpretation of quantum mechanics were divergent and diffuse to the extent that no unique interpretation could have been built on them and that almost any of the present day interpretations of quantum mechanics can be argued as being a systematic development of some of the 'Copenhagen views'. B o h r . According to Bohr the key to the understanding of the quantum theory was the viewpoint of complementarity which he advocated and developed in a series of articles published during the years 1927- 1962. Rather than attempting to give a systematic exposition of Bohr's views and to argue which of the isms Bohr favoured we provide a few citations from the writings of Bohr which, we think, are revealing with respect to Bohr's account of the measurement process. [Bohr 28] The quantum theory is characterised by the acknowledgment of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests essentially upon the classical physical concepts.

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• .. Quantum postulate implies a renunciation as regards the causal spacetime co-ordination of atomic processes. [But] the idea of observation belongs to the causal space-time way of description. • .. the complementary character of the description of atomic phenomena • .. appears as an inevitable consequence of the contrast between the quantum postulate and the distinction between object and agency of measurement, inherent in our very idea of observation. [Bohr 39] A clarification of the situation as regards the observation problem in quantum theory ... was first achieved after the establishment of a rational quantum mechanical formalism. • .. In the first place, we must recognise that a measurement can mean nothing else than the unambiguous comparison of some property of the object under investigation with a corresponding property of another system, serving as a measuring instrument, and for which this property is directly determinable according to its definition in everyday language or in the terminology of classical physics. [Bohr 48] An adequate tool for the complementary mode of description is offered by the quantum-mechanical formalism. The idea of the complementarity of observables first conceived by Bohr is expressed in the present approach to quantum measurements by the mutual exclusiveness of state transformers according to Sects. III.2.1 and III.8.2. For Bohr, the referents of quantum mechanics are observed phenomena, and the notion of an individual microsystem is meaningful only within the context of the whole macroscopic experimental setup defining the phenomenon under observation. The classical description of measuring devices is reproduced here by the requirement of the pointer objectification and of the classical properties of the pointer observable (Sect. III.6.3). H e i s e n b e r g . The breakthrough for the proper understanding of the formalism of quantum mechanics lay, according to Heisenberg [Heis 27], in the discovery of the uncertainty relations. These relations showed, in a quantitative way, the limitations on the applicability of classical concepts to the description of atomic phenomena. As mentioned at the end of Sect. III.8.2, the uncertainty relations at the same time may be interpreted as a relaxation of the complementarity. Though Heisenberg's starting point was more formal than Bohr's, Heisenberg later acknowledged Bohr's viewpoint of complementarity as the fundamental basis of quantum mechanics. With respect to an analysis of the measuring process this appears in the fact that, like Bohr, Heisenberg emphasised the methodological necessity for clearly distinguishing the object under investigation and the applied measuring apparatus. Whereas the object was to be described in terms of quantum mechanics, the measuring apparatus

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had to be described in classical terms. The following extracts from Heisenberg's writings illustrate these remarks.

[Heisenberg 1930] It must also be emphasised that the statistical character of the relation [between the values of two quantum mechanical observables - authors' note] depends on the fact, that the influence of the measuring devices on the system to be measured, is treated in another way than the mutual influence of the parts of the s y s t e m . . . . If one were to treat the measuring device as a part of the system ... then the changes of the states of the system considered above as indeterminate would become determinate. But no use could be made of this determinateness unless our observation of the measuring device were free of undeterminateness. For these observations, however, the same considerations are valid, and we should be forced, for example, to include our own eyes as part of the system, and so on. Finally, the whole chain of cause and effect could be quantitatively verified only if the whole universe were incorporated into the s y s t e m - but then physics has vanished and only a mathematical scheme remains. The partition of the world into observing and observed system prevents a sharp formulation of the law of cause and effect. [authors' emphasis] (The observing system need not always be a human being; it may also be an apparatus, such as a photographic plate, etc.)

[Heisenberg 1958] In natural science we are not interested in the universe as a whole, including ourselves, but we direct our attention to some part of the universe and make that the object of our s t u d i e s . . . , it is important that a large part of the universe, including ourselves, does not belong to the object. • .. [Before or at least] at the moment of observation our object has to be in contact with the other part of the world, namely the experimental arrangement [which is to be described in terms of classical physics]. • .. After this interaction has taken place, the probability function contains the objective element of tendency and the subjective element of incomplete knowledge, even if it has been a 'pure' case b e f o r e . . . The observation itself changes the probability function discontinuously; it selects of all possible events the actual one that has taken p l a c e . . . . • .. the transition from the 'possible' to the 'actual' ... takes place as soon as the interaction of the object with the measuring device, and thereby with the rest of the world, has come into play; it is not connected with the act of registration of the result by the mind of the observer. The discontinuous change in the probability function, however, takes place with the act of registration, because it is the discontinuous change of our knowledge in

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the instant of registration that has its image in the discontinuous change of the probability function. • .. quantum theory corresponds to the ideal of objective description of the world as far as possible. Certainly quantum theory does not contain genuine subjective features, it does not introduce the mind of the physicist as a part of the atomic event. But it starts with the division of the world into the 'object' and the rest of the world, and from the fact that at least for the rest of the world we use the classical concepts in our description. The partition of the world into an observed system S and a measuring apparatus ,4 which is inevitable for an objective description of the physical system is reformulated in the present report by the reduction of the state of the compound system S + j t into the reduced states of 8 and j[, respectively (Sect. II.1.2). Within the framework of our approach the apparatus ,4 should be described by means of classical concepts, since only in this way can the objectification of the measuring results be achieved (Sect. III. 6). von N e u m a n n a n d Pauli. von Neumann did not accept Bohr's view, shared by Heisenberg, of the necessity of classical language in the description of atomic phenomena. On the contrary, according to von Neumann, quantum mechanics is a universally valid theory which applies equally well to the description of macroscopic measuring devices as to microscopic atomic objects. It may well be that a detailed description of a measuring apparatus is highly complicated but, according to von Neumann, no limitation in principle is known for a description of a measuring apparatus and thus of the whole measuring process in quantum mechanics. As is well known, and already noted in Chap. III, von Neumann developed the quantum mechanical theory of the measuring process in a way that still meets today's standards of rigour. Within his approach von Neumann clearly faced the problem that the object system and the measuring apparatus had to be separated after the premeasurement and that this problem could not be solved within quantum mechanics. To solve the dilemma, von Neumann introduced what is known as the projection postulate, and he argued that the final termination of any measuring process is in the conscious observer, in his becoming aware of the measurement result. Von Neumann also argued that - apart from the fact that in the course of the development of physical theories the borderline between a physical object, like the measuring apparatus, and the conscious observer is as if shifted in the direction of the latter - the conscious observer cannot be included in the domain of any physical theory. We let von Neumann speak (in the 1955 translation), using, however, the technical notations of the present text.

[von Neumann 1932] We therefore have two fundamentally different types of interventions which can occur in a system $ . . . . First, the arbitrary changes by measurements

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T ~ Zu(R)(T).

Second, the automatic changes which occur with the passage of time (2)

T ~ U~TUt.

• .. In the measurement we cannot observe the system S by itself, but must rather investigate the system S + ,4, in order to obtain (numerically) its interaction with the measuring apparatus A. The theory of measurement is a statement concerning S + ,4, and should describe how the state of S is related to certain properties of the state of ,4 (namely, the positions of a certain pointer, since the observer reads these). • .. the measurement or the related process of the subjective perception is a new entity relative to the physical environment and is not reducible to the latter. • -. But in any c a s e , . - . , at some time we must say: and this is perceived by the observer. • .. Now quantum mechanics describes the events which occur in the observed portions of the world, so long as they do not interact with the observing portion, with the aid of process 2, but as soon as such an interaction occurs, i.e., a measurement, it requires the application of process 1. The dual form is therefore justified. Another member of the Copenhagen school who developed measurement theory in a systematic way was Wolfgang Pauli. His analysis of the measurement problem is almost literally the same as the one given by von Neumann. Indeed Pauli writes as follows (we here give the 1980 translation):

[Pauli 1933] The measurement . . . generates in general ... out of a pure case ... a mixture -.. [cf. T ~-~ Zu(R)(T)]. This r e s u l t . . , is of decisive importance for the consistent interpretation of the concept of measurement in quantum mechanics. For this result shows that we arrive at consistent results concerning the system, whatever be the way in which the division between the system to be observed (which is described by wave functions) and the measuring apparatus is made. (Cf. J.v. Neumann (1932), where in Chap. VI this question is discussed in detail.) • -. It is possible to express the fact that a definite measuring apparatus will be used in the mathematical formalism of quantum mechanics directly. On the contrary, this is not possible with the stipulation that the measurement should give a definite result ... Any statement about a physical fact made with the help of a measuring device (observer or the registration apparatus) which is not counted as part of the system cannot (from the standpoint of mathematical formalism which describes directly

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101

only probabilities) present a particular, scientifically not pre-determined act which is to be taken into account by a reduction of the wave-packets... [cf. T ~ Zu(R)(T) ~ [Zu(X)(T)]-1Zu(X)T]. We need not be surprised at the necessity for such a special procedure if we realise that during each measurement an interaction with the measuring apparatus ensues which is in many respects intrinsically uncontrollable. The theory presented in Chap. III can be seen as a further elaboration of the treatment by von Neumann and Pauli, with the important difference that here the measurement problem is not 'solved' with a reference to a conscious observer. S o m e e l a b o r a t i o n s . The analysis of the measuring process as given by von Neumann remained more or less unknown for a long time, perhaps due to its then advanced mathematical presentation. Some authors, like London and Bauer [Lon 39], appreciated this work of von Neumann but they felt the need for a 'concise and simple' treatment of the problem. London and Bauer developed the theory of the quantum mechanical measurement process just in accordance with von Neumann. However, they introduced one significant change into von Neumann's description, namely, by including the conscious observer as a part of the quantum mechanical description of the measurement process. Indeed they considered quantum mechanically a system consisting of the object system S, the measuring apparatus ,4, and the observer (9. To solve the objectification problem, London and Bauer went on to assume that the observer can, by 'introspection' and with his 'immanent knowledge', always rightly create his own objectivity, and thus identify his own pure state. 'I am in the state P[pk]' so that, due to correlations, the measuring apparatus ,4 is in the state P[¢k] and thus the object system $ is in the state P[Tk]. The London-Bauer interpretation of the measurement according to which the objectification of the measuring result is provided only by the activities of the observer's consciousness was criticised by Wigner [Wig 61]. A philosophical analysis of the London-Bauer approach in the light of Wigner's consideration was given by Shimony [Shi 63]. With his 'friend paradox' Wigner gave an illustration that after the measurement interaction between S, ,4 and O the observer (.9 (Wigner's friend) will always be in a state which indicates a definite measuring result. On the other hand, if the total compound system S + A + (.9 were treated by means of ordinary quantum mechanics in Hilbert space, then the observer £9 would be in a superposition of possible final states, that is, in a state of 'suspended animation'. For Wigner [Wig 63] this result appears absurd and he concludes that the quantum mechanical equations of motion cannot be linear, if macroscopic objects like ,4 and (.9 are described. However, no explicit proposal for a nonlinear modification of the SchrSdinger equation was made. Systematic investigations in this direction started only around 1976 (Sect. 4.3).

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Later Wigner changed his mind and adopted the point of view expressed by Zeh [Zeh 70] according to which the interaction of the macroscopic system (,4 or (9) with the environment destroys the nondiagonal elements in the density matrix of this system (cf. Sect. 5.3). Wigner proposed a modified von Neumann-Liouville equation which describes the (exponential) decrease in time of the nondiagonal elements in the density matrix of macroscopic systems and thus should lead automatically to the desired objectification of the measuring result. The consciousness of the observer, which in the London-Bauer approach was assumed to solve this problem is no longer needed for the objectification. Moreover, according to Wigner [Wig 83], the consciousness of the human observer is beyond the scope of quantum physics and classical physics.

IV.3. E n s e m b l e and H i d d e n Variable Interpretations According to the optimistic epistemological position mentioned in Sect. 1, physical objects and their properties are just as they are observed in the macroscopic world; the classical deterministic realism thus maintained leads one to conclude that the objectification problem is merely an illustration of the incompleteness of quantum mechanics. This theory should therefore be regarded as a statistical framework allowing one to describe the outcomes of measurements performed on an ensemble of equally prepared objects. The rise of the so-called ensemble and hidden variable interpretations of quanturn mechanics is much due to the critique of Albert Einstein and Erwin Schr5dinger on quantum mechanics as a fundamental theory of individual atomic objects. The probabilistic nature of quantum mechanics and the incompleteness argument of Einstein, Podolsky, and Rosen [Eins 35] led, on the one hand, to a consideration of quantum mechanics only as a statistical theory of atomic objects, and, on the other hand, to the development of proper completions of the theory. The critique of the projection postulate, as put forward by Margenau [Marg 36,63], was a further impulse for developing the ensemble interpretations of quantum mechanics. Though much of this critique was later found to be unjustified, these interpretations still have their advocates. The statistical interpretations of quantum mechanics can be divided into two groups, the measurement statistics and the statistical ensemble interpretations (Sects. III.3.2-3). These interpretations rely explicitly on the relative frequency interpretation of probability, and in them the meaning of probability is often wrongly identified with the common method of testing probability assertions. In the measurement statistics interpretation the quantum mechanical probability distributions, such as pA, are considered only epistemically as the distributions for measurement outcomes. The concept of state is taken to characterise conceptual infinite sequences FT A of measurement outcomes at,, a l 2 , ' " such that

IV.3. Ensemble and Hidden Variable Interpretations

103

n pA(x) = limn--.o~ ~1 ~-']~i=1 Xx (at,). In this pragmatic view quantum mechanics is

only a theory of measurement outcomes providing convenient means for calculating the possible distributions of such outcomes. It may well be that such an interpretation is sufficient for some practical purposes; but it is outside the interest of this treatise to go into any further details, for example, to study the presuppositions of such a minimal interpretation. The measurement problem is simply excluded in such an interpretation. The statistical nature and the alleged incompleteness of quantum mechanics are apparent in the ensemble interpretation of quantum mechanics. In this interpretation a state T characterises a conceptual infinite collection FST of identical, mutually noninteracting systems $1, S2,.... The probability measures pT A defined by a state T describe the distribution of the values of an observable A among the members Si of the ensemble FST. Accordingly, the number pA(x) is the relative abundance of systems $i in FST having the value of A in the set X. The ensemble interpretation of quantum mechanics describes individual objects only statistically as members of ensembles. This interpretation is motivated by the idea that each physical quantity has a definite value at all times. Thus no measurement problem would occur in this interpretation. Some merits of the ensemble interpretation of quantum mechanics are put forward, for example, in [Bal 70,88, d'Espagnat 1971]. But these merits seem to consist only of a more or less trivial avoiding of the conceptual problems, like the measurement problem, arising in a realistic approach. In fact it is only in the hidden variable approaches that one tries to take seriously the idea of the value-definiteness of all quantities. The basic idea of the hidden variable interpretations of quantum mechanics was to 'complete' the probabilistic state description of quantum mechanics with some 'hidden variables' to obtain a dispersion-free state description providing the quantum mechanical probabilities as statistical averages over the 'hidden variables'. Strictly speaking, this type of approach sets out to formulate a more extensive theory than quantum mechanics in that it tries to give a wider mathematical framework than the Hilbert space formalism in order to allow for the hidden variables. Much work has gone into this project, and a number of no-go-theorems emerged restricting the possible forms of such interpretations. It follows from these theorems that only the so-called contextual non-local forms of the hidden variable interpretations are tenable at all. The model of Bohm and de Broglie, also called the causal interpretation of quantum mechanics, is the best known example in this direction [Bohm 52,66,89] It provides a picture of the quantum mechanical reality in which individual particles have definite positions at all times in such a way that they follow trajectories governed by a deterministic Newtonian law of motion. Quantum mechanics and its non-local features are taken into account by the presence of the so-called quantum potential. The underlying deterministic equation of motion generating the trajec-

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tories can be taken in the form of the Liouville equation and combined with the quantum equilibrium condition, that is, the initial condition that at some fixed (initial) time the distribution function coincides with the squared modulus of some wave function. Then quantum mechanics can be recovered along with its minimal interpretation. This claim, a proof of which was outlined by Bohm, has been substantiated considerably by extensive mathematical investigations in the 'Bohmian mechanics' [Diirr 92,93] approach, which has experienced a remarkable renaissance in the last years. Physical applications of the causal interpretation have been elaborated in numerous examples, providing impressive demonstrations of its intuitive appeal. Holland [Holland 1993] gives a fair and comprehensive account of the merits and difficulties of this approach. The causal interpretation selects the position as a preferred variable to which it attributes value-definiteness in all states. For all other observables it can be shown that due to the definiteness of the positions, measurements always lead to definite outcomes which are causally determined and which occur with the frequency predicted by the quantum mechanical probability. The measurement problem is thus solved in this approach. The nature of this solution becomes apparent if one realises that the Bohm theory is formally equivalent with quantum mechanics: the 'hidden variable' is suggested by the form of the Schrhdinger equation itself, and the quantum equilibrium condition ensures that no statements arise that may be in conflict with quantum mechanical propositions. Moreover, this interpretation maintains the unitary Schrhdinger evolution also in measuring processes, so that it is only the value assignment to the position observable by which it differs from the realistic interpretation of Sect. 1. Thus it turns out that what had been intended to be a hidden variable augmentation of quantum mechanics is a variant of the modal interpretations" the notions of reality and objectivity are modified so as to allow value assignments for a preferred observable in all states [Bub 96] (see Sect. 5.2 below). It should be noted that selecting a particular observable as having definite values stands quite in contrast to the symmetric treatment of all observables in the realistic interpretation fixed in Chap. II. Therefore, in adopting the Bohmian or the modal point of view, one should be able to give good physical reasons as to why there should be such a 'preferred' observable. The Bohm theory is deterministic but non-classical in the sense that the equation of motion for the definite position variable contains the non-local quantum potential. There is a class of hidden variable theories that are genuinely classical in trying to account for the quantum phenomena by introducing local interactions with a fluctuating background medium. Some more recent variants of these approaches are the stochastic mechanics reviewed in [Nelson 1985] or the stochastic electrodynamics (e.g., [Boy 80, Cole 90]). The classical stochastic theories are ruled out in principle as being incompatible with quantum mechanics by the KochenSpeaker-Bell theorems (see, e.g., [Peres 1993]). But as far as experimental testing

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105

is concerned, there is still a debate on whether the apparent agreement between quantum mechanics and the Einstein-Podolsky-Rosen experiments could be due to the low detector efficiencies and whether better experiments might produce results in favour of a classical stochastic theory. As long as the 'efficiency loophole' has not been closed, it is interesting to note that there is completely independent experimental evidence which conclusively rules out a large class of stochastic hidden variables and requires at least that such theories must be endowed with nonlocal features similar to those of the Bohm theory. In fact Baublitz [Baub 93] pointed out that quantum mechanics and classical stochastic theories give different predictions for the average change of energy (AE) per electron in the tunnelling taking place in field emission experiments. Analysing the data of experiments performed since the 1970s, Baublitz [Baub 95] was able to show that the experimental values of (AE) are in good agreement with quantum mechanical calculations but differ significantly from the values required by some local stochastic theories. There are some further recent hidden variable formulations which may not easily be attacked by the known counter arguments. We mention two such approaches, without going into details. The first one, proposed by the late Asim Barut [Barut 90], follows closely the original ideas of SchrSdinger on the matter wave interpretation of the C-function. The other approach, elaborated by Gudder [Gudder 1988], is based on the notion of quantum transition amplitudes and allows for value-assignments at the price of non-standard modifications of the probability theory.

