Lectures on quantum probability. Sociedad Matematica Mexicana

A.M.Chebotarev LECTURES ON QUANTUM PROBABILITY SOCIEDAD MATEMATICA MEXICANA 2000

Contents 1 Introduction 1.1 Classical and quantum events 1.2 Violation of Bell's inequality 1.3 Violation of the triangle inequality 1.4 Violation of partial order by a convex map 1.5 Parallelism between quantum and classical probability theories 2 Probabilities and mean values in QP 2.1 States and observables 2.2 Probability measures generated by states 2.3 Tensor product and partial trace 2.4 Creation and annihilation operators in l2 2.5 Canonical unitary isomorphism 2.6 Example 2.7 Exercises 3 Evolution equations 3.1 Evolution of states and observables 3.2 Equivalent characterizations of CP-maps 3.3 Master Markov equation 3.4 Examples of solvable master equations 3.5 Perron-Frobenius theorem for CP-maps 3.6 Appendix 4 Conditional complete positivity 4.1 Characterization of CCP-maps 4.2 Nonuniqueness of coefficients of a CCP-map 4.3 Integral form of MME 4.4 Equation XG + G*X + λX = B 4.5 Exercises 5 Minimal solution of MME 5.1 Domain assumptions 5.2 Continuity properties 5.3 A priori bounds and continuity property 5.4 Semigroup property of the minimal solution 5.5 Domain of the minimal infinitesimal map 5.6 Exercises 6 Nonexplosion conditions for MME 6.1 Nonexplosion conditions for a minimal QDS 6.2 Range of the minimal resolvent 6.3 Conservation rules for QDS 6.4 Conservative extensions of QDS 6.5 Conditions sufficient for explosion

1 1 4 7 8 8 17 17 22 25 27 29 32 36 39 39 42 48 56 60 63 67 67 70 72 76 80 81 81 83 86 91 92 101 103 103 110 115 122 125

6.6 Exercises and open problems 7 Sufficient nonexplosion conditions 7.1 Jensen inequalities for CP-maps 7.2 Regularization of self- adjoint operators 7.3 Nonexplosion criteria 7.4 Resolvent analysis of conservativity 7.5 Appendix 7.6 Exercises 8 Applications to Markov processes 8.1 Jensen inequalities for expectations 8.2 Regular Markov jump processes 8.3 Normal distribution of jumps 8.4 Cauchy distribution of jumps 8.5 Representation of Feynman integrals 8.6 Regularity of the Azema-Emery process 8.7 Nonexplosion criteria for diffusion 8.8 Exercises 9 Evolution in operator algebras 9.1 Characterization of self-adjointness 9.2 Two-level atoms interacting with field 9.3 A model for heavy ion collision 9.4 Pauli processes in l2 9.5 Exercises and open problems 10 Quantum stochastic processes 10.1 Diffusion process and random shifts 10.2 Fock space arid annihilation operators 10.3 Quantum extension of the Wiener process 10.4 Number process in Fock space 10.5 Noncommutative Ito multiplication table 10.6 Adapted operators and definition of QSDE 11 Quantum stochastic differential equations 11.1 Example of a solvable QSDE 11.2 Example of the strong resolvent limit 11.3 QSDE in ω-representation 11.4 Ito and Stratonovich forms of QSDE 11.5 QSDE as a strong resolvent limit 11.6 Exercises and open problems 12 Symmetric quantum boundary value problem 12.1 Fock building 12.2 Symmetric boundary value problem 12.3 Localization of solutions 12.4 Nonexplosion conditions 12.5 Derivation of QSDE and MME

127 129 129 134 137 139 145 147 149 149 151 155 157 159 162 164 167 169 169 177 180 182 182 187 187 190 196 198 201 205 209 209 217 219 225 234 242 243 243 246 256 260 265

12.6 Construction of self-adjoint extensions 12.7 Open problems

268 270

Index A adapted operator, 205 adapted stochastic integral, 229 annihilation operator, 27 annihilation process, 194 B Baker-Hausdorff rule, 28 Bell inequality, 5 boson Fock space, 191 C Carleman condition, 177 CCP-map, 67 chamber-adapted operator, 244 chaos expansion, 196 closable operator, 30 closed operator, 20 coherent vector, 27, 191 compatibility conditions, 81 complete positive map, 41 conditional expectation, 26 conservative semigroup, 40 contractive map, 41 core, 31, 148 core of CPn-map, 47 creation operator, 27 creation process, 194 D D-extension, 171 domain assumptions, 73 domain of adjoint operator, 18 E empty chamber, 244 essential domain, 31, 148 expectation, 23 exponential vector, 27, 191 F fermion Fock space, 191 Fock building, 244 Fock chamber, 244 Fock vector, 190

G graphic norm, 31 Green's formula, 271 H Heisenberg picture, 40 Hermitian operator, 18 I indicator function, 2, 23 isometry, 20 J Jordan product, 48 K Kadison inequality, 43 M master Markov equation, 48, 51 maximal algebra IQ , 110 mixed state, 18 MME, 9, 48 monotone map, 134 N nonstandard generator, 76 normal operator, 20 normal state, 18 number process, 198 O observable, 18 order of chambers, 259 P parameters of Markov jump process, 151 partial trace, 25 polar decomposition, 20 polarization identity, 19, 37 positive map, 42 positive state, 18 pure state, 18 Q QSDE, 12 quantum dynamical semigroup, 41 quantum probability space, 3

R representation of a normal state, 18 resolution of identity, 19 resolvent equation, 105 S σ-weak topology, 22 Schrodinger picture, 40 semigroup property of regularization, 134 spectral decomposition, 18 spectral family, 2 standard chamber, 244 standard generator, 76

standard QDS, 76 state, 2, 18 Stinespring theorem, 46 strong topology, 18 submonotone map, 83, 134 T total subset, 191 trace-form, 141 U ultraweak topology, 40 W weak topology, 18 weak-