IV.4. Modifying Quantum Mechanics Quantum mechanics does not seem to be easily reconciled with a classical, deterministic conception of reality. Thus, if one accepts the completeness of quantum mechanics and still intends to maintain the goal of objectification (0), then one is forced to question the universal, unrestricted validity of the theory: either one pursues the option ( C P ) in order to ensure the required classical properties of some macroscopic systems, or one searches for a modified description of the dynamics of quantum systems (MD) so that a unification of von Neumann's two types of state changes under one common law is achieved. The first option has gained some general support on the basis of abstract foundational studies (Sect. 4.1) while it was also studied in terms of models (Sect. 4.2). The latter possibility has been dealt with mainly in terms of ad hoc proposals for nonlinear or stochastic modifications of the SchrSdinger equation (Sect. 4.3).

IV.4.1 Operational Approaches and the Quantum-Classical Dichotomy Common to all the variants of Copenhagen interpretations is the emphasis of the operational aspect that quantum machanics makes statements about objects as they appear under the conditions of observation, or measurement. Furthermore,

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the necessity of describing the macroscopic aspects of a measurement in terms of classical physics was always pointed out and seen in contrast to the phenomenon of nonobjectivity, or indeterminacy. This situation forces one to consider the possibility that quantum mechanics needs to be complemented with an essentially classical theory of macroscopic systems in order to account for the classical behaviour of measuring devices and the occurrence of definite pointer readings. A systematic reflection of these points has been undertaken in various approaches aiming at an operational reconstruction of the structure of quantum mechanics. These approaches set out to investigate the pragmatic and physical preconditions for the constitution of physical objects, including the microscopic objects of quantum mechanics. In the quantum logic approach it is found that the formal structures that can be justified in this way are open to the possible existence of (partly) classical systems. Hence according to this approach there are no a priori operational reasons inherent in the general language structure of a physical theory, which would either exclude or require the existence of superselection rules. This is a demonstration of the fact that the measurement problem is indeed specific to the irreducibility of the lattice of propositions of ordinary Hilbert space quantum mechanics. One may thus be led to consider formal extensions of ordinary quantum mechanics by stipulating the presence of superselection rules (Sect. 4.2). Another type of operational approach starts with an analysis of the general statistical structure of physical theories [Ludwig 1983]. This procedure allows one to investigate, in very general terms, the relationship between objective, deterministic theories for macroscopic phenomena and quantum mechanics as a theory for microscopic systems. If it can be shown that the former cannot be derived from a many-body extrapolation of the latter, then the universal validity of quantum mechanics is lost, and there is no reason to expect that measuring devices belong to the domain of quantum mechanics [Lud 83]. Contrary to the usual textbook wisdom it has been argued as a result of careful analysis that macrophysical theories, like thermodynamics or classical mechanics, cannot be derived as approximate limiting cases of quantum mechanics. Such theories should rather be regarded as theories in their own right. Yet as macroscopic systems are aggregates of microscopic systems, it is important to understand the interrelation between quantum mechanics and macroscopic theories. This programme has been carried out to quite a large extent in the form of an embedding of macrotheories into a quantum mechanical many-body theory [Barc 82, Ludwig 1987]. The effect of such an embedding can be described, roughly speaking, as restricting the sets of states and observables of the many-body quantum theory so that the remaining structure allows for the emergence of a classical (deterministic) behaviour of the macroscopic quantities. Some concrete examples of embeddings illustrating the abstract approach can be found in the contributions of Lanz and Melsheimer, and Neumann in [Cologne 1993].

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107

The conclusion to be drawn from these investigations is that the reconciliation of quantum mechanics with an objective macroscopic description causes difficulties even to such an epistemologically cautious approach, which abstains from farreaching realistic aspirations and uses only a minimal interpretation. The classical behaviour of macroscopic observables cannot be derived from quantum mechanics; it is only shown that the classical features of macroscopic systems can be consistently described in an approximate way within quantum mechanics [Lud 93]. Even then, the macroscopic quantities must be represented as unsharp observables in order to ensure the consistency of the embedding in question (cf. [Ludwig 1987], Chap. X.2.5). In this way the measurement problem has been dissolved at the expense of giving up the universal validity of quantum mechanics.

IV.4.2 Classical Properties of the Apparatus Sometimes it has been argued that the object system S itself, as well as the apparatus system A, have naturally restricted sets of observables so that not all selfoadjoint operators correspond to observables. Such limitations on measurability may be due to fundamental conservation laws [Ara 60, Wig 52], or due to the limited number of actually existing interactions [Pro 71] (cf. Sect. III.8.3). While in this way one circumvents the implication ( o e ) by exploiting a potential violation of (QS), it is an open question whether objectification can be achieved in this way. Such a solution would not have the status of a theoretically well-founded approach either but would rest upon accidental facts as long as there were no theory of the fundamental interactions. This, admittedly vague, option is mentioned here just for completeness as one logical possibility of trying to deal with (OP). A more effective attempt at tackling the objectification problem is the (more or less ad hoc) consideration of superselection rules for the apparatus. That is, one assumes that the pointer, being an observable of a macroscopic system, should be a classical observable [Beltrametti and Cassinelli 1981, Jau 64,67,68, Wan 80]. Then the objectification (0) is ensured since the pointer observable is objective throughout. However, the dynamical consistency problem described in Sect. III.7 still persists and requires a modification of the dynamics at least to the extent of accepting that the interaction Hamiltonian is no observable [Beltrametti and Cassinelli 1981, Wan 80]. While superselection rules can easily be incorporated into the Hilbert space framework of quantum mechanics, they nevertheless represent an element which does not follow from the standard axiomatics. Thus the problem remains of providing a theoretical explanation for such restrictions on the sets of observables and states. Furthermore, the consistency problem of Theorem 7.2.1 may be interpreted as an indication of the fact that the irreversibility inherent in measurement has not been properly taken into account. In fact, the unitary evolution on a (separable) Hilbert space is quasiperiodic [Perc 61] and generally time inversion invariant, while irreversibility requires a breaking of this fundamental time inversion symme-

108

-

IV. Objectification and Interpretations

try. Accordingly there are many model considerations aiming at an explanation of the effectively irreversible evolution of the system S ÷ ,4 into a state equivalent to that required by objectification. Usually the equivalence of states is stipulated with respect to a restricted class of macroscopic observables of ,4. The effective irreversibility is achieved by taking account of the macroscopic nature and ergodic properties of ,4 [Cini 79, Dan 62, Haa 68, Ludwig 1961, Ros 65, Weid 67]. In this way one obtains an approximate description of the dynamics in terms of a Markovian master equation, which would be sufficient as far as the macroscopic observables (and their functions) are concerned. Moreover, these approaches allow one to investigate the thermodynamic limit, thus affording a bridge to theories dealing with infinite systems (Sect. 5.4). A more recent attempt along these lines considers irreversibility via dynamical instability properties on the level of the von Neumann-Liouville equation from which an effective restriction of the set of states can be derived; in this way the two important features of the measurement problem irreversibility and classical properties - are shown to be interrelated [Mis 79].

As mentioned above, the incorporation of superselection rules needs to be supplied with some physically convincing motivation. The only known physical mechanism for producing strict superselection rules is by means of spontaneous symmetry breaking, which may occur in systems having infinitely many degrees of freedom. There are further arguments pointing in the same direction. In fact approaches based on discrete superselection rules are somewhat artificial as they do not allow for continuous trajectories of the classical (pointer) observables involved. Concrete models of partly classical systems inevitably involve some continuous observables such as position, which are to be considered as classical observables. An early well-elaborated example is that by Sherry and Sudarshan [Sher 78,79] in which the measurement process is described as an interaction between the quantum mechanical object system S and the classical measuring device .A. The latter system is given a quantum mechanical description by means of embedding its observables into the set of self-adjoint operators of some Hilbert space. The classical nature of ,4 is preserved by stipulating that its trajectory variable corresponds to a (continuous) classical (hence superselection) observable. In this theory the dynamical problem of the Hamiltonion not being an observable is taken into account and leads to a requirement of classical integrity of the localisation variable. This approach does not aim at justifying the classical nature of the measuring device, but it provides a very instructive example of how to reconcile classical and quantum mechanical descriptions in the context of measurement. Interesting new contributions in this direction are found in [Bla 93,95, Jad 95]. The 'many-Hilbert spaces' theory by Machida and Namiki [Mach 80] is an attempt to explain the emergence of classical properties of measuring devices due to their macroscopic nature. It is argued that the energy and the number of constituents of an apparatus ,4, being a macroscopic and therefore open system, are

IV.4. Modifying Quantum Mechanics

109

not well-defined. Hence the state of A should be a mixture of states with different definite particle numbers, these numbers being distributed in a relatively narrow range around the macroscopically large mean number no. In the limit no ~ c~ one obtains a state operator which can be represented as an average with respect to a continuous size parameter. In this way one effectively performs a transition to a direct integral of Hilbert spaces in which macroscopic observables are defined as continuous averages of microscopic observables. As pointed out by Araki [Ara 80], this procedure leads to macroscopic observables inducing continuous superselection rules. Measurement models involving such continuous superselection rules have been subsequently developed further by other authors (see, e.g., [Fuk 90]). The many-Hilbert spaces theory and its current elaborations are reviewed in [Nam 93]. In this work interesting similarities and differences between this theory and the decoherence approach (Sect. 5.3) become apparent; in particular it seems that the Machida-Namiki theory tries to explain only a rather weak, statistical form of decoherence. Apart from its ad hoc nature, the introduction of classical properties in the description of physical systems faces some other obstacles. In some cases (though not always), the implementation of continuous superselection rules in realistic models forces one into the unfortunate situation of having to deal with nonseparable Hilbert spaces [Ara 80, Gue 66, Piron 1976]. In addition the dynamical consistency problem (Theorem 7.2.1) still persists. The fact that in the case of continuous superselection rules the Hamiltonian generator of the dynamical group cannot be an observable was noticed already by Piron [Piron 1976]. A theoretical framework for systems which are partially classical is offered by the C*-algebraic approach to quantum mechanics. In this theory it is made clear that the best available justification for a restriction of the set of observables has to be based on the limited possibilities of physical observers. This amounts to admitting that objectification cannot be achieved in the strong realistic sense but that only a contextual or relational conception of definite values is tenable (Sect. 5.4).

IV.4.3 Modified Dynamics The goal of objectification may be maintained within a theory of proper quantum systems by considering some modification of the dynamical axiom of quantum mechanics which would lead to dynamically induced superselection rules. For example, one may assume a spontaneous stochastic process to supersede the ordinary unitary evolution. The corresponding generalisation of the Schrhdinger equation is interpreted as representing the autonomous dynamics of isolated systems. The stochastic process in question should become noticeable only for large systems so as to ensure that certain macroscopic observables, like pointers, are practically always found to have (nearly) sharp values. The need for giving up the linearity of dynamics in order to reach objectification was recognised by Wigner [Wig 63] but no explicit proposal for a nonlinear

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IV. Objectification and Interpretations

SchrSdinger equation was made at that time. Systematic investigations into modified quantum dynamics started only around 1976 (see, e.g., [Bia 76, Dio 89, Fre 90, Gis 84, Ghi 86, Pea 76]). One of the first elaborated examples, and perhaps the best known, of this type of approach is the unitied dynamics theory of Ghirardi, Rimini and Weber [Ben 87, Ghi 86,90a]. In this theory the von Neumann-Liouville equation for a quantum mechanical n-particle system is modified by adding a linear term which models a spontaneous localisation process taking place at random times: n

ih~T

= [H,T]_ ÷ ZAk(/RAqkTAq+d q - T).

(1)

k=l

The integral term is known in the context of the theory of unsharp measurements as the nonselective operation representing the reduced state of the object system after an unsharp position measurement [¢f. nq. (III.2.43)]. The operators Aqk are of the form Aqk = X / - ~ e x p ( - a(Qk- qk)2), where Qk represents (a component of) the position operator of the k-th particle. Equation (1) represents a quantum dynamical semigroup with a non-self-adjoint generator, so that the irreversibility needed for measurement is built in from the outset. The mean rates ()~k) of the localisation processes can be chosen small enough so that systems having only a few number of constituents practically evolve according to the SchrSdinger equation, while the localisation events become noticeable only for systems with a macroscopically large number of constituents. In this way the localisation observable of a macroscopic system is dynamically objectified practically at any instant of time. In the measurement context, the pointer of an apparatus ¢4 becomes thus an effectively classical observable without any ad hoc restriction of the set of states of ,4. The spontaneous localisation theory and its succeeding continuous dynamical reduction variants provide an interesting step towards a unified quantum mechanical description of microscopic and macroscopic systems. Still they are facing a number of problems. First it would be desirable toestablish a theoretical basis for its new dynamical principle, in analogy to and generalising the axiomatic characterisation of the unitary SchrSdinger dynamics [Sim 76]. Some results in this spirit have been obtained for the case of possibly nonlinear or irreversible evolution equations [Dani 89, Gis 83,86]. Second, it should be observed that the spontaneous localisation process relates to an unsharp position observable, so that the induced Gemenge is a continuous family of nonorthogonal and only unsharply localised states (wave functions). This 'tail problem' [Albert 1992, Pea 94] seems to indicate that also in the modified dynamics approach only some form of unsharp objectification may be reached. Another problem consists in the possibility that the spontaneous localisation theory may overshoot its aim: the very stochastic reduction process needed for explaining the classical behaviour of pointer observables could also erase any macroscopic

IV.5. Changing the Concept of Objectification

111

quantum effects. That this need not be the case has been indicated for the phenomenon of superconductivity in [Buf 95]. Third, the question still has been raised whether the theory can be formulated in a Lorentz covariant way [Bell 87,89]. The recourse to nonlinearity in quantum mechanics bears with itself some dangerous consequences, such as the possible existence of superluminal signals [Gis 90, Pea 86]. However, it does not seem impossible to formulate such theories in accordance with special relativity and causality [Gis 89]. Recent work provides steps towards an incorporation of stochastic reduction processes into the framework of relativistic quantum theories [Ghi 90b, Pea 94]. Finally we note that the experimental testing of dynamical objectification and of nonlinear evolution equations in general is a difficult and still largely open problem [Pea 84, Shi 79, Wein 89], although it seems that recent optical realisations of quantum jumps may provide a good case in favour of stochastic dynamics (see below). In the last years, systematic and more general investigation of stochastic modifications of the unitary dynamics have been taken up (e.g., [Gis 89,90, Pea 89,93,94]. In particular, it has been realised that equations of the type (1) can be equivalently written as stochastic evolution equations for vector states. A theory of primary state diffusion has been developed as a unified framework of various approaches and models discussed in the literature (for concise accounts, see [Gis 93a, Perc 94]). While these results surely provide valuable and practically useful insights, they still do not satisfy the desire for a principal justification of generalising the quantum dynamics. By contrast, those familiar with the quantum stochastic calculus will point out that state diffusion and stochastic reduction equations can be derived as subdynamics from the unitary evolution of the system of interest combined with some environment system [Bela 95a]. In this way there emerges a technical link between the present approaches based on autonomous stochastic processes and the environmental decoherence approaches to be reviewed in Sect. 6.3. In the present open situation it seems wise to let the two approaches coexist and interact with each other and suspend, as long as necessary, the decision whether the stochastic element is fundamental or derived. In fact this attitude has already led to fruitful exchanges of techniques between the pragmatic modelling of quantum jump equations and the state diffusion approach [Gar 94].

IV.5. Changing the Concept of Objectification There is an interpretation of quantum mechanics which radically takes the objectification problem (OP) as implying that no objectification takes place at all in a measurement process. This extreme form of (MO) is adopted by the many-worlds interpretation (Sect. 5.1). By contrast, the modal interpretations try to maintain a one-world view on the basis of certain modifications of the concepts of reality

112

IV. Objectification and Interpretations

and objectivity in the sense of essentially contextual, or relational notions of value attributions (Sect. 5.2). The many-worlds interpretation shares a common formal feature with some modal interpretations. They are concerned with unitary premeasurements M ~ of discrete sharp observables A, for which the final state U(~o ® ¢) of the compound system ,~ + ,4 assumes a biorthogonal decomposition with respect to the final component states "~i and ¢i of S and ,4 for any initial state ~ of S (cf. Sect. III.2.5): ® ,) =

®

=

®,,

(1)

ij According to Theorem III.4.4.1, this situation occurs exactly when the premeasurement M ~ produces strong correlations between the component states "yi and ¢i. We shall review several modal interpretations which consider in different ways the state U(~ ® ¢) in its biorthogonal decomposition (1) and ascribe different roles to the reduced states

Ts(~) := ns(P[U(~o ® ¢)1) = Z p ~ (Aa i ) P['Yi] , A

T~(~) := TC~(P[U(~ ® ¢)]) = ~ p ~ ( a ~ ) P[¢,].

(2)

(3)

In these interpretations the strong correlation premeasurement is taken to be the whole measuring process. The many-worlds interpretation has been the starting point for another line of research, known as the decoherence approach. In this approach one tries to explain in terms of quantum mechanics how and why the macroscopic world shows (perhaps only to conscious observers) a largely deterministic, classical behaviour. Thus, definite properties, such as pointer positions, should arise due to a process of decoherence induced by interactions of the system under consideration with its environment. It turned out that decoherence in this sense is crucial in understanding certain features of the many-worlds and modal interpretations. On the other hand, the decoherence approach has itself initiated the search for new interpretations in the spirit of (MO) (Sect. 5.3). The measurement models studied in the decoherence approach and the modal interpretations all seem to encounter, at some stage, the problem of explaining how the 'preferred pointer basis' (that is, {¢~}) is singled out and made dynamically robust in nature. It seems that the algebraic theory of superselection sectors can provide valuable insights into the necessary formal structures of the systems involved and the processes they undergo (Sect. 5.4). Still it turns out that the preferred pointer basis can at best be determined in an approximate sense for finite systems; that is, both the pointer value-definiteness and pointer mixture property for ,4 may be only approximately realisable. It may therefore turn out that unsharp objectit]cation is all one can ultimately hope to achieve (Sect. 5.5).

IV.5. Changing the Concept of Objectification

113

IV.5.1 M a n y - W o r l d s I n t e r p r e t a t i o n The first attempt to interpret the quantum mechanical formalism without additional assumptions concerning the objectification was made by Everett [Eve 57] and Wheeler [Whe 57]. Everett investigated unitary premeasurements of sharp discrete observables A and studied the decomposition (1) of the state • = U ( ~ ® ¢ ) of the compound system S + ,4 into a sum of products of two states 7i and ¢i, one referring to the object system S and one referring to the apparatus ,4, the 'observer'. The 'measurement' is then nothing else but the correlation between the respective state ~/i of the system and the 'relative state' ¢i of the observer who is aware of the system's state ~/i. The large variety of alternatives which coexist in the state • has later been interpreted by some authors as an ensemble of 'really' existing 'worlds' an idea which has given rise to the name many-worlds interpretation [DeW 70,71, Whe 57]. According to Everett's analysis the state (1) which provides strong statecorrelations between S and ,4, can be considered already as a description of the complete measuring process. Any definite measurement outcome is described by a product state ~i = 7i ® ¢i, where 7i is the state of the system and ¢i represents the observer (that is, the apparatus and some registration devices) as aware that the system S is in the state ~/i. Hence the state vector • - ~ v/pA(ai)~i describes the complete variety of possible final states of a measurement. Assume next that Ad~ is even a strong value-correlation measurement of A, so that the vectors 7i are eigenvectors of A: A~i = aiTi for any i = 1, 2 , . . . and all ~. Then the minimal interpretation and the probability reproducibility condition require that the coefficients pA(ai), which appear in the decomposition (1), are the probabilities for the outcomes ai in the/~na/states of S and ,4. This postulate was justified in the sense of the relative frequency interpretation of probabilities (Sect. III.3). However, in the present case such an interpretation cannot be given. The A (hi) cannot be interpreted here as relative frequencies of outcomes ai numbers p~, in the two Gemenge { (pA(ai), P[-yi]) • i = 1, 2 , . . . } and { (pA(ai), P[¢i]) • i = 1, 2 , . . . } which correspond to the decompositions (2) and (3) of the reduced states Ts(~o) and TA(~), simply because these mixed states play no role in the description of a measurement in the many-worlds interpretation. The whole system ,9 + jI evolves according to the unitary coupling T ® ¢ ~-, U(T ® ¢) and no state reduction or objectification will take place. The numbers p~A (ai) constitute also in the present situation a probability measure in the formal sense. But now the results of Sect. III.3.2 cannot be used for the justification of a relative frequency interpretation. Yet, even in this 'interpretation without objectification' the relative frequency interpretation of the probabilities p~,A (ai) can be justified in the following sense. Let ,S (n) = $1 + . . . + Sn be a compound system of n identical, equally prepared systems with states ~o and denote the compound state by ~(n). If on each system the observable A is measured, that

114

IV. Objectification and Interpretations

is, the observable A (n) is measured on S (n), the observer will register a sequence -- (/1,-..,In) of n index numbers lk indicating the values ark and store it in her memory. The relative frequency of index numbers i in a 'memory sequence' are denoted f~n)(l). If one defines a relative frequency operator F (n) as in Eq. (III.3.6), it is obvious that T(n) is not an eigenstate of F (n). However, one obtains the following 'large n' result due to Finkelstein [Fin 62] and Hartle [Sar 68]. THEOREM 5.1.1. Let {81,..., ,.qn} be a set of n equally prepared identical systems S} with states ~oand let ~o(n) be the state of the compound system S (~). If F (n) is the relative frequency operator for the value ai appearing in the eigenvalue (al, , . . . , al~)

of A (~), then A

nlirnoo(~0(n)[ (F (n) - p ~ ( a i ) )

2

~o(n)) = 0

for any i = 1, 2,... and for all ~o. It has been pointed out by Ochs [Ochs 77] that the strong and somewhat unrealistic premises of Hartle's proof (pure state ~o, equally prepared and independent systems) can be considerably relaxed and replaced by more realistic assumptions. Furthermore Ochs emphasised that Theorem 5.1.1 is essentially a 'law of large numbers' which shows that quantum probabilities fulfil this important requirement. The theorem means that for a given observer, that is, an apparatus plus a registration device, the relative frequency f~n)(g) of values ai will (for large n) be very close A to the probability p~(ai) in almost every memory sequence ~ - (/1,... ,ln) [DeW 71]. At first glance this result is somewhat surprising since none of the n systems 8i with preparation ~o possesses an A-value ak in an objective sense. However, this argument does not invalidate the relevance of the mentioned result, since within the present interpretation for the measuring process the objectification of the measured observable is not required in any sense. Theorem 5.1.1 has been referred to as 'the meta-theorem' of quantum mechanics, with the reading that quantum mechanics implies its own interpretation. This view has been criticised by many authors (cf., e.g., [Squires 90, Squ 90], and references therein). In fact Theorem 5.1.1 assigns (Hilbert space norm) 'measure zero' to non-random sequences of measurement outcomes with relative frequencies deviating from the quantum mechanical value. This does not entail by itself that the corresponding set of worlds would 'occur with probability zero' ([Des 85], p. 20). Such an interpretation would presuppose the probability interpretation to apply on the ensemble level. Therefore it is necessary to add an extra interpretational axiom to the qantum formalism, such as the one proposed in IDes 85]: 'the world consists of a continuously infinite-measured set of universes', which then ensures the required probability interpretation. On the basis of this short account of the many-worlds interpretation we mention some of its consequences relating to the measuring process and illustrating

IV.5. Changing the Concept of Objectification

115

its advantages and disadvantages. For details we refer to the literature [DeW 71]. First, for the description of the measuring process the consciousness of the observer is not needed. Automata for the registration of memory sequences are completely sufficient. Several automata can be used for the formation of a measurement chain without thereby changing the registered results in the respective memory sequences. The results of the automata persist throughout the whole chain. Moreover two automata are also allowed to communicate with each other since this exchange of information will never lead to a paradoxical situation like the paradox of Wigner's friend. In the many-worlds view, quantum theory in Hilbert space does not describe the reality which we usually have in mind but some reality which is composed of many distinct worlds. The observer is only in one of these worlds. For this reason she is unable to recognise the full determinism of the totality of many worlds which are described by ~, but only one part of it which is governed by a probability law, the justification of which is given by Theorem 5.1.1. It follows from these arguments that quantum mechanics can be applied to an individual system. Every automaton, that is, a measuring apparatus with a memory sequence, describes the system correctly according to the probability laws. For this reason quantum theory can be applied even to the universe, its formation and evolution. The state of the universe corresponds to a decision tree with an enormous number of branches. Every measuring process splits the world again into many alternative components, each representing mutually exclusive alternatives; but the whole description is completely consistent. This interpretation of the branching or splitting occurring in measurement-like processes applies only to situations where the biorthogonal decomposition (1) is unique. But this is the case only if all the initial probabilities p~A (ai) are different from each other. Otherwise, there seems to be no sufficient reason why the relevant 'pointer basis' should be singled out physically. This problem disappears if besides the object and registration system the common environment is taken into account, as is the case in the existential interpretation [Zur 93] formulated in the context of the decoherence approach (Sect. 5.4). Under these circumstances one is dealing with a triorthogonal expansion, which is always unique [Cli 94, Elby 94]. The many-worlds interpretation thus seems to have the status of a proper interpretation of quantum mechanics with its usual axioms. It does not presuppose the pointer objectification and the value objectification, thereby avoiding the problem (OP). Accordingly, it can make use of Theorem 5.1.1 as a legitimation of the probability interpretation, since this theorem need not be understood here in the sense of an ignorance interpretation. The price for this interpretation of quantum measurement theory is a very strange ontology: quantum mechanics does not describe any longer the one world in which we are living but at the same time the totality of all possible worlds.

116

IV. Objectification and Interpretations

Modulo the qualifications mentioned above, the basic merit of the many-worlds interpretation consists in the observation that the quantum mechanical formalism provides an interpretation of the theory, including the relative frequency interpretation of the formal probabilities; this view is valid if the strong correlations between the object state and the memory state of the observer are considered as already representing the 'measurement'. However, the additional statement that the possible 'worlds' really exist seems to be rather irrelevant for the following two reasons: first, the 'many-worlds hypothesis' is not compatible with any physically acceptable conception of reality (cf., for example, [dEsp 89]); second, the existence of the alternative worlds cannot be falsified by any quantum mechanical experiment. Accordingly there have since been a variety of attempts at formulating alternative ontologies that would allow one to maintain the no-objectification spirit of the many-worlds interpretation and to explain the apparent occurrence of definite events to observers. Along with the development of the decoherence theories (Sect. 5.3) there have emerged pictures of one deterministic quantum world in which the appearance of different measurement outcomes is associated with the 'branching' of the observer's consciousness, where stable correlations between pointer states and brain states are established by processes of decoherence [Albert 1992, Alb 88, Lockw 89, Sau 93,95, Squ/res 1990, Stapp 1993]. It does not seem clear yet whether the resulting ontologies are less bizarre in each case than the many-worlds picture. The mathematical structures required for the physical description of observers needed in these 'one-world', or 'many-minds' interpretations of quantum mechanics have been investigated in a series of papers by Donald [Don 90,92,95].

IV.5.2 Modal Interpretations Rather than abandonning the goal of objectification altogether, one may try to formulate alternative consistent value-attributions and thus a conception of the reality of properties that allows for an understanding of definite measurement outcomes within quantum mechanics. This is the task pursued in the so-called modal interpretations of quantum mechanics, several variants of which have already been put forward [Bub 92,94; Di 89,94; Healey 1989; Koch 85, vFra 81,90, van Fraassen 1991]. They all aim at going beyond the minimal interpretation in order to provide a language for quantum mechanics in accord with the realistic intentions (A), (B), (C), (D), which is weak enough to evade the objectification problem (OP). In this section we discuss some of these interpretations in the context of measurement theory, and we show that they imply particular specifications of the measurement process [Cas 93,94,95]. Consider any physical system S, be it the object system, a measuring apparatus, or their composition. With any state T of the system we associate two sets of

IV.5. Changing the Concept of Objectification

117

(sharp) properties:

~:~1(T) • = { P E P(?-l) • tr [TP] - 1 } ;

(4)

P#o(T) "= {P E P(7-l) • tr[TP] ¢ 0}.

(5)

If P E P l ( T ) , then a yes-no measurement of P is determined to yield the yesresult, and we say that the property P is real in the state T. On the other hand, if P E P#0(T), then a yes-no measurement of P can lead to the yes-result, but it does so only with the probability t r [ T P ] . The objectification problem is now essentially due to the strong formulation of the objectification requirement in terms of the following two postulates: the requirement that a measurement leads to a definite result, understood as the assumption that the pointer observable has the corresponding value after the measurement; and the stipulation that an observable has a value exactly when the system is in an eigenstate of that observable. The very idea of the modal interpretations is to modify this reality criterion so as to give room for the possibility that a system in (a mixed or entangled) state T could 'have' other properties R E P#o(T), be it in addition to the properties P E :P~ (T) or alternative to them. Such properties would pertain to the system With the corresponding measurement outcome probability. Therefore, in the first place, a modal interpretation aims at defining a set of properties Pm.~.(T) that the system may possibly possess in the state T, without T necessarily representing a Gemenge of eigenstates of these properties. There are a variety of proposals for Pm.i.(T) currently under discussion in the literature, with no conclusive agreement in sight; on the contrary, some of the modalists maintain a more or less exploratory attitude as to what algebraic structures such a set Pm.i. (T) should have. It will depend on the decisions made in this respect whether the proposed set :Pm.i. (T) admits a consistent value attribution in the sense of a structure-preserving map (i.e., an appropriate homomorphism) onto the set {0, 1} (cf. Sect. II.2.2). First systematic investigations into these questions are reported in [Bacc 95, Cli 95, Bub 96, Bub 1996]. Instead of trying to cover each of the known proposals, we formulate some natural candidates for Pm.i. (T) as they arise from the decomposability properties of the state T and from its support projection. We consider first some examples which satisfy the constraint P l ( T ) c_ Pm.i. (T) C_ P#o(T),

(6)

so that the ensuing reality and objectivity criteria are relaxations of those formulated in Sect. II.2.4. The support projection of a state T, PT, is the smallest projection to which the state assigns probability one, tr[TPT] -- 1. Thus, for any P e P(7-/), P e Pl (T) if and only if PT <_ P, or equivalently PT = PTP = PPT. On the other hand, for any unit vector ~ E 7-/, P[~] < PT if and only if PT~ -- ~, or equivalently, ~ is in

118

IV. Objectification and Interpretations

the closure of the range of T, that is, ~ E r-~(T). (For vector states P[~] we write ~'1 (T) for ~'1 (P[~o]).) It follows that for any P E P(7-/), if P E Ps (T), then also P E ~1 (~0), for all ~ E ~ ( T ) . Let Pcra,(T) := U{:Pl(~o) • ~o E r-Wfi(T)},

(7)

where cran refers to the defining property of this set, the closure of the range of T. This set satisfies (6), and it is the set of properties which the system could have in state T according to the Copenhagen variant of the modal interpretation [van Fraassen 1991]. Clearly, ~I(T) = ~cran(T) if and only if T is a vector state, whereas ~cran(T) -- P#0(T) exactly when PT = I. Recall next that a vector state ~ is a convex component of a state T whenever T = +kP[~] + (1 - +k)T',

(8)

for some weight 0 < A < 1 and for some state T ~. This occurs exactly when ~ is in the range of the square root of T, that is, ~ E ran(T 1/2) "- (T1/2~o • ~ E 7-/} [Had 81]. One. may define

~rsq(T)

"-- U{~Ol(~)" ~0 E rail(T1/2)},

(9)

with rsq standing for the range of the square root of T. This set is also of type (6), and could form a basis of another variant of the modal interpretation. A state T can always be decomposed into vector states such that T = ~AiP[¢i].

(10)

If T is a mixed state, there are uncountably many such decompositions. A decomposition (10) is irreducible if for each i, the vector ¢i does not belong to the closure of the linear span of the other vectors Cj, j ¢ i, that is, for all i, ¢i ~ l i n ( ¢ l , . - - , ¢ i - 1 , ¢ i + 1 , ' " } . We say that a vector state ~o is an irreducible convex component of T if it participates in an irreducible decomposition of T. It is known [Had 81] that this is the case exactly when ~ is in the range of T, that is, ~ E ran(T) "= (T~o" ~ E 7~}. If the decomposition (10) is orthogonal, that is, ( ¢ ~ { ¢ j ) = 5~j, then it is also irreducible, but not necessarily the other way round. Such an orthogonal decomposition arises from the spectral decomposition of T = ~-~i ti E T ( ( t i } ) , and it is unique whenever all the eigenvalues ti of T are nondegenerate. Both the irreducible decompositions as well as the spectral decomposition of T suggest further possible sets of possessed properties obeying (6)" Pran (T)"= U {~1 (~0)" ~0 E ran(T)} Psd(T) "= U i { P E P(TI) " P >_ E T ({ti}) }

(11)

(12)

IV.5. Changing the Concept of Objectification

119

where the abbreviations ran, and sd stand for the range and the spectral decomposition of T, respectively. The set Psd(T) is basic to those variants of the modal interpretations which build on the polar decomposition of the final entangled objectapparatus state [Di 89,94, Healey 1989, Koch 85]. For any state T, one has [Cas

95]: VI(T) c V d(T) C V o (T)c

c V o (T)c V¢0(T).

(13)

There are two types of intuitive ideas behind the above sets of possible possessed properties ~m.i. (T). First, the motivation for considering the sets Reran(T) and ~sd(T) is that in addition to the smallest property which the system has in the state T with certainty, the property PT, it could also have some stronger (smaller) properties. In the case of :PercH(T) one allows for the possibility that any atomic property P[~] contained in PT could also pertain to the system in that state. In the case of Psd(T) this possibility is restricted only to the spectral properties ET({ti}). The motivations for the sets Prsq(T) and Pran(T) are slightly different, referring to the decompositions of the state T. In the first case one argues that when the system is in a mixed state T, it could, in fact, be in one of the vector states occurring as a convex component of T, whereas in the second case this is restricted to the irreducible, or even orthogonal, vector state components of the state. In order to explain the role and implications of the modal interpretations in the framework of measurement theory, we assume now that the state in question is the state of the apparatus after the measurement. To be more specific, we consider a unitary measurement M u of an observable E with respect to a fixed reading scale 7~. For any initial state ~ of the object system, the final apparatus state is the reduced state TA(~) = T~A(P[U(~® ¢)]). The basic stipulation of the modal interpretations then is the following: if the probability is nonzero for an E-measurement to lead to a result in a set Xi, then the pointer observable P ~ could have the corresponding value i after the measurement, and it would have that value with the probability p~E (Xi). The assumption that the pointer observable could have such a value is formalised by the requirement that the corresponding property is in the set Pm.~. (TA(~)). This amounts to posing the following modal conditions on the measurement [Cas 95]: MC. For any unitary measurement .Mu of an observable E, any reading scale T~, all P[~] e S(?-ls), i e I,

tr[TA(~)PA (f-I (Xi))] - p~E (X,);

(MC1)

if p~S(xi) ¢ O, then PA(f-l(Xi)) e 7~m.i.(TA(~)).

(MC2)

The modal condition (MC1) is just the probability reproducibility condition and is thus always satisfied in any measurement. However, in general, the set Pm.i. (TA(~))

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is a proper subset of the set :P#0(TA(~)). Therefore condition (Me2) is an additional constraint on the measurement process. In the class of the minimal unitary measurements of a discrete sharp observable the modal conditions have exhaustively been characterised for various choices of the set ~m.i.(TA(~P)) [Cas 94]. Let {¢ij} be the generating set of vectors of a measurement ~ 4 ~ of an observable A = ~ aiPi, as described in Theorem III.2.5.1. If ~ is the initial object state, then A Ts(cp) = Zp~,(ai)P["/i]

(14)

TA(cp) = Z V/pA(ai)pA(aj)(~i I"Yj ) lCj )(¢il

(15)

are the final states of 8 and A, respectively (Sect. III.2.5). In order to discuss the solutions of the condition (MC2) for the cases (7), (9), (11), and (12) in this model, we assume that the vectors ~i are such that lin{~l,...,'yi,.. "} - 7-/. This assumption simplifies the discussion without implying any loss of generality. The following points are then obtained [Cas 94]. First, the linear independence of the vectors ~/i is necessary for the condition (MC2) with Pm.i. (TA(~)) = P c r a n (TA(~p)). However, it is not sufficient to ensure (MC2) except when 7-/is finite dimensional. To this end some stronger requirements are to be posed on the set {-yi}. The weakest possible 'topological' strengthening of the linear independence of {~/i} is the gl-linear independence: for each sequence of complex numbers (ci) with ~ Icil < oo, the relation ~ ci~/, = 0 implies that ci = 0 for all i. Then one can prove: P[¢i] e Pcra,~(T~a(~P)) for any i and ~ with pA(ai) ~ 0 if and only if {-yi} is ll-linearly independent. Second, one can show that P[¢i] e 7~rsq(TA(~)) for each i and ~, such that p~A (hi) ~ O, if and only if Ts(~) - ~'~pA(ai)P[~/i] is an irreducible decomposition. This is to say that each ¢i, with pA(ai) ~ O, is a possible convex component of the final apparatus state TA(~) if and only if the final pointer component states Z(ai)(P[~]) /p~(~) A a -- P[~/~] form an irreducible decomposition of the final object state Ts (~). Third, the sequence {~/i} has the/initeness property if it is linearly independent and for each i, ~/i = Oi + ~-~Ml,j#iaj~/j for some Oi E li---n{'yl,'" ,'yi-l,~/i+l," "}±. One then obtains the following: P[¢i] e Pran(TA(~)) for each i and ~, such that p~A (hi) ~ O, if and only if {~/~} has the finiteness property. Finally, if {-y~} is an orthonormal sequence, then the decompositions (11) and (12) are the spectral decompositions, in which case the condition (MC2) is satisfied for 7~m.i.(TA(~)) -- 7~sd(TA(~)). Conversely, if for that choice of 7~m.i. (TA(~)), (MC2) is fumned for each i and ~ for which pA(ai) # O, then {~i} is orthonormal [Lah 90]. From Sect. III.4.4 we recall that the orthogonality of the vectors 7i is equivalent to Az[~ being a strong-state correlation measurement.

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To summarise, the following characterisations of the modal conditions (MC2) on measurements are given: for any P[~o] e S(7-ls), and for each i = 1, 2,..-, such that p~A (hi) ~ O, P[¢i] e Pcran(T.4(qo)) if and only if {'yi} is/1-independent;

(16)

P[¢i] e 7)rsq(TA(qo)) if and only if {~i} is irreducible;

(17)

P[¢i]

e

~ran (TA(~O))

if and only if {-yi} has the finiteness property;

P[¢i] e :Psd(TA(qo)) if and only if {-yi} is orthonormal.

(18) (19)

There are two further version of the modal interpretations, which we briefly mention here. They arise from the polar and from an orthogonal decomposition of the entangled vector state • - U(qo®¢) of the compound object-apparatus system. Let • = ~ v / - ~ i ® ~i be the polar decomposition of • (which, for simplicity, we assume to be unique). One may then consider the set :=

®

(20)

where pd stands for the polar decomposition. We note that ~i)pd(~) C ~i)y£O(~), whereas P l ( ~ ) is comparable with Ppd(~) if and only if the two sets coincide, which is the case only when • is of the product form • = ~ ® ~/. Ppd(~) is the set of properties which the system could have in the state • according to the modal interpretations which are based on the polar decomposition. Due to the biorthogonality of that decomposition the reduced states obtain the spectral decompositions T~s(~) = ~-]~wiP[~i] and nA(~) = ~,wiP[~l~]. It then follows that {I ® R • R e P~d(T~(~)) } C_ Ppd(~), which shows to what extent the property attributions to a compound system and its subsystems according to Eqs. (20) and (12) are consistent. A similar consistency result is not obtained in the Copenhagen variant of the modal interpretation, where it typically occurs that P[¢i] e : P c ~ (TA(qo)), but I ® P[¢i] ~t : P c ~ (V(~o ® ¢)) = Pl (V(~o ® ¢)). In that interpretation this special feature is ascribed to the holistic nature of a quantum mechanical composite system [van Fraassen 1991]. To introduce the other modal interpretation based on a particular orthogonal decomposition of the entangled state V, let Po(~) := {P e 7)(H) : ( ~ I P V ) = 0}, and recall that the s e t ~i)obj(~ ) =- Pl(~I/) [.J ~O0(~I/) is the set of (sharp) properties which are objective in the state ~. If (P~) is a sequence of mutually orthogonal projection operators such that E Ri = I, we write • = E P ~ - E II Ri~ II ~i, with the agreement that ~i = O, whenever Ri~ = O. We define:

bj po(R,)

:=

'obj(v ) ~(~)

(21)

Clearly, for any P ~ P(?-ls ® 7-l.a), P ~ , obj (~) if and only if P[~i] <_ P or o(a~) P _< I - P [ ~ ] , for each i. Thus, in particular, each R~ belongs t o , obj (~)" By

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IV. Objectification and Interpretations

construction, it is also obvious that the sets, obj (@) and :P#0(@) are incomparable. Whereas the set Pobj (~) contains properties which are objective in the state @, the s e t , obj (~) is intended to represent those properties which could be objective in together with the given properties Ri [Bub 92,94,96]. In the measurement content these properties are assumed to be the pointer properties Ri - I ® P A ( f - I ( X i ) ) associated with the possible measurement outcomes with respect to a given reading scale. What this modal interpretation lacks is any hint of a physical explanation as to why there should be such 'preferred' observables singled out to be properties possessed by the system. Posing now the modal condition (MC2) for the set (20), one can prove in the frame of the premeasurements J~4~ of an observable A - ~ aiPi that for each initial object state ~ and each i - 1, 2,-.. with pA(a~) ¢ O, the following holds [Lah

9o]: I @ P[¢i] e Ppd(U(~ ® ¢)) if and only if ( ~ ) is orthonormal.

(22)

In other words, the polar decomposition theories imply that the measurement must be a strong-state correlation measurement. Interestingly, condition (MC2), with ~)m.i. (U(~ ® ¢)) :,~D(I®P[¢'])obj(V(~ ® ¢)) implies no restrictions at all on the meas-

urement. To conclude, we have analysed several versions of modal interpretations that have been developed in order to offer a language, or a way of speaking, in which it appears as if the measured observable would have a value after the measurement, and thus as if the objectification had been achieved. The modified 'reality' or 'objectivity' criteria of these interpretations are formalised in so weak a way that they all are compatible with the minimal interpretation of quantum mechanics; but they do add structures to the measurement process beyond the probability reproducibility feature. In particular, none of these interpretations adheres to the idea that a projection postulate type of a state transformation T ~ Ts(i,T) or T,4 ~ T~4(i, T) would actually take place in a measurement [van Fraassen 1991]. In addition to the structural properties investigated here, the modal interpretations are forced to address further important physical problems. In particular, they need to explain why definite pointer positions are preserved in the course of time. It is at this point that the polar decomposition theories invoke decoherence arguments to the effect that the robustness of pointer properties is ensured by interactions with the environment of ,4 (e.g., [Di 94]). The robustness problem just mentioned shows that in addition to property attributions at a single time, one also needs to consider sets of properties possessed by a system in the course of its time evolution. While this problem has been taken up only rather recently in the modal interpretations [Di 94, Ver 95], related investigations had been carried out independently in the consistent histories approach [Gri 84; Omn 88a-c, 92, Oran,s 1993], and the decohering histories approach [Gel190,93]. Again the goal is to formulate some weakened objectivity criterion for properties

IV.5. Changing the Concept of Objectification

123

in such a way that there is consistency in the assignments at different times; it is hoped ultimately to arrive at an explanation for the emergence of a classical domain in the quantum world. The consistency conditions are requirements of decoherence ensuring that the corresponding sequential probabilities form joint probabilities. In the weakest version one stipulates the vanishing of interference terms (cf. Sect. II.2.4). It has been pointed out by Dowker and Kent [Dow 96] that the stipulation of decoherence as a consistency feature calls for a justification in terms of a theory of observation; in other words, the 'objectivity' achieved in this type of approach is not absolute but fundamentally observer-related and contextual. The decoherent histories idea has inspired a rapidly increasing activity in developing models (e.g., [Hal 93a,b]) and has led to a renewed interest in sequential quantum logics as a systematic framework for quantum histories [Ish 94a,b].

IV.5.3 Decoherence via Environment-Induced Superselection It has been argued that due to its macroscopic nature a measuring apparatus ,4 must be considered as an open system. According to Zeh [Zeh 70], its interaction with the environment cannot be neglected but should rather be made responsible for the emergence of classical properties and irreversibility as required for measurement. Incorporating the environment g into the description, the unitary Hamiltonian dynamics of S + .A + g [cf. (O)] leads to a nonunitary evolution of the subsystem S + ,4 for which the necessary objectification conditions (PVD) and (PM) may be fulfilled. In this way the undesired consequences of the theorem (OP) would appear to be avoided by violating (U) as long as only ,S + A is concerned. As pointed out by Zurek [Zur 81,82,84], the interaction of .A with the environment should be responsible for specifying the pointer observable (the 'pointer basis') and effecting the transition to the 8 + A-mixture required for objectification [Eq. (III.6.8)]. By means of idealised models, it is possible to show that a quantum nondemolition type of interaction between Jt and g leads to a kind of monitoring of the pointer through the environment, which makes the reduced state of S + ,4 approximately (or quasi-) diagonal in the pointer basis. This process takes place within a very short relaxation time [Zur 84] and with large Poincar@ recurrence times if the environment possesses a large number of constituents. The coherence originally present in the state of S + ,4 after the measurement interaction is not destroyed but dislocalised into the many degrees of freedom of the environment [Joos 85]. In other words, the interaction of the macroscopic apparatus with the environment leads to an effective decoherence of the ,S + A state into the desired mixture. Accordingly the present line of reasoning is commonly referred to as the decoherence approach towards the objectification problem. The fact that the apparatus A is always found in a mixture of (approximate) pointer states is interpreted as the appearance of an environment-induced superselection rule. In recent years a variety of realistic models in this spirit have been devised, showing the emergence of classical properties in macroscopic systems due to interactions with their

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IV. Objectification and Interpretations

environment (eg., [Gla 86, Haa 87,93, Joos 85, Khal 87, Wal 85]). These models exploit the dissipative or amplifying influences of ~ on the (apparatus) system ,4 under consideration. In the measurement context, the reduced S q-j[ dynamics may be modelled in the form of a quantum dynamical semigroup similar to those investigated in the stochastic dynamical theories (cf. also [Wig 83]). The decoherence approach is also concerned with the general cosmological problem of explaining the emergence of the (quasi-)classical domain within a quantum world in the large, of which the occurrence of definite measurement outcomes would be just a special instance (for recent reviews, cf. [Giulini et al 1996, Mazagon 1991, Zur 93]). It has inherited this concern from the many-worlds interpretation and shares it thus with the more recent one-world/many-minds theories, the modal interpretations, and the consistent, or decoherent histories approaches. Like these theories, the decoherence approach does not aim at reaching the goal of objectification (O) since the effective environment-induced superselection rules are obtained by voluntarily neglecting the - practically unobservable - degrees of freedom of the environment. Indeed the obstacle (OP) reappears if understood as a statement about the total system S -b ,4 + t~. In particular, the resulting S + A-mixture does not admit an ignorance interpretation with respect to the pointer basis. Neglecting the environment then amounts only to an apparent abandoning of the postulate (U) of unitary dynamics. Hence what has been achieved in this way could at best be called quasi-objectification. The advocates of the decoherence approach are well aware of the qualifications just mentioned. For instance, Joos and Zeh [Joos 85], as well as Zurek [Zur 82], emphasise explicitly that the quantum mechanical coherence is not destroyed but only displaced. This process of displacement can be described in information theoretical terms [Part 89], showing again the observer-related nature of the decoherence. However, the decoherence theorists draw different conclusions as to what should be regarded an appropriate interpretation of quantum mechanics. Some adhere essentially to the many-worlds interpretation, while others apparently tend to a many-minds view. Also the 'existential interpretation' outlined by nurek [Zur 93] seems to be understood in this spirit in that it makes explicit reference to a splitting of the 'apparatus' into a measuring device and an information recording system. The decoherence approach provides important insights into the framework of interpretations without objectification. First, it shows possible ways of explaining the selection of a preferred pointer basis and thus towards an understanding of the robustness of macroscopic properties. It is interesting to observe that for finite systems the decoherence process does lead only to approximate realisations of (PM) and (PVD); hence even the quasi-objectification can at best be 'unsharp'. Yet, insofar as the models studied in the decoherence approach allow for taking thermodynamic-type limits, they may also serve as prototypical examples for any

IV.5. Changing the Concept of Objectification

125

potential solution of the objectification problem that builds on infinite systems. In fact an infinite environment is capable of yielding strict decoherence in finite times, infinite recurrence times, and strict superselection rules. Thus the idea of environmental decoherence and the dislocalisation of correlation information may ultimately turn out to offer the only satisfactory explanation of the status of superselection rules in physics. This is becoming increasingly clear from recent developments in algebraic quantum mechanics to which we turn next.

IV.5.4 Algebraic Theory of Superselection Sectors From a formal point of view, the continuous-superselection theories mentioned in Sect. 4.2 are based on the choice of a particular representation of the C*-algebra of observables associated with the system under consideration. The C*-algebraic formulation of quantum mechanics was developed primarily as a rigorous frame for dealing with systems having infinitely many degrees of freedom in the same way as with finite systems. It has proved fruitful in relativistic quantum field theories as well as in statistical physics. For concise presentations of the Mgebraic quantum mechanics as well as surveys of the relevant literature the reader may refer to the monographs [Primas 1983] and [Hang 1992]. The possibility of solving the objectification problem ( O P ) by means of classical properties of the apparatus A rests on the description of A as a system having a quasi-local C*-algebra of observables. Such an algebra has infinitely many inequivalent (Gelfand-Neumark-Segal) representations. Instructive and 'elementary' illustrations of this phenomenon are given in [Bub 88] and [Sewell 1986]. Related to this is the existence of disjoint states on these representations, which serve as candidates for a pointer basis [Hepp 72]. The interaction between A and some quantum mechanical object system S is represented by means of a one-parameter group of automorphisms acting on the tensor product of the respective W*-algebras. The measurement models elaborated in [Hepp 72] demonstrate that, by a suitable choice of the interaction, probability reproducibility as well as repeatability can be achieved via strong value-correlation. Furthermore, due to the disjointness of the pointer states, the dynamics lead, in the infinite-time limit, to a state which is equivalent to the mixture required by the objectification condition. Again, the disjointness of the pointer states guarantees that the ignorance interpretation can be applied to this mixture. The macroscopic observable associated with the disjoint pointer states is given as the space-average of a microscopic local observable, thus in a similar way as in the many-Hilbert-spaces model (Sect. 4.2). Systematic formulation of the measurement conditions in the C*-algebraic framework as well as further elaborations of similar models are found in [Bona 80, Liu 95, Sun 93, Whi 76]. It must be noted, however, that the time evolution, being automorphie and time-inversion invariant, can lead to objectification only in the infinite-time limit. Indeed for any finite time one can find an observable of .A which shows significant

126

IV. Objectification and Interpretations

interference terms proving the persisting coherence of the S + .A state [Bell 75]. This fact demonstrates that the algebraic approach towards measurement has its own dynamical problem resembling the one of Sect. III.7. In order to achieve a gradual approach within finite time towards the objectification, one has to break the time-inversion symmetry of the evolution. Lockhart and Misra [Lock 86] propose to incorporate the required irreversible behaviour of ,4 by means of a suitable selection of the von Neumann algebra of observables of ,4 in such a way that a certain causality requirement is satisfied. The theory developed by these authors thus interconnects the phenomena of objectification and irreversibility, indicating that an understanding of the latter provides also an explanation of the former in the framework of algebraic quantum mechanics. Technically, the dynamical automorphism group on the full algebra is seen to act in a non-automorphic, and thus irreversible, way on the chosen representation. The status of the dynamical problem whether it is acceptable that the Hamiltionian cannot be an observable has remained controversial [Breu 93a,b, Wan 93]. In recent years Primas [Pri 90a,b] has strongly advocated the approach sketched out here and elaborated a programme towards an individual description of the measurement process. Defining a quantum object as a system with no EinsteinPodolsky-Rosen correlations with its environment, it follows that a proper quantum system is an object exactly when its environment is a classical system IRa 82] Accordingg to Primas, the measurement problem then consists of finding a suitable representation of the algebra of observables of S + A such that S 'lives' in a classical environment. In this picture the reduced dynamics of S turns out to be a stochastic process, described by a nonlinear stochastic Schrbdinger equation (cf. Sect. 4.3). Due to enormous technical difficulties, the development of models along these lin~ has remained in a rather preliminary stage [Ama 93, Pri 90c, Zao 90]. The crucial question in the algebraic approach to the objectification problem is how the choice of a representation of a C*-algebra, with its ensuing emergence of superselection sectors, can be justified [Bona 93, Lan 91]. The stipulation of t h e desired appropriate causal irreversible behaviour mentioned above may be regarded as somewhat ad hoc. It has been pointed out by various authors [Bela 94, Lan 91, Pri 90b] that the distinction of the underlying C*-algebra of a physical system from its possible representations is fundamental in the following sense: each representation, which fixes a particular von Neumann-algebra of observables, arises from the full specification of the context of the set of experimental procedures available to the observers in a given situation. Thus the C*-algebra contains all quantities pertaining to a system that may become observables in an appropriate context; these potential observables are sometimes called beables (in Bell's sense, cf. [Lan 91]). But due to the limited possibilities of actual observers, who necessarily observe the physical world from inside, the von Neumann algebra of actua/observables will be restricted. This explanation of the possible emergence of classical observables sug-

IV.5. Changing the Concept of Objectification

127

gests natural ways of linking the algebraic approach with the decoherence theory [Lan 91]. This observation has led to new rigorous investigations of the robustness and approximate nature of the 'pointer basis' [Breu 93a,b]. The necessary observerrelatedness of the algebra of observables has been expressed formally in terms of the so-called nondemolition principle, which states that the very possibility of observing an object requires the existence of a conditional e x p e c t a t i o n - and thus a reduced algebra of observables - on the side of the apparatus [Bela 94,95b]. Algebraic quantum mechanics entails powerful mathematical tools for formulating quantum theories of large systems. In particular, it seems to provide a framework wide enough to allow for a consistent and thorough theory of the measurement process. However, taking literally the actual infinity (of the number of constituents, or of the degrees of freedom) in order to solve foundational problems of a physical theory appears hardly acceptable in itself. In our view, the consideration of infinite limits in a physical theory is an idealisation made in a state where more detailed knowledge about the situation under investigation is lacking; it is an admissible idealisation as long as no inconsistency or conflict with observation arises. Whatever position one adopts with respect to this question, it should be kept in mind that taking such limits is a well-exercised technique for approaching a simplified description of a physical problem, yielding hints for a more comprehensive solution. In this sense the measurement models of algebraic quantum mechanics furnish valuable and promising contributions to a resolution of the objectification problem, serving as guides for a better understanding of the nature of macroscopic observables and their - possibly only approximate - classical behaviour.

IV.5.5 Unsharp Objectification The idea of unsharp objectification arises if one intends to leave quantum mechanics intact and still tries to maintain a notion of real and objective properties as close as possible to the strong concepts of Sect. II.2.4. Then, what is left to be reconsidered in view of the statement (OP), is the formulation of the postulates (PVD) and (PM). Indeed it has been realised in virtually all the approaches discussed so far that sharp pointer values and an exact mixture of pointer states are hardly ever obtained, if not beyond reach, in realistic measurement situations. A precise formulation of the sort of approximations involved here can be based on the notion of unsharp pointer observables. Indeed 'unsharp objectification' refers to a situation where pointer readings would correspond not to definite values but rather to approximately real properties. This relaxation of (PVD) would immediately necessitate a corresponding modification of the pointer mixture condition (PM). It is an open question whether the insolubility theorem III.6.2.1 can be extended to cover the case of such genuinely unsharp pointers. If this were not so, then there would exist measurement schemes whose couplings would lead the object-plus-apparatus system into a mixture of 'near-eigenstates' of the pointer. If in addition it could be ascertained that this mixture admits an ignorance interpretation, then a solution

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IV. Objectification and Interpretations

to the objectification problem would have been achieved quite in the realistic spirit put forth in Sects. II.2.4 and III.6. On the other hand, if this strong form of unsharp objectification cannot be obtained, the notion of unsharp pointers and the ensuing approximate realisations of the pointer value-definiteness and the pointer mixture property are still likely to be useful tools for the other approaches to the objectification problem. In the sequel we provide some examples supporting the conjecture that the emergence of classical behaviour in certain macroscopic quantities is best described in terms of unsharp observables. It has often been argued that classical properties of quantum systems should emerge if these systems are macroscopic in some sense. If this view turned out to be true, then it would render the ad hoc recourse to superselection rules (Sect. 4.2) an admissible idealisation. Yet the usual textbook technique of taking the limit h --- 0 is far too crude a procedure. It is merely a mathematical operation without any operational justification. Certainly the 'smallness' of h / n some sense is important, but its precise meaning should follow from physical requirements. An operational criterion should result from a characterisation of 'approximately classical properties', 'nearly classical behaviour', etc. Again, it seems difficult to provide precise definitions of these terms. One may require (approximately) deterministic trajectories, either within certain subspaces of 'classical' states, or for expectation values of 'macroscopic' observables. Examples of the first option are classical descriptions based on (generalised) coherent states; for a brief survey and concise definitions, cf. [Pri 90a]. The second method is employed, for instance, by Ludwig [Ludwig 1987] or by Daneri, Loinger and Prosperi [Dan 62, Pro 74]. Instead of reviewing the various attempts, we present a few concrete examples illustrating, in a self-explanatory way, that quantum mechanics does allow for a quasi-classical level of description. The essential new point in these examples is the decisive role ascribed to unsharp observables which are represented by positive operator valued measures. An important quasi-classical feature of unsharp quantum observables is the existence of coexistent sets of noncommuting observables. This coexistence can be achieved by means of introducing a sufficiently large degree of unsharpness. The resulting blurring of potential interference can be utilized in the explanation of an approximately classical behaviour of macroscopic observables. The present procedure underlines and illustrates the general abstract result mentioned in Sect. 4.1, namely, that consistency between quantum mechanics and the objective description of macrosystems requires macroscopic quantities to be represented as unsharp quantum observables. As a first example we mention the possibility of joint unsharp measurements of quantum mechanical position and momentum observables [Ali 85, Busch, Grabowski, Lahti 1995, Davies 1976, Holevo 1982, PrugoveSki 1986, Schr 81]. Such quantum mechanical phase space measurements behave like classical trajectory determinations provided the involved position and momentum unsharpnesses are macroscop-

IV.5. Changing the Concept of Objectification

129

ically large [Bus 82]. Hence classical trajectories can be approximately realised in terms of unsharp quantum observables [Kaka 90, Tza 87,88]. It has been shown in a phase space measurement model that quantum observables with a large unsharpness admit nearly non-disturbing measurements, as one would expect in a classical measurement situation [Sche 94,95]. Next, it may be recalled that Bell's inequalities are satisfied within quantum mechanics if coexistent triples of unsharp observables are considered [Bus 89a, Khal 87]. Thus, in the Einstein-Podolsky-Rosen experiment one may assume, without running into contradictions, the simultaneous unsharp objectivity of noncommuting spin observables. Finally, a 'surprising quantum effect' discovered recently displays a classical rotation behaviour of quantum mechanical spin observables in the course of a sequence of 'weak' measurements [Aha 87]. This classical feature is found to be a consequence of an Einstein-Podolsky-Rosen type correlation between the involved spin N system and the weak measurement device [Bus 88]. Due to the huge measurement unsharpness the entanglement between object and apparatus is not completely destroyed but, on the contrary, guarantees the appearance of the surprising quantum effect: the occurrence of apparently forbidden readings lying far beyond the spectrum of the observables measured. The large number of constituents of the spin N system ensures that noncommuting spin components simultaneously have relatively well-defined values. Taking into account that such a weak measurement defines an unsharp observable, one finds a natural explanation of the phenomenon under consideration [Bus 88], along with the lesson that the emergence of classical features may be interpreted as a macroscopic quantum effect. The measurement unsharpness represented by unsharp observables must not be confused with a mere inaccuracy; on the contrary, it is rooted in a genuine quantum indeterminacy inherent in the measuring device [Busch, Grabowski, Lahti 1995]. In the light of the above examples it seems appropriate to understand macroscopic observables as unsharp observables with a large degree of intrinsic unsharpness. The magnitude of this unsharpness induces a kind of natural coarse-graining in the sense that it fixes a scale on which the macroscopic observable assumes fairly well-defined values [Qua 92,94]. In this context another observation is of great importance: the existence of

quasi-classical states. In a model of a quantum mechanical amplifier Glauber [Gla 86] described a very peculiar type of 'macroscopic' oscillator states which have the property of being quasi-diagonal with respect to various observables: position, momentum, energy, or phase-space observables. The term quasi-diagonality refers to the exponential decay of nondiagonal elements with increasing separation from diagonal terms. It is plausible to assume that measurements of macroscopicaUy unsharp observables performed on such states do not allow one to detect interference effects proving the nonobjectivity of some of these observables. Hence the above-

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IV. Objectification and Interpretations

mentioned noncommuting observables may be assumed to be unsharply objective since this assumption presumably cannot be disproved. Applying these ideas to the measurement context leads to the following scenario. Presumably any realistic pointer observable is an intrinsically and macroscopically unsharp observable, hence a macroscopic observable in the sense explained above. It may turn out that the interaction of the measuring device with the environment forces the apparatus to remain in some quasi-classical states in which, in particular, the pointer observable is quasi-diagonal. Then it becomes impossible to 'see' (in the sense of macroscopic observation) the apparatus in a superposition of macroscopically distinguishable states. The 'objectification' reached in this way is an essentially unsharp one and certainly does not require classical properties in the sense of superselection rules. Some of the models referred to in the preceding subsections seem to fit in well with the spirit of unsharp objectification [Gla 86, Haa 87,93, Khal 87, Kuda 89,90, Wal 85]. In these models the quasi-classical behaviour of a macroscopic quantum system ,4 is shown to result from interactions with its environment 8. The pointer states are essentially nonorthogonal, though still almost orthogonal. This is related to the fact that they refer to a macroscopically unsharp pointer observable. It appears likely that a closer analysis of these models will show that the resulting A-states are indeed quasi-classical states. The unsharp objectification proposal leaves a number of questions to be settled. Some clarifications concerning the operational meaning of unsharpness have been obtained in recent investigations, and these lead to a conception of an unsharp quantum reality [Busch, Grabowski, Lahti 1995, Schroeck 1996]. The resulting generalised measurement theoretical concepts, such as that of approximate repeatability, allow for an extension of the quantum nondemolition idea to continuous and even unsharp observables. In this way a precise understanding of the nondemolition monitoring of coherent pointer states through some environment can be achieved. Unsharp objectification is intended to be a relaxation of the objectification requirement. Accordingly this option copes with (OP) not by requiring classical observables of the apparatus in the strict sense but rather by exploring the emergence of approximately classical features of macroscopic quantum observables. It may be expected that the theory of unsharp observables offers conceptual tools supporting further developments in the decoherence approaches.

V. C o n c l u s i o n In this treatise we have tried to develop a systematic exposition of the quantum theory of measurement. The operational language of general observables and state transformers allows for a concise definition of the notion of measurement, from which it is possible to derive necessary and sufficient conditions for the realisability of quantum mechanical measuring processes. The quantum theory of measurement is motivated by the idea of the universal validity of quantum mechanics, according to which this theory should be applicable, in particular, to the measurement process. Hence one would expect, and most researchers in the foundations of quantum mechanics have done so, that the problem of measurement should be solvable within quantum mechanics. The long history of this problem shows that, in spite of many important partial results, there seems to be no straightforward route towards its solution. This general impression is confirmed in the present work by means of a number of no-go-theorems. After an attempt to localise the measurement problem within a variety of interpretations of quantum mechanics (Chap. I), we described the phenomenon of nonobjectivity (Chap. II) which raises the question of what the term measurement could possibly mean in quantum mechanics (Survey of Chap. III). The notion of measurement proposed in this work (Sect. III.1) is based on two fundamental requirements, probability reproducibility and objectification. The former is needed in order to specify which processes can be regarded as premeasurements of a given observable. The latter condition stipulates that a measurement should lead to a definite result, which is problematic in view of the phenomenon of nonobjectivity. The probability reproducibility condition, as well as other measurement theoretical postulates such as strong correlations, repeatability or ideality, can be formally incorporated into the quantum theory of measurement in a consistent manner (Sects. III.2-5). The fulfilment of these postulates in actual measurements will generally meet obstacles due to dynamical and other limitations on measurability (Sects. III.7-8). However, the heart of the quantum mechanical measurement problem must be seen in the second fundamental requirement, objectification (Sects. III.2.4, III.6). According to the Insolubility Theorem III.6.2.1, objectification cannot be reached in a unitary premeasurement of a nontrivial observable if the object and apparatus are proper quantum systems. This statement includes measurement schemes with possibly unsharp pointer observables as long as they allow for the necessary objectification requirements of pointer value-definiteness and the pointer mixture property for S + ,4. It follows that the quantum theory of measurement seems to be incapable of reaching its goal: while it yields a description of measurement processes, it does not seem able to provide an explanation of their realisability.

132

V. Conclusion

The question now is how to evaluate this remarkable conclusion and its implications. A systematic and complete account of the possible reactions to and attitudes towards the measurementproblem is offered by the logical structure of this problem as summarised in Sect. IV.1. We summarise the discussions of Chap. IV, adding some conjectures and speculations concerning promising future ventures. The most radical options of dealing with the measurement problem are perhaps those which question the very basis of the whole argumentation. This attitude seems to be tentatively adopted in the various Copenhagen interpretations (Sect. IV2): the pioneers of quantum mechanics refrain from firmly proclaiming the universal validity of quantum mechanics and do instead grant the classical description of macroscopic measuring devices a prominent status. In addition, their position is one of extreme epistemological precaution in that it regards physical phenomena as being observed not in themselves but only as they appear under the conditions of observation. The obligation is thus taken away from quantum mechanics to provide a complete account of the measurement process; instead, this theory is taken to specify the limits of the applicability of classical concepts and descriptions. In some of its variants, the Copenhagen interpretation leaves open the possibility that the process of objectification is only concluded in the conscious observers, of which quantum mechanics may not be able to give a proper account. In our view it is desirable and possible to entertain the intentions of a more realistic interpretation of quantum mechanics, thereby going beyond the minimal interpretation. The epistemologically opposite extreme option to the Copenhagen one is pursued in the hidden variable approaches which seek to adhere to a classical and perhaps deterministic realism (Sect. IV.3). The phenomenon of nonobjectivity is not accepted as a feature of physical reality but rather taken as evidence against the completeness of quantum mechanics, with the implication that the objectification problem becomes irrelevant within quantum mechanics. However due to the known no-go theorems the hidden variable approach can be said to have failed to hflfil its original intention. In other words, quantum reality is fundamentally different from, and incompatible with, classical reality. It so turns out that maintaining a realistic attitude in view of the insolubility theorem requires either a modification of quantum mechanics, or a reconsideration of the notions of objectivity and objectification. The first possibility amounts to denying the universal validity of quantum mechanics and giving room for theories of macrosystems which display partly classical behaviour. This can be achieved either by admitting classical observables from the outset (Sect. IV.4.1-2) or by introducing stochastic dynamical equations leading to effective classicality on the macroscopic level (Sect. IV.4.3). A partly classical theory may be understood as a many-body quantum theory with a restricted set of states and observables, the restriction being defined by means of some superselection rules. Considering the pointer of a measurement device as

v. Conclusion

133

a classical observable bears some problems which we review next. In the context of minimal unitary premeasurements of discrete sharp observables - a situation which is most frequently considered in the l i t e r a t u r e - it was found that objectification can be obtained exactly when the pointer is a classical observable (Theorem III.6.3.1). Hence in this case the only way to meet the objectification requirement would indeed be to admit the presence of superselection rules for the measuring apparatus. However, the presence of a classical, discrete and nondegenerate pointer observable entails that the respective apparatus cannot be a quantum system as constituted in (Galilei or Poincar@) spacetime. To reconcile the constitution of a measuring apparatus via systems of imprimitivity with the objectification requirement seems to require continuous superselection rules so that separable Hilbert spaces would no longer provide a sufficient basis for quantum mechanics (Sect. III.6.3). The option of considering superselection rules becomes even more problematic if the simultaneous fulfilment of the probability reproducibility and objectification requirements is attempted on the basis of a measurement dynamics generated by an observable of the object plus apparatus compound system. In these circumstances it turns out that the assumption of a classical pointer observable excludes the realisation of the probability reproducibility (Theorem III.7.2.1). Hence one is facing the remarkable conclusion that the very concept of a premeasurement, as characterised by the probability reproducibility condition, precludes its realisation as a measurement in the sense of objectification. In this way the two basic measurement postulates turn out to contradict each other in the context of the standard measurement model, but most likely also in more general situations. This result leads to yet another disturbing consequence: insofar as the preparation of physical systems requires the possibility of performing measurements, it seems impossible to understand the process of preparation within quantum mechanics (Sect. III.8). The only escape from these conclusions would be to find good arguments why the generator of the time evolution would not have to be an observable. Finally, it does not seem satisfactory to accept an ad hoc stipulation of superselection rules. There is no sharp borderline in nature, but rather a smooth transition, between the realms of quantum and classical mechanics. One would thus feel more comfortable if some physical motivation could be given for the emergence of classical observables. However, the known explanations, such as spontaneous symmetry breaking, lead to approximate, or effective superselection only (as far as systems with finitely many degrees of freedom are concerned). Hence theories with strict superselection rules can at best be regarded as idealised descriptions and should therefore not be taken as a basis for solving the measurement problem. In the present course of speculating about possible future venues it may be worth mentioning one promising proposal for a fundamental physical explanation of a superselection rule. Christian [Chri 94] has argued that a formulation of 'nonrel-

134

V. Conclusion

ativistic' quantum mechanics, that takes into account from the outset the principle of equivalence, should be based not on the Galilean space-time theory but rather on the dynamic Newton-Cartan theory. He has provided heuristic evidence showing that in this frame the Bargmann superselection rule for the mass of a particle emerges in a generalised form, excluding superpositions of particle configurations that induce significantly different spacetime metrics. If this approach came to be established as a proper nonrelativistic limit of a future theory of quantum gravity, then it would be the coupling between matter and space-time via the gravitational field equations that could be made responsible for definite positions of macroscopic objects such as pointers. If all the difficulties encountered with the incorporation of superselection rules could be overcome, then the dynamics of a compound object-apparatus system in a measurement would turn out to effect a stochastic process, forcing the pointer to 'jump' randomly into its final position. In general, an individual description of the interaction between a proper quantum system and a partly classical system would be appropriately based on a stochastic dynamical equation, while the original SchrSdinger equation would only provide the probabilities for transitions between different superselection sectors. In this sense it may be said that the classical pointer option forces one into accepting a modified dynamics as well. By contrast, modifying the dynamical axiom would allow one to maintain the basic Hilbert space structure of quantum mechanics, without requiring superselection rules. It is found that stochastic processes of spontaneous localisation can be modelled such as to lead to an effectively permanent, or quasi-continuous, objectification of the positions of macroscopic systems. At the present stage the models offered suffer from an element of arbitrariness since no systematic foundation, nor any cogent uniqueness arguments are known. An interesting perspective is opened up again by attempts at linking the stochastic element to gravitational effects (e.g., [Dio 89]). This idea has been reviewed and developed recently by R. Penrose [Penrose 1989, 1994] who suggests searching for a dynamical reduction theory in which superpositions of different configurations of a massive body become unstable as the corresponding gravitational fields become significantly different. The spontaneous reduction rate would increase with the size of the bodies in question. At time scales large compared to that of the reduction process, the localisation observable would thus appear to have definite values. In this sense, dynamical reduction theories could be viewed as another way of explaining (effective) superselection rules. If one does not like to question the universal validity of quantum mechanics, the only remaining alternative consists of reconsidering the conception of the reality and objectivity of properties (Sect. IV.5). The most radical way of taking seriously quantum mechanics and the insolubility theorem would consist of denying that the objectification takes place at all as a real physical process. This is done in the socalled many-worlds interpretation and its successors, the one-world/many-minds

v. Conclusion

135

approaches (Sect. IV.5.1). Despite the amazing coherence obtained in this way, these proposals face their own problems; at the least they are found to build on rather bizarre ontologies in their attempts to explain why the world appears to be as objective as it does to observers being part of it. A more positive attitude towards the reality of the process of objectification appears therefore desirable, as it is entertained in the modal interpretations (Sect. IV.5.2). These interpretations seek to establish new criteria for value attributions to observables that are consistent with the formal structure of quantum mechanics but still provide an understanding of definite measurement outcomes. Since the occurrence of definite readings is not just a matter of interpretation but rather the result of physical processes, it may well turn out that the modal interpretations finally go hand in hand with the environmental decoherence theories (Sect. IV.5.3) [Lan 95]. Again, definite pointer readings and more generally, the quasi-classical domain of quantum mechanics, would emerge instances of effective superselection due to the limitations of localised observers, and thus the relevant dynamics would be best described in terms of effective stochastic processes as indicated above (Sect. IV.5.4). Recent developments in the theory of unsharp observables have made conceivable a new option within quantum mechanics, that of unsharp objectification, which is based on the possible emergence of quasi-classical properties of macroscopic quantum systems (Sect. IV.5.5). It is an open question whether the insolubility theorem would break down in the case of intrinsically unsharp pointers. If it did (which we doubt would be the case), one would be facing a solution to the measurement problem as 'close' to the ideal goal as one might wish: there would exist dynamics leaving the apparatus in the desired Gemenge of 'near-eigenstates' of the unsharp pointer. Otherwise, 'unsharp objectification' would still provide the modal and decoherence approaches with the conceptual tools for dealing with the only approximate nature of the diagonalisation of density matrices typically achieved in realistic models involving finite systems. At this point it seems appropriate to note the striking analogy between the quantum mechanical measurement problem and the problem of irreversibility in statistical mechanics. The latter problem consists of explaining the observed Jrreversible and stochastic behaviour of macroscopic systems in view of the timeinversion symmetry of the underlying microscopic dynamical equation. Similarly, the measurement process appears to be a stochastic process, leading irreversibly to definite outcomes, while according to quantum mechanics the underlying evolution of the object plus apparatus system is governed by a SchrSdinger equation. In accordance with this parallelism, some of the approaches to the measurement problem resemble very much certain techniques belonging to the realm of statistical physics.

136

v. Conclusion

The relevance of the macroscopic nature of the measuring apparatus to the measurement problem was envisaged in several instances in the present work. One may show that the difficulties in detecting interferences between macroscopically distinct (pointer) states increase with an increasing number of constituents of the apparatus [Kaka 91, Leg 80]. In analogy with the strategy of ergodic theory it was claimed that quantum chaos, as the source of irreversibility, is necessary for ensuring the consistency of a solution, within quantum mechanics, of the the measurement problem [Peres 80,87]. Further connections between irreversibility and objectification are exploited in the approaches to the measurement problem reviewed in Sects. IV.4.1-2 and IV.5.4. In fact the recourse to macroscopic or infinite systems has much in common with the thermodynamic limit procedures of statistical physics (e.g., [Gav 90, L~v 77, Lock 86, Mis 79, Omn 88c, Pri 90b-d, Pro 74]). The parallelism between the objectification and irreversibility problems allows one to take advantage of the experiences gathered from the latter century-old problem, in an estimation of the possibilities for dealing with the former. This immediately suggests caution in considering any ad hoc changes of quantum mechanics. On the other hand, this may also be taken as an invitation to consider both problems in a framework more general than that of ordinary quantum mechanics. Perhaps the most elaborated analysis of the question of macroscopic systems in this general spirit is given by Ludwig [Ludwig 1987] (cf. our Sect. IV.4.1). The present account of approaches to the objectification problem illustrates the lively development going on in the quantum theory of measurement. Increasing fruitful interactions and convergence, at least on the technical side, can be witnessed in current research into the two main realistic branches of the decision tree, those aiming at modified quantum mechanics or new conceptions of objectification. Some of the excitement in the present discussion is captured in two chains of very inspiring communications in the literature. The first, initiated by Zurek's popular account on decoherence, is concerned with 'negotiating the tricky border between quantum and classical' [Zur 91]. The other focusses on the question, raised by SchrSdinger and reemphasised by Bell [Bell 87], whether there are quantum jumps. This question epitomises the problem of what is regarded as 'real' in a physical theory. The proponents of modified dynamics theories take objectification as a real stochastic 'jump' process and blame the many-worlds interpretation of entertaining a huge redundancy in the form of infinitely many unobserved world-components [Gis 92,93b, Shi 91]. On the other side, the decoherence theorists take the quantum mechanical state vector as the entity determining what is 'real' and therefore conclude that it "thus appears becoming evident that our classical concepts describe mere shadows on the wall of Plato's cave in which we are living. Using them for describing reality must lead to 'paradoxes'" [Zeh 93]. - We shall refrain here from suggesting a decision in favour of one of these options.

v. Conclusion

137

With these remarks we feel we have indicated important issues for future research into the physics underlying quantum measurements and at the same time illustrated the philosophical relevance of our theme. General philosophical discussions of the problem of measurement, together with an evaluation of some of the approaches reviewed in Chapter IV can be found in the works by d'Espagnat [dEsp 87,89] and Shimony [Shimony 1993], which both aim at elucidating the quantum mechanical conception of reality. We have not touched upon a methodological problem related to the self-referential nature of a quantum theory of measurement [dChi 77, Peres 82, Pri 90d]. In fact from a methodological point of view the measuring process does not belong to the domain of quantum mechanics but rather serves to constitute the semantics of this theory. It is the requirement of the semantical completeness of quantum mechanics which stipulates that the very (measuring) processes providing operational definitions of the concepts of the theory must be describable in terms of the theory. This semantical completeness, which is illustrated by the pseudo-realistic figure below, induces a logical situation similar to that encountered with GSdel's theorem [GSd 31]. To avoid inconsistencies within a universally valid quantum mechanics, it is argued, the theory cannot be applied to yield a complete description of a measurement situation. Rather one has to accept that part of the process of measurement remains unanalysed [Breu 95, Peres 82]; in other words, according to this point of view one has to distinguish between two levels of description: the endophysical (ontic) and the exophysical (epistemic) levels [Pri 90c,d], analogously to the distinction between object language and metalanguage in logic. On the other hand, no Goedel-type propositions were formulated in quantum mechanics up to now. In our opinion these ideas deserve to be taken seriously; but they also require further elaboration towards rigorous formalisation before their far-reaching implications can be properly estimated.

Figure 1. Illustration of the semantical completeness of quantum mechanics. Adapted from L. S. Penrose, R. Penrose, The British Journal of Psychology 49 (1958) 31.

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1.2, IV.3

IV.3

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IV.3

IV.3

IV.3

Bauer, E. see [Lon 39] Belavkin, V. P. [Bela 94] Nondemolition Principle of Quantum Measurement Theory. Foundations of Physics 24, 1994, 685-714. [Bela 95a] A Dynamical Theory of Quantum Measurement and Spontaneous Localization. Russian Journal of Mathematical Physics 3(1), 1995, 3-24. [Bela 95b], and Melsheimer, O., A Hamiltonian Solution to Quantum Collapse, State Diffusion and Spontaneous Localisation. In Nottingham 1994, pp. 201-222.

IV.5.4

IV.4.3

IV.5.4

Author Index and Reference

Bell, J. S. [Bell 66] On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern Physics 38, 1966, 447-452. [Bell 75] On Wave Packet Reduction in the Coleman-Hepp Model. Helvetica Physica Acta 48, 1975, 93-98. [Bell 87] Are There Quantum Jumps? In: SchrSdinger. Centenary of a Polymath, Cambridge University Press, Cambridge. (Reprinted in: J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987.) [Bell 89] Towards An Exact Quantum Mechanics. In: Themes in Contemporary Physics. Essays in Honor o£ Julian Schwinger's 70th Birthday, S. Deser, R. J. Finkelstein, eds., World Scientific, Singapore 1989, pp. 1-26.

147

1.2 IV.5.4

IV.4.3, V

IV.4.3

Beltrametti, E. see Bibliography 1.2, II.1, II.2.2, II.2.5, III.4.7, III.7.2, IV.4.2 [Belt 90] , Cassinelli, G., and Lahti, P. J., Unitary Measurements of Discrete Quantities in Quantum Mechanics. Journal o£ Mathematical Physics 31, 1990, 91-98. III.2.2, III.2.5, III.4.4, III.6.3, III.7.2, III.8.3, IV.4.2 Benatti, F. [Ben 87] , Ghirardi, G. C., Rimini, A., and Weber, T., Quantum Mechanics with Spontaneous Localization and the Quantum Theory of Measurement. I1 Nuovo Cimento 100 B, 1987, 2741.

IV.4.3

Berberian, S. see Bibliography

III.4.2

Biaiynicki-Birula, I. [Bia 76], and Mycielski, J., Nonlinear Wave Mechanics. Anna/s of Physics 100, 1976, 62-93.

IV.4.3

Blanchard, Ph. [Bla 93], and Jadczyk, A. Classical and quantum intertwine, in Cologne 1993, pp. 65-76. On the interaction between classical and quantum systems. Physics Letters A 175, 1993, 157-164 [Bla 95], and Jadczyk, A., Event enhanced quantum theory and

IV.4.2

148

AuthorIndex and Reference piecewise deterministic dynamics. Annalen der Physik 4, 1995, 583-599. Event-Enhanced Formalism of Quantum Theory or Columbus Solution to the Quantum Measurement Problem. In Nottingham 1994, pp. 223-233.

Bohm, D [Bohm 52] A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables". Part I: Physical Review 85, 1952, 166-179; Part II: Physical Review 85, 1952, 180-193. [Bohm 54], and Vigier, J. P., Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations. Physical Review 96, 1954, 208-216. see [Aha 61] [Bohm 66], and Bub, J., A Proposed Solution of the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory. Reviews of Modern Physics 38, 1966, 453-469. [Bohm 89], and Hiley, B. J., Non-Locality and Locality in the Stochastic Interpretation of Quantum Mechanics. Physics Reports 172, No. 3, 1989, 93-122. Bohr, N. [Bohr 28] The Quantum Postulate and the Recent Development of Atomic Theory. Nature, 121, (1928), pp. 580-590. [Bohr 39] Causality Problems in Atomic Physics. In: New Theories in Physics, Paris 1939, pp. 11-45. [Bohr 48] On the Notions of Causality and Complementarity. Dialectica 1, 1948, 312-319. [Bohr 49] Discussion with Einstein on Epistemological Problems in Atomic Physics. In: Albert Einstein: Philosopher-Scientist, P. A. Schilpp, ed., Library of Living Philosophers, Evanston, Illinois 1949, pp. 663-688. Bona, P. [Bona 80] A Solvable Model of Particle Detection in Quantum Theory. Acta F.R.N. Univ. Comen. Physica XX, 1980, 65-95. [Bona 93] Selfconsistency and objectification. In Cologne 1993, pp. 98-105.

IV.4.2

1.2, IV.3

1.2

IV.3

IV.3

1.2, IV.2 IV.2 IV.2

1.2

IV.5.4 IV.5.4

Author Index and Reference

149

Born, M. [Born 26] Zur Quantenmechanik der Stoflvorg/inge. Zeitschrift ffir Physik 37, 1926, 863-867.

IV.2

Boyer, T. H. [Boy 80] A Brief Survey of Stochastic Electrodynamics. In: Foundations of Radiation Theory and Quantum Electrodynamics. Ed. A. O. Barut, Plenum Press, New York 1980, pp. 49-63.

IV.3

Breuer, T. [Breu 93a], Amann, A., and Landsman, N. P., Disjoint final states in robust quantum measurements. In: Cologne 1993, pp. 118124. [Breu 93b], Amann, A., and Landsman, N. P., Inaccuracy and Spontaneous Symmetry Breaking in Quantum Measurements. Journal of Mathematical Physics 34, 1993, 5441-5450. [Breu 95] The Impossibility of Accurate State Self-Measurements. Philosophy of Science 62, 1995, 197-214. Brown, H. R. [Bro 86] The Insolubility Proof of the Quantum Measurement Problem. Foundations of Physics 16, 1986, 857-870.

IV.5.4

IV.5.4 V

III.6.2

Bub, J. see [Bohm 66], [Elby 94] [Bub 88] How to Solve the Measurement Problem of Quantum IV.5.4 Mechanics? Foundations of Physics 18, 1988, 701-722. [Bub 92] Quantum mechanics without the projection postulate. IV.5.2 Foundations of Physics 22, 1992, 737. [Bub 94] On the structure of quantal proposition systems. FounIV.5.2 dations of Physics 24, 1994, 1261-1279. [Bub 96], and Clifton, R. A Uniqueness Theorem for No Collapse Interpretations of Quantum Mechanics. Studies in History IV.5.2, IV.5.2 and Philosophy of Modern Physics, in press 1996. Buffa, M. [Buf 95], Nicrosini, O., and Rimini, A. (1995). Dissipation and Reduction of Superconducting States Due to Spontaneous Localization. In Nottingham 1994, pp. 235-244.

IV.4.3

150

Author Index and Reference

Busch, P. see Bibliography 1.2, II.1.1,111.2.6, 111.8, 111.8.1,111.8.2, IV.5.5 [Bus 82] Unbestimmtheitsrelation und simultane Messungen in der Quantentheorie. Dissertation, Cologne 1982. English translation: Indeterminacy Relations and Simultaneous Measurements in Quantum Theory. International Journal of Theoretical Physics 24, 1995, 63-92. IV.5.5 [Bus 85] Momentum Conservation Forbids Sharp Localisation. Journal of Physics A 18, 1985, 3351-3354. III.8.3 [Bus 88] Surprising Features of Unsharp Quantum Measurements. Physics Letters A 130, 1988, 323-329. IV.5.5 [Bus 89a], and Schroeck, F. E., Jr., On the Reality of Spin and Helicity. Foundations of Physics 19, 1989, 807-872. III.8.3, IV.5.5 [Bus 89b], and Lahti, P. J., Some Remarks on Unsharp Quantum Measurements, Quantum Non-Demolition and All That. Annalen der Physik 47, 1990, 369-382. III.9.2 [Bus 90a], and Lahti, P. J., Completely Positive Mappings in Quantum Dynamics and Measurement Theory, Foundations of Physics 20, 1990, 1429-1439. III.2.2 [Bus 90b], Cassinelli, G., and Lahti, P. J., On the Quantum Theory of Sequential Measurements. Foundations of Physics 20, 1990, 757-775. III.4.6, III.4.7, III.8.1 [Bus 90c] Macroscopic Quantum Systems and the Objectification Problem. In: Joensuu 1990, pp. 62-76. III.6.3 see [Lah 91] [Bus 93], Kienzler, P., Lahti, P., and Mittelstaedt, P., Testing Quantum Mechanics Against a Full Set of Bell Inequalities. Physical Review A 47, 1993, 4627-4631. II.2.6 see [Sche 95] [Bus 95a], Grabowski, M., and Lahti, P., Repeatable Measurements in Quantum Theory: Their Role and Feasibility. Foundations of Physics 25, 1995, 1239-1266. III.4.6 [Bus 95b], Cassinelli, G., and Lahti, P., Probability Structures for Quantum State Spaces. Reviews in Mathematical Physics 7, 1995, 1105-1121. [Bus 96a], and Lahti, P. J., Individual Aspects of Quantum Measurements. Preprint 1996. III.2.3 [Bus 96b], and Lahti, P. J., Correlation Properties of Quantum Measurements. Journal of Mathematical Physics, in press 1996. III.2.4, III.4.2, III.4.3, III.4.4

Author Index and Reference

[Bus 96c], and Shimony, A., Insolubility of the Quantum Measurement Problem for Unsharp Observables. Studies in History and Philosophy of Modern Physics, in press 1996. [Bus 96d] Can 'Unsharp Objectification' Solve the Quantum Measurement Problem? Preprint 1996.

151

III.6.2 III.6.2

Casher, A. see [Aha 87] Cassinelli, G. see Bibliography: Beltrametti [Cas 84], and Zanghi, N., Conditional Probabilities in Quantum Mechanics II. I1 Nuovo Cimento 79 B, 1984, 141-154. [Cas 89], and Lahti, P. J., The Measurement Statistics Interpretation of Quantum Mechanics: Possible Values and Possible Measurement Results of Physical Quantities. Foundations of Physics 19, 1989, 873-890. III.3.1, see [Belt 90] see [Bus 90b] [Cas 93], and Lahti, P., The Copenhagen variant of the modal interpretation and quantum theory of measurement Foundations of Physics Letters 6, 1993, 533-544. [Cas 94], De Vito, E., and Lahti, P., Properties of the range of a state operator. Reports on Mathematical Physics 34, 1994, 211-224. [Cas 95], and Lahti, P., Quantum theory of measurement and the modal interpretations of quantum mechanics. International Journal for Theoretical Physics 34, 1995, 1271-1281.

III.2.4

III.3.2

IV.5.2

IV.5.2

IV.5.2

Christian, J. J. [Chri 94] On Definite Events in a Generally Covariant Quantum World h Newton-Cartan Theory. Preprint, University of Oxford 1994.

V

Cini, M. [Cini 79], De Maria, M., Mattioli, G., and Nicolo, F., Wave Packet Reduction in Quantum Mechanics: A Model of a Measuring Apparatus. Foundations of Physics 9, 1979, 479-500.

IV.4.2

152

AuthorIndex and Reference

Clifton, R. [Cli 94] The Triorthogonal Uniqueness Theorem and Its Irrelevance to the Modal Interpretation of Quantum Mechanics. In Helsinki 1994, pp. 45-60. [Cli 95] Why Modal Interpretations of Quantum Mechanics Must Abandon Classical Reasoning About Physical Properties. International Journal of Theoretical Physics 34, 1995, 13031312. Cole, D.C. [Cole 90] Classical Electrodynamic Systems Interacting with Classical Electromagnetic Random Radiation. Foundations of Physics 20, 1990, 225-240. Coleman, S. [Cole 95], and Lesniewski, A. Unpublished results 1995.

IV.5.1

IV.5.2

IV.3

III.3.3

Collet, M. J. see [Wal 85] Courbage, M. see [Mis 79] dalla Chiara, M. L. [dChi 77] Logical Self-Reference, Set Theoretical Paradoxes and the Measurement Problem in Quantum Mechanics. Journal of Philosophical Logic 6, 1977, 331-347.

V

Daneri, A. [Dan 62], A., Loinger, A., and Prosperi, G. M., Quantum Theory of Measurement and Ergodicity Conditions. Nuclear Physics 33, 1962, 297-319. (Reprinted in Wheeler and Zurek 1983.) IV.4.2, IV.5.5 Daniel, W. [Dani 89] Axiomatic description of irreversible and reversible evolution of a physical system. Helvetica Physica Acta 62, 1989, 941-968.

IV.4.3

Author Index and Reference

153

Davies, E. B. see Bibliography 1.2, II.1, II.21, III.2.1, III.2.2, III.4.6, III.4.7, IV.5.5 [Dav 70], and Lewis, J. T., An Operational Approach to Quantum Probability. Communications in Mathematical Physics 17, 1970, 239-259. III.4.2, III.4.6, III.8.1 de Broglie, L. see Bibliography

1.2

De Maria, M. see [Cini 79] Deutsch, D. [Deu 85] Quantum Theory as a Universal Physical Theory. International Journal of Theoretical Physics 24, 1985, 1-41.

IV.5.1

De Vito, E. see [Cas 94] DeWitt, B. S. see Biblography 1.2, III.3.3 [DeW 70] Quantum Mechanics and Reality. Physics Today 23, 1970, 30-35. IV.5.1 [DeW 71] The Many-Universes Interpretation of Quantum Mechanics. In: Foundations of Quantum Mechanics, B. d'Espagnat, ed., Academic Press, New York 1971. IV.5.1 Dieks, D. [Di 89] Quantum mechanics without the projection postulate and its realistic interpretation. Foundations of Physics 19, 1989, 1397-1423. [Di 94] The Modal Interpretation of Quantum Mechanics, Measurements, and Macroscopic Behavior. Physical Review A 49, 1994, 2290-2300. Di6si, L. [Dio 89] Models for Universal Reduction of Macroscopic Quantum Fluctuations. Physical Review A 40, 1989, 1165-1174.

IV.5.2

IV.5.2

IV.4.3

154

Author Index and Reference

Donald, M. J. [Don 90] Quantum Theory and the Brain. Proceedings of the Royal Society of London A 427, 1990, 43-93. [Don 92] A Priori Probability and Localized Observers. Foundations of Physics 22, 1992, 1111-1172. [Don 95] A Mathematical Characterization of the Physical Structure of Observers. Foundations of Physics 25, 1995, 529-571. IV.5.1 Dowker, H.F. [Dow 96], and Kent, A., On the Consistent Histories Approach to Quantum Mechanics. Journal of Statistical Physics 82, 1996, 1575-1646. Diirr, D. [Diirr 92], Goldstein, S., and Zanghi, N., Quantum Equilibrium and the Origin of Absolute Uncertainty, Journal of Statistical Physics 67, 1992, 843-907. [Diirr 93], Goldstein, S., and Zanghi, N., A global Equilibrium as the Foundation of Quantum Randomness, Foundations of Physics 23, 1993, 721-738.

IV.5.1 IV.5.1

IV.5.2

IV.3

IV.3

Einstein, A. [Eins 35] Einstein, A., Podolsky, B., and ROSEN, N., Can Quantum-Mechanical Description of Physical Reality be Considered Complete? Physical Review 47, 1935, 777-780. 1.2, III.l.1, IV.3 [Eins 36] Physics and Reality. Journal of the Franklin Institute 221, 1936, 349-382. 1.2 Elby, A. [Elby 94], and Bub, J., Triorthogonal Uniqueness Theorem and Its Relevance to the Interpretation of Quantum Mechanics. Physical Review A 49, 1994, 4213-4216.

IV.5.1

Emch, G. G. see [Whi 75] d'Espagnat, B. see Bibliography

IV.3

Author Index and Reference [dEsp 87] Empirical Reality, Empirical Causality and the Measurement Problem. Foundations of Physics 17, 1987, 507-529. [dEsp 89] Are there Realistically Interpretable Local Theories? Journal of Statistical Physics 56, 1989, 747-766.

155

V IV.5.1, V

Everett, H. [Eve 57] The Theory of the Universal Wave Function. In: DeWitt and Graham (1973). 1.2, III.3.1, III.5, III.5.2, IV.5.1 Feinstein, A. see Bibliography

III.5.1

Fine, A. [Fine 69] On the General Quantum Theory of Measurement. Proceedings of the Cambridge Philosophical Society 65, 1969, 111-122. [Fine 70] Insolubility of the Quantum Measurement Problem. Physical Review D 2, 1970, 2783-2787.

III.1.2 III.6.2

Finkelstein, D. [Fin 62] The Logic of Quantum Physics, Transactions of the New York Academy of Science 25 1962+63, 621-637. III.3.3, IV.5.1 Foulis, D. [Fou 78], and Randall, C. H., The Operational Approach to Quantum Mechanics. In: The Logico-Algebraic Approach to Quantum Mechanics III, C. A. Hooker, ed., Reidel, Dordrecht 1978, pp. 167-201.

1.2

Frenkel, A. [Fre 90] Spontaneous Localizations of the Wave Function and Classical Behavior. Foundations of Physics 20, 1990, 159-188.

IV.4.3

Fukuda, R. [Fuk 90] Macrovariables and the Theory of Measurement. Tokyo 1989, pp. 124-134.

IV.4.2

In:

G arraway, B. M. [Gar 94], and Knight, P. L., A comparison of quantum state diffusion and quantum jump simulation of two-photon processes

156

AuthorIndex and Reference in a dissipative environment. Physical Review A 49, 1994, 1266-1274.

IV.4.3

Gaveau, B. [Gav 90], and Schulman, L. S., Model Apparatus for Quantum Measurements. Journal of Statistical Physics 58, 1990, 12091230.

V

Gell-Marm, M. [Gell 90], and Hartle, J. B., Quantum Mechanics in the Light of Cosmology. In: Tokyo 1989, pp. 321-343. [Gell 93] Gell-Mann, M., and Hartle, J.B., Classical equations for quantum systems. Physical Review D 47, 1993, 3345-3382.

IV.5.2 IV.5.2

G hirardi, G. C. [Ghi 86], Rimini, A., and Weber, T., Unified Dynamics for Microscopic and Macroscopic Systems. Physical Review D 34, 1986, 470-491. III.7.2, IV.4.3 see [Ben 87] [Ghi 90a], and Rimini, A., Old and New Ideas in the Theory of Quantum Measurement. In: Erice 1989, pp. 167-191. IV.4.3 [Ghi 90b], Grassi, R., and Pearle, P., Relativistic Dynamical Reduction Models: General Framework and Examples. In: Joensuu 1990, pp. 109-123. IV.4.3 Gisin, N. [Gis 83] Irreversible quantum dynamics and the Hilbert space structure of quantum kinematics. Journal of Mathematical Physics 24, 1983, 1779-1782. [Gis 84] Quantum Measurement and Stochastic Processes. Physical Review Letters 52, 1984, 1657-1660. III.7.2, [Gis 86] Generalisation of Wigner's theorem for dissipative quantum systems. Journal of Physics A: Mathematical and General 19, 1986, 205-210. [Gis 89] Stochastic Quantum Dynamics and Relativity. Helvetica Physica Acta 62, 1989, 363-371. [Gis 90] Weinberg's Non-linear Quantum Mechanics and Supraluminal Communications. Physics Letters A 143, 1990,1-2. [Gis 92], and Percival, I.C., Wave function approach to dissipative

IV.4.3 IV.4.3

IV.4.3 IV.4.3 IV.4.3

Author Index and Reference

157

processes: are there quantum jumps? Physics Letters A 167, 1992, 315-318. [Gis 93a], and Percival, I. C., The quantum state diffusion picture of physical processes. Journal of Physics A 26, 1993, 22452260. [Gis 93b], and Percival, I. C., Stochastic wave equations versus parallel world components. Physics Letters A 175, 1993, 144145.

V

IV.4.3

V

Giulini, D. see Bibliography

IV.5.3

Giuntini, R. see Bibliography

1.2

Glauber, R. J. [Gla 86] Amplifiers, Attenuaters and the Quantum Theory of Measurement. In: Frontiers of Quantum Optics. Eds. E. R. Pike and S. Sarkar, Adam Hilger, Bristol 1986. IV.5.3, IV.5.5 Gleason, A. M. [Glea 57] Measures on the Closed Subspaces of a Hilbert Space. Journal of Mathematics and Mechanics 6, 1957, 885-893.

1.2, II.2

GSdel, K. [GSd 31] Uber formal unentscheidbare S/itze der Principia Mathematica und verwandter Systeme I. Monatshefte f'dr Mathematik und Physik 38, 1931, 173-198.

V

**

Goldstein, S. see [D/irr 92, 93] Grabowski, M. see Bibliography: Busch [Grab 90] Quantum Measurement and Dynamics. Annalen der Physik 47, 1990, 391-400. see [Bus 95a] Graham, N. see Bibliography: DeWitt

1.2

158

AuthorIndex and Reference [Gra 73] The Measurement of Relative Frequency. In: DeWitt and Graham 1973, pp. 229-253.

III.3.3

Griffiths, R. B. [Gri 84] Consistent Histories and the Interpretation of Quantum Mechanics. Journal of Statistical Physics 36, 1984, 219-272.

IV.5.2

Groenewold, H. J. [Gro 71] A Problem of Information Gain by Quantum Measurements. International Journal of Theoretical Physics 4, 1971, 327-338.

III.5.1

Gudder, S.P. see Bibliography

IV.3

Guenin, M. [Gue 66] Axiomatic Foundations of Quantum Theories. Journal of Mathematical Physics 7, 1966, 271-282.

IV.4.2

Gutman, S. [Gut 95] Using Classical Probability To Guarantee Properties of Infinite Quantum Sequences. Archive quant-ph/9506016, 1995.

III.3.3

Haag, R. see Bibliography

IV.5.4

Haake F. [Haa 68], and Weidlich, W., A Model for the Measuring Process in Quantum Theory. Zeitschrift f'dr Physik 213, 1968, 451-465. IV.4.2 [Haa 87], and Walls, D. F., Overdamped and Amplifying Meters in the Quantum Theory of Measurement. Physical Review A 36, 1987, 730-739. IV.5.3, IV.5.5 [Haa 93], and Zukowski, M., Classical motion of meter variables in the quantum theory of measurement. Physical Review A 47, 1993, 2506-2517. IV.5.3, IV.5.5 Hadjisavvas, N. [Had 81] Properties of Mixtures of Non-Orthogonal States. Letters on Mathematical Physics 5, 1981, 327-332. II.2.5, IV.5.2

Author Index and Reference

Halliwell, J. J. [Hal 93a] Quantum-mechanical histories and the uncertainty principle: Information-theoretic inequalities. Physical Review D 48, 1993, 2739-2752. [Hal 93b] Quantum-mechanical histories and the uncertainty principle II: Fluctuations about classical predictability. Physical Review D 48, 1993, 4785-4799.

159

IV.5.2

IV.5.2

Harrison, F. E. see [Wan 93] Hartle, J. B. [Har 68] Quantum Mechanics of Individual Systems. American Journal of Physics 36, 1968, 704-712. III.3.3, IV.5.1 see [Gell 90, 93] Healey, R. see Bibliography

IV.5.2

Heisenberg, W. see Bibliography [Heis 27] Uber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik. Zeitschrift f'dr Physik 43, 1927, 172-198.

1.2, IV.2

Hellwig, K. E. [Hell 71] Measuring Processes and Additive Conservation Laws. In: Foundations of Quantum Mechanics, B. d'Espagnat, ed., Academic Press, New York 1971, pp. 338-345.

III.8.3

Helstrom, C. W. see Bibliography Hepp, K. [Hepp 72] Quantum Theory of Measurement and Macroscopic Observables. Helvetica Physica Acta 45, 1972, 237-248.

IV.2

III.5

IV.5.4

160

AuthorIndex and Reference

Hiley, B. J. see [Bohm 89] Holevo, A. S. see Bibliography

1.2, IV.5.5

Holland, P. R. see Bibliography

IV.3

Ingarden, R. ling 76] Quantum Information Theory. Reports on Mathematical Physics 10, 1976, 43-72.

III.5

Isham, C. I. [Ish 94a] Quantum Logic and the Histories Approach to Quantum Theory, Journal of Mathematical Physics 35, 1994, 2157-2185. [Ish 94b], and Linden, N., Quantum Temporal Logic and Decoherence Functionals in the Histories Approach to Generalised Quantum Theory. Journal of Mathematical Physics 35, 1994, 5452-5476.

IV.5.2

Jadczyk, A. see [Bla 93, 95] [Jad 95] Particle Tracs, Events and Quantum Theory. Progress of Theoretical Physics 93, 1995, 631-646.

IV.4.2

Jammer, M. see Bibliography

IV.5.2

1.2

J auch, J. M. see Bibliography 1.2, II.1, II.2.2, II.2.6, IV.4.2 [Jau 64] The Problem of Measurement in Quantum Mechanics. Helvetica Physica Acta 37, 1964, 293-316. IV.4.2 [Jau 67], Wigner, E. P., and Yanase, M. M., Some Comments Concerning Measurements in Quantum Mechanics. I1 Nuovo Cimento 48 B, 1967, 144-151. IV.4.2 [Jau 72], and B£ron, G., Entropy, Information and Szilard's Paradox. Helvetica Physica Acta 45, 1972, 220-232. III.5.2

Author Index and Reference

Joos, E. see Bibliography: Giulini [Joos 85], and Zeh, H.D., The Emergence of Classical Properties through Interaction with the Environment. Zeitschrift f'dr Physik B 59, 1985, 223-243. Kakazu, K. see [Kuda 89] [Kaka 90], and Matsumoto, S., Stability of Particle Trajectories and Generalized Coherent States. Physical Review A 42, 1990, 5093-5102. [Kaka 91] Equivalence Classes and Generalized Coherent States in Quantum Measurement. I1 Nuovo Cimento B 106, 1991, 1173-1185.

161

IV.5.3

IV.5.5

V

Kent, A. see [Dow 96] Khalfin, L. A. [Khal 87] , and Tsirelson, B. S., A Quantitative Criterion of the Applicability of the Classical Description within the Quantum Theory. In: Joensuu 1987, pp. 369-401. IV.5.3, IV.5.5 Kiefer, C. see Bibliography: Giulini Knight, P. L. see [Gar 94] Kochen, S. [Koch 67] and SPECKER, E. P., The Problem of Hidden Variables in Quantum Mechanics. Journal of Mathematics and Mechanics 17, 1967, 59-87. [Koch 85] A New Interpretation of Quantum Mechanics. In: Joensuu 1987, pp. 151-169. Kraus, K. see Bibliography

1.2 IV.5.2

1.2, II.1, II.l.1, III.2.1, III.2.2

162

AuthorIndex and Reference

Kudaka, S. [Kuda 89], Matsumoto, S., and Kakazu, K., Generalized Coherent State Approach to Quantum Measurement. Progress of Theoretical Physics 82, 1989, 665-681. [Kuda 90], and Matsumoto, S., Quantum Measurement and Generalised Coherent State. In: Joensuu 1990, pp. 190-202.

IV.5.5 IV.5.5

Kupsch, J. see Bibliography: Giulini, D. Lahti, P. J. see Bibliography: Busch [Lah 85] Uncertainty, Complementarity and Commutativity. In: Cologne 1984, pp. 61-80. see [Cas 89] see [Bus 89b] see [Belt 90] see [Bus 90a,b] [Lah 90] Quantum Theory of Measurement and the Polar Decomposition of an Interaction. International Journal of Theoretical Physics 29, 1990, 339-350. [Lah 91], Busch, P. and Mittelstaedt, P., Some Important Classes of Quantum Measurements and their Information Gain. Journal of Mathematical Physics 32, 1991, 2770-2775. see [Cas 93b, 94, 95] see [Bus 96a,b] Landsman, N. P. see [Breu 93a, 93b] [Lan 91] Algebraic Theory of Superselection Sectors and the Measurement Problem in Quantum Mechanics. International Journal of Modern Physics A 6, 1991, 5349-5371. [Lan 95] Observation and Superselection in Quantum Mechanics. Studies in History and Philosophy of Modern Physics 26 B, 1995, 45-73. Lanz, L. see [Barc 82]

III.8.2

IV.5.2

III.4.7

IV.5.4

IV.5.4, V

Author Index and Reference

163

[Lanz 93], and Melsheimer, O., Quantum mechanics and trajectories. In Cologne 1993, pp. 233-241.

IV.4.1

Leggett, A. [Leg 80] Macroscopic Quantum Systems and the Quantum Theory of Measurement. Supplement of the Progress of Theoretical Physics 69, 1980, 80-100.

V

Lesniewski, A. see [Cole 95] L~vy-Leblond, J. M. [L~v 77] Towards A Proper Quantum Theory. In: Quantum Mechanics, A Haft Century Later, J. Leite Lopes and M. Paty, eds., D. Reidel, Dordrecht 1977, pp. 171-206.

V

Lewis, J. T. see [Dav 70] Lindblad, G. see Bibliography III.5 [Lin 72] An Entropy Inequality for Quantum Measurements. Communications in Mathematical Physics 28, 1972, 245-249. III.5.1 [Lin 73] Entropy, Information and Quantum Measurements. Communications in Mathematical Physics 33, 1973, 305-322. III.5, III.5.1, III.5.2 Linden, N. see [Ish 94b] Liu, S.-J. [Liu 95], and Sun, C.-P., Generalization of Cini's model for quantum measurement and dynamical realization of wavefunction collapse. Physics Letters A 198, 1995, 371-377.

IV.5.4

Lockhart, C. M. [Lock 86], and Misra, B., Irreversibility and Measurement in Quantum Mechanics. Physica 136 A, 1986, 47-76.

IV.5.4, V

164

AuthorIndex and Reference

Lockwood, M. see Bibliography

IV.5.1

Loinger, A. see [Dan 62] London, F. [Lon 39] and Bauer, E., La Thdorie de l'Observation en Mdchanique Quantique. Hermann, Paris 1939. English translation ("including a new paragraph by Professor Fritz Bauer") in Wheeler and Zurek (1983).

1.2, IV.2

Lower, M. see [Alb 88] Liiders, G. [Liid 51] Uber die Zustands~inderung durch den Meflprozess. Annalen der Physik 8, 1951, 322-328. III.2.5, III.5.3 Ludwig, G. 1.2, II.1, II.l.1, IV.4.1 see Bibliography: Ludwig 1983 III.7.1, IV.4.1, IV.5.5, V see Bibliography: Ludwig 1987 [Lud 61] GelSste und ungel6ste Probleme des Mei~prozesses in der Quantenmechanik. In: Werner Heisenberg und die Physik unserer Zeit, F. Bopp, ed., Vieweg, Braunschweig 1961, pp. IV.4.2 150-181. [Lud 83] The Connection between the Objective Description of Macrosystems and Quantum Mechanics of "Many Particles". In: Old and New Questions in Physics, Cosmology, Philosophy and Theoretical Biology, A. van der Merve, ed., Plenum IV.4.1 Press, New York 1983, pp. 243-263. [Lud 93] The minimal interpretation of quantum mechanics and the objective description of macrosystems. In Cologne 1993, pp. 242-250. IV.4.1 Luczack, A. [Lucz 86] Instruments on yon Neumann algebras, Institute of Mathematics, L6di University, Poland 1986.

III.4.6

Author Index and Reference

165

Machida, S. [Mach 80] Theory of Measurement - A Mechanism of Reduction of Wave Packet. Progress of Theoretical Physics 63, 1980, IV.l, IV.4.2 1457-1473 (Part I) and 1833-1847 (Part II). Mackey, G. see Bibliography Margenau, H. [Marg 36] Quantum Mechanical Description. Physical Review 49, 1936, 240-242. [Marg 63] Measurement and Quantum States. Philosophy of Science 30, 1963, 1-16.

1.2

1.2, IV. IV.3

Matsumoto, S. see [Kuda 89, 90], [Kaka 90a] Mattioli, G. see [Cini 79] Milburn, G. J. see [Wal 85] Misra, B. [Mis 79], B., Prigogine, I. and Courbage, M., Lyapunov Variable: Entropy and Measurement in Quantum Mechanics. Proceedings of the National Academy of Sciences of the U. S. A. 76, 1979, 4768-4772. (Reprinted in Wheeler and gurek (1983).) see [Lock 86] Mittelstaedt, P. see Bibliography [Mitt 90] The Objectification in the Measuring Process and the Many Worlds Interpretation. In: Joensuu 1990, pp. 261-279. see [Lah 91] Mycielski, J. see [Bia 76]

IV.4.2, V

1.2 III.3.3

166

Author Index and Reference

Namiki, M. see [Mach 80] [Nam 93], and Pascazio, S., Quantum Theory of Measurement Based on the Many-Hilbert-Space Approach. Physics Reports 232(6), 1993, 301-414. Nelson, E. see Bibliography Neumann, H. [Neum 93] Macroscopic properties of photon quantum fields. In: Cologne 1993, pp. 303-308.

IV.4.2

IV.3

IV.4.1

von Neumann, J. see Bibliography 1.2, II.1, III.l.1, III.2.1, III.2.5, III.4.6, III.4.7, III.5, III.5.1, III.6.2, III.8.1, IV.2 Nicolo, F. see [Cini 79] Nicrosini, O. see [Buf 95] Ochs, W. [Ochs 77] On the Strong Law of Large Numbers in Quantum Probability Theory. Journal of Philosophical Logic, 6, 1977, 473-480. III.3.3, IV.5.1 [Ochs 80] Gesetze der Groi~en Zahlen zur Auswertung quantenmechanischer Met]reihen. In: Grundlagen der Quantentheorie, P. Mittelstaedt and J. Pfarr, eds., Bibliographisches Institut, Mannheim 1980, pp. 127-138. III.3.1 Ohya, M. see Bibliography Omn~s, R. see Bibliography [Omn 88a] Logical Reformulation of Quantum Mechanics I. Foundations. Journal of Statistical Physics 53, 1988, 893-932.

III.5

IV.5.2 IV.5.2

Author Index and Reference

[Omn 88b] Logical Reformulation of Quantum Mechanics II. Interference and the Einstein-Podolsky-Rosen Experiment. Journal of Statistical Physics 53, 1988, 933-955. [Omn 88c] Logical Reformulation of Quantum Mechanics III. Classical Limit and Irreversibility. Journal of Statistical Physics 53, 1988, 957-975. [Omn 92] Consistent interpretations of quantum mechanics. Reviews in Modern Physics 64, 1992, 339-382.

167

IV.5.2

IV.5.2, V IV.5.2

Ozawa, M. [Oza 84] Quantum Measuring Processes of Continuous Observables. Journal of Mathematical Physics 25, 1984, 79-87. III.2.1, III.2.1, III.4.6 [Oza 86] On Information Gain by Quantum Measurements of Continuous Observables. Journal of Mathematical Physics 27, 1986, 759-763. III.5.1 Partovi, H. M. [Part 89] Irreversibility, Reduction and Entropy Increase in Quantum Measurements. Physics Letters A137, 1989, 445-450.

IV.5.3

Pascazio, S. see [Nam 93] Pauli, W. see Bibliography

1.2, III.l.1, III.4.5, III.8.2, IV.2

Pearle, P. M. [Pea 76] Reduction of the state vector by a nonlinear SchrSdinger equation. Physical Review D 13, 1976, 857-868. [Pea 84] Experimental Test of Dynamical State-Vector Reduction. Physical Review D 29, 1984, 235-240. [Pea 86] Stochastic Dynamical Reduction Theories and Superluminal Communication. Physical Review D 33, 1986, 22402252. [Pea 89] Combining Stochastic Dynamical State-Vector Reduction with Spontaneous Localization. Physical Review A 39, 1989, 2277-2289. [Pea 93] Ways to describe state vector reduction. Physical Review A 48, 1993, 913-923.

IV.4.3 IV.4.3

IV.4.3

IV.4.3 IV.4.3

168

AuthorIndex and Reference [Pea 94] Reality Checkpoint. In: New York 1992, pp. 187-204.

Penrose, R. see Bibliography

IV.4.3

V

Percival, I. C. [Perc 61] Almost Periodicity and the Quantal H-Theorem. Journal of Mathematical Physics 2, 1961, 235-239. see [Gis 92], [Gis 93b] [Perc 94] Primary State Diffusion. Proceedings of the Royal Society of London A 447, 1994, 189-209. Peres, A see Bibliography [Peres 80] Can we undo quantum measurements? Physical Review D 22, 1980, 879-883. Reprinted in Wheeler and Zurek, 1983. [Peres 82], and Zurek, W. H., Is Quantum Theory Universally Valid? American Journal of Physics 50, 1982, 807-810. [Peres 87] Quantum Chaos and the Measurement Problem. In: Quantum Measurement and Chaos, E. R. Pike and S. Sarkar, eds., Plenum Press, New York 1987, pp. 59-80.

IV.4.2

IV.4.3

1.2, IV.3 V V

V

Petz, D. see Bibliography: Ohya Piron, C. see Bibliography

1.2, III.6.3, IV.4.2

Podolsky, B. see [Eins 35] Prigogine, I. see [Mis 79] Primas, H. see Bibliography 1.2, IV.5.4 [Pri 90a] Induced Nonlinear Time Evolution of Open Quantum Objects. In: Erice 1989, pp. 259-280. IV.5.4, IV.5.5 [Pri 90b] The Measurement Process in the Individual Interpre-

Author Index and Reference

tation of Quantum Mechanics. In: Rome 1989, pp. 49-68. IV.5.4, V [Pri 90c] Necessary and Sufficient Conditions for an Individual Description of the Measurement Process. In: Joensuu 1990, pp. 332-346. [Pri 90d] Mathematical and Philosophical Questions in the Theory of Open and Macroscopic Quantum Systems. In: Erice 1989, pp. 233-257. Prosperi, G. M. see [Dan 62] [Pro 71] Macroscopic Physics and the Problem of Measurement in Quantum Mechanics. In: Foundations of Quantum Mechanics, B. d'Espagnat, ed., Academic Press, New York 1971, pp. 97-126. [Pro 74] Models of the Measuring Process and Macro-Theories. In: Foundations of Quantum Mechanics and Ordered Linear Spaces, A. Hartk/imper and H. Neumann, eds., SpringerVerlag, Berlin 1974, pp. 163-198. see [Barc 82] Prugove~ki, E. see Bibliography Quadt, R. [Qua 92] VeraUgemeinerte Entropiekonzepte und Anwendungen in der Theorie dynamischer Systeme und in der Informationstheorie. Doctoral Dissertation, Cologne 1992, Verlag Shaker, Aachen. [Qua 94], and Busch, P., Coarse-graining and the quantum-classical connection. Open Systems and Information Dynamics 2, 1994, 129-155. Raggio, G. [Rag 82] States and Composite Systems in W*-Algebraic Quantum Mechanics. Dissertation, ETH Z/irich 1982. Randall, C. H. see [Fou 78]

169

IV.5.4, V

V

IV.4.2

IV.5.5, V

IV.5.5

IV.5.5

IV.5.5

IV.5.4

170

Author Index and Reference

Reed, M. see Bibliography

II.1

Rimini, A. see [Ghi 86], [Ben 87], [Ghi 90a], [Buf 95] Rosen, N. see [Eins 35] Rosenfeld, L. [Ros 65] The Measuring Process in Quantum Mechanics. Supplement, Progress of Theoretical Physics, 1965, pp. 222-231.

IV.4.2

Saunders, S. [Sau 93] Decoherence, Relative States, and Evolutionary Adaptation. Foundations of Physics 23, 1993, 1553-1585. [Sau 95] Time, Quantum Mechanics, and Decoherence. Synthese 102, 1995, 235-266.

IV.5.1

Scherer, H. [Sche 94] Quantemnechanische Modelle von mLscharf pr~iparierenden und von schwach stSrenden Messungen. Doctoral Dissertation, Cologne 1994, Verlag Shaker, Aachen. [Sche 95], and Busch, P., Weakly Disturbing Phase Space Measurements in Quantum Mechanics. In: Nottingham 1994, pp. 155-163.

IV.5.5

SchrSdinger, E [Schr5 35] Die gegenw~rtige Situation in der Quantenmechanik. Die Naturwissenscha~en 23, 1935, 807-812, 824-828, 844-849. [Schr5 36] Probability Relations between Separated Systems. Proceedings of the Cambridge Philosophical Society 32, 1936, 446-452. Schroeck, F. E., Jr. see Bibliography [Schr 81] A Model of a Quantum Mechanical Treatment of Measurement with a Physical Interpretation. Journal of Mathematical Physics 22, 1981, 2562-2572. see [Bus S9a]

IV.5.1

IV.5.5

1.2, III.l.1

1.2

1.2, IV.5.5

IV.5.5

Author Index and Reference

Sch~flman, L. S. see [Gav 90] [Schu 91] Definite quantum measurements. (New York) 212, 1991, 315-340.

171

Annals of Physics

Sewell, G. L. see Bibliography Sherry, T. [Sher 78], and Sudarshan, E. C. G., Interactions Between Classical and Quantum Systems: A New Approach to Quantum Measurement I. Physical Review D 18, 1978, 4580-4589. [Sher 79], and Sudarshan, E. C. G., Interactions Between Classical and Quantum Systems: A New Approach to Quantum Measurement II. Physical Review D 20, 1979, 857-868. Shimony, A. see Bibliography [Shi 63] Role of the observer in quantum theory. American Journal of Physics 31, 1963, 755-773. see [Ste 71] [Shi 74] Approximate Measurement in Quantum Mechanics, II. Physical Review D9, 1974, 2321-2323. [Shi 79] Proposed Neutron Interferometer Test of Some Nonlinear Variants of Wave Mechanics. Physical Review A 20, 1979, 394-396. [Shi 91] Desiderata for a modified quantum dynamics. In: Philosophy of Science Association 1990. Reprinted in Shimony 1993, 1991, pp. 55-67. see [Bus 96c] Simon, B. see Bibliography: Reed [Sire 76] Quantum dynamics: from automorphism to Hamiltonian. In: Studies in Mathematical Physics in Honor of V. Bargmann. Eds. E. Lieb, B. Simon, and A. S. Wightman, Princeton University Press, Princeton 1976, pp. 327-349.

IV.1

IV.5.4

IV.4.2

IV.4.2

V IV.2

III.6.2

IV.4.3

V

IV.4.3

172

Author Index and Reference

Specker, E. P see [Koch 67] Squires, E. see Bibliography [Squ 90] On an Alleged "Proof' of the Quantum Probability Law. Physics Letters A145, 1990, 67-68.

IV.5.1 IV.5.1

Stamatescu, I.-O. see Bibliography: Giulini, D. Stapp, H. P. see Bibliography Stein, H. [Ste 71] Limitations on Measurement. In: Foundations of Quantum Mechanics, B. d'Espagnat, ed., Academic Press, New York 1971, pp. 56-76.

111.8.3

Stinespring, W. F. [Sti 55] Positive Functions on C*-Algebras. Proceedings of the American Mathematical Society 6:I, 1955, 211-216.

III.4.6

Sudarshan, E. C. G. see [Sher 78, 79] Sun, C.-P. [Sun 93] Quantum Dynamical Model for Wave-function Reduction in Classical and Macroscopic Limits. Physical Review A 48, 1993, 898-906. see [Liu 95]

IV.5.4

Takesaki, M. see Bibliography

III.7.2

Thirring, W. see Bibliography

III.5

Author Index and Reference

173

Tsirelson, B. S. see [Khal 87] Tzara, C. [Tza 87] Fuzzy Measurements in Quantum Mechanics and the Representation of Macroscopic Motions. I1 Nuovo Cimento 98 B, 1987, 131-143. [Tza 88] Emergence of a Classical Motion from a Quantum State: A Further Test of a Theory of Fuzzy Measurements. Physics Letters A 127, 1988, 247-250.

IV.5.5

IV.5.5

Vaidman, L. see [Aha 87] van Fraassen, B. see Bibliography [vFra 79] Foundations of Probability: A Modal Frequency Interpretation. In: Problems in the Foundations of Physics, G. Toraldo di Francia, ed., North-Holland Publishing Corporation, Amsterdam 1979, pp. 344-394. III.3.1, [vFra 81] A Modal Interpretation of Quantum Mechanics. In: Current Issues in Quantum Logic, E. Beltrametti and Bas C. van Fraassen, eds., Plenum Press, New York 1981, pp. 229258. [vFra 90] The Modal Interpretation of Quantum Mechanics. in: Joensuu 1990, pp. 440-460. Varadarajan, V. S. see Bibliography

III.3.2

IV.5.2 IV.5.2

1.2, II.1.1, III.6.3, III.8.1

Vermaas, P.E. [Ver 95] Unique Transition Probabilities in the Modal Interpretation. Studies in History and Philosophy of Modern Physics 26 B, 1995. Vigier, J. P. see [Bohm 54]

IV.5.2

IV.5.2

174

AuthorIndex and Reference

Walls, D. F. [Wa185], Collet, M. J., and Milburn, G. J., Analysis of a Quantum Measurement. Physical Review D 32, 1985, 3208-3215. IV.5.3, IV.5.5 see [Haa 87] Wan, K. K. [Wan 80] Superselection Rules, Quantum Measurement and the SchrSdinger's Cat. Canadian Journal of Physics 58, 1980, 976-982. [Wan 93], and Harrison, F. E., Superconducting Rings, Superselection Rules, and Quantum Measurement Problems. Physics Letters A 174, 1993, 1-8.

IV.4.2

IV.5.4

Weber, T. see [Ghi 86], [Ben 87] Weidlich, W. [Weid 67] Problems of the Quantum Theory of Measurement. Zeitschrift ffir Physik 205, 1967, 199-220. see [Haa 68] Weinberg, S. [Wein 89] Testing Quantum Mechanics. Annals of Physics 194, 1989, 336-386. Wheeler, J. A. see Bibliography [Whe 57] Assessment of Everett's "relative" State Formulation of Quantum Theory. Reviews of Modern Physics 29, No. 3, 1957, 463-465. Whitten-Wolfe, B. [Whi 76], and Emch, G. G., A Mechanical Quantum Measuring Process. Helvetica Physica Acta 49, 1976, 45-55.

IV.4.2

IV.4.3

1.2, III.8.3

IV.5.1

IV.5.4

Wigner, E. P. [Wig 52] Die Messung Quantenmechanischer Operatoren. Zeitschrift fiir Physik 133, 101-108. III.8.3, IV.4.2

Author Index and Reference

175

[Wig 61] Remarks on the Mind-Body Question. In: The Scientist Speculates, I. J. Good, ed., W. Heinemann, London, 1961. 1.2, IV.2 [Wig 63] The Problem of Measurement. American Journal of Physics 31, 1963, 6-15. III.6.2, IV.2, IV.4.3 see [Jau 67] [Wig 83] Review of the Quantum Mechanical Measurement Problem. In: Munich 1981, pp. 43-63. IV.2, IV.5.3 WSlfel, J. [WS1 87] ~)ber Folgen quantenmechanischer Messungen und deren informationstheoretische Behandlung. Docotoral Dissertation, Cologne 1987.

III.5.3

Yanase, M. see [Ara 60], [Jau 67] Zanghi, N. see [Cas 84], [Diirr 92, 93] Zaoral, W. [Zao 90] Towards a Derivation of a Nonlinear Stochastic SchrSdinger Equation for the Measurement Process from Algebraic Quantum Mechanics. In: Joensuu 1990, pp. 479-486.

IV.5.4

Zeh, H.D. see Bibliography: Giulini, D. [Zeh 70] On the Interpretation of Measurement in Quantum TheIV.2, IV.5.3 ory. Foundations of Physics 1, 1970, 69-76. see [Joos 85] [Zeh 93] There are No Quantum Jumps, Nor are There Particles! V Physics Letters A 172, 1993, 189-192. Zukowski, M. see [Haa 93] Zurek, W. H. see Bibliography: Wheeler [Zur 81] Pointer Basis of Quantum Apparatus: Into What Mixture does the Wave Packet Collapse? Physical Review D 24,

176

AuthorIndex and Reference

1981, 1516-1525. see [Peres 82] [Zur 82] Environment-induced Superselection Rules. Physical Review D 26, 1862-1880. [Zur 84] Pointer Basis and Inhibition of Quantum Tunneling by Environment-Induced Superselection. In: Tokyo 1983, pp. 181-189. [Zur 91] Zurek W.H., Decoherence and the Transition from Quantum to Classical. Physics Today, October 1991, 36-44. Discussion in Letters: Negotiating the tricky border between quantum and classical. Physics Today, April 1993, 13-15, 8190. [Zur 93] Preferred States, Predictability, Classicality, and the Environment-Induced Decoherence. Progress of Theoretical Physics 89, 1993, 281-302. IV.5.1,

IV.5.3

IV.5.3

IV.5.3

V

IV.5.3

Notation

Xx 42 Xx ,e 43 E

Pl (T) 117 P~o(T) 117 36 7~A( W ) 1 2 7~s(W) 12 relf 45 p(~, ~2; ~) 50

7, 9

E A 8, 9 E n 36 F (n) 46 I~i> <~Pj[ 12 $ (7-/)14 g (7"/s)10 Ep( 7 / s ) 1 0 el, e2, el2

S(ns) s S(T) 65 q l , if2 50

T.a(X,T) 35 T.a(i,T) 36

50

7~ 39 ~/ 7 7/('0 46 7/A 11, 28 7/s 8

T(2) (a, T) 46 T~ 74 TL 67 Ts(X, T) 35 Ts(i,T) 36 T(n) (f~, T) 46

7/s ®~A 11 H(E, T; 7~) 69 Z 33 Z L 40 /?~ 31 ZAN 41 Zu 39

Tvg 67 T(n)

T(U)t s T(Tts ® 7"/A) 11 trITE(X)I 9

L(n) 7

TI~-T2

11 M 31 34 (n) 36

A4u 33 M ~ 39

8

18 Ti ~ o T2 18 /~t 13 UL 40 1)t 13

(TtA, PA, TA, V) 29

pT A S P~E 8

29
P[~]

(7/A, PA, ¢, U) 33 <7/A, A.a, ¢, U) 39 <7t~, AA, ¢, UL> 40

pE S p~(x) s 7

P~ 28 P~ 29

(a,y) 7 (a~,y~) 29

P ~ 36

(R,B(R))7

Subject Index algebraic quantum mechanics 125 beable 126 biorthogonal (polar) decomposition 23, 27, 112, 119 calibration condition 34 classical system 18, 20, 79 coarse-grained version (of an observable) 30, 36 coexistent effects 10 coexistent observables 10, 86 complementary observables 86, 87 compound system 12 consistent histories 122 correlation (anti-) 49, 50 strong observable- 52, 54 strong state- 52 strong value- 55 decision tree (Table 1) 6, 91, 92 decoherence 112, 123 decohering histories 122 dynamics 13 modified 92, 109, 110 nonunitary 14 reduced 13 unitary 13, 81 E-discrimination 30 effect 10, 14 E-filter 30 entangled systems 13 entanglement 13, 23 entropy of a state 65 environment 14, 27 equivalent measurement schemes 32 equivalent states 18 final component states 35, 57 Gemenge 19, 22, 23, 36, 48 Gleason's theorem 15 ignorance interpretation 16, 19, 22, 74

information 69 deficiency of 69 potential gain of 69 (average) relative 71 insolubility theorem 76 interference (effect, term) 20 interpretation Bohr 96 causal 103, 104 Copenhagen 93, 95, 105 existential 115, 124 Heisenberg 97 hidden variable 93, 103, 105 London-Bauer 101 many-minds 116 many-worlds 93, 111, 113 measurement statistics 44, 102 minimal 9, 95, 107 modal 111, 116 Pauli 100 realistic 62, 92 statistical ensemble 44, 46, 102 stochastic 104 von Neumann 99 Liiders transformer 32, 41, 62 operation 32, 55, 58 (see also 21, 35, 36, 75, 76) (see measurement) measurement 9, 28, 30; 1-137 d-ideal 62 first kind 52, 58 ideal 60, 61 Liiders 41, 58, 70, 87 nondegenerate 62 maximal state-entropy 66, 67 p-ideal 61 repeatable 37, 52, 59, 67, 70, 84

180

Subject Index

measurement (cont.) value reproducible 60 von Neumann 41, 70 measurement coupling 29, 91 measurement scheme 29 measuring apparatus 28, 29 mixture 9 nondemolition principle 127 nonunique decomposability 21 (of mixed states) objectification dynamical 110 pointer 30, 74, 80 quasi- 124 unsharp 76, 112, 127 value 30, 74 objectification problem 28, 75, 76, 91 objectification requirement 30, 73, 91 observable 9 associate 33 classical 17, 18, 77, 83 continuous 84 discrete 11, 38, 59 genuinely unsharp 76, 127 informationaUy complete 30, 90 informationaUy equivalent 30 nonobjective 18, 20 objective 16, 20 pointer 28, 36 sharp 9, 10, 17, 19, 38 superselection (see classical) trivial (nontrivial) 32, 55, 76 unsharp 10, 76, 107, 127 operation 31 partial trace 12 pointer function 29 pointer observable 28 classical 77, 92, 107 minimal 39 pointer value-definiteness 36, 60, 76, 91

pointer mixture condition for .,4 36, 37 for S + A 76,91 positive operator valued (POV) measure 7, 10, 11 premeasurement 30, 34 minimal 39, 67, 77 unitary 33, 34, 77, 88 (see measurement) probability measure 8, 9, 14 dispersion-free 15, 17, 18 generalised 14, 15 probability reproducibility condition 29, 83, 91 projection operator valued (PV) measure 7, 8, 9 proper quantum system 17, 91 property 10 nonobjective 20 objective 16, 19 real 19 sharp 10 unsharp 10 quantum jumps 111 quantum stochastic calculus 111 quantum-classical dichotomy 105 reading 79 reading scale 35 finite 36, 54 refinement (of an observable) 30 registration 79 relative frequency 45 relative frequency operator 46, 114 SchrSdinger equation 13, 101, 110 spontaneous localisation 110 standard model 41 (of a measurement scheme) state 8 component 35 entangled 23, 27, 119

Subject Index state (cont.) mixed 9, 21 pure 8, 18 reduced 12, 29 vector 8 state diffusion 111 state preparation 89 state transformation 29, 31, 65, 85 state transformer 31, 33, 35 E-compatible 32, 33 completely positive 33 Liiders 32, 41 von Neumann 41

statistical dependence 50 superposition 8 superselection rule 17, 22 Bargmann 134 discrete 18 continuous 108, 109, 125 dynamically induced 109 environment-induced 123 value attribution 16, 112, 116 value space (of an observable) 9 von Neumann-LiouviUe equation 13

181

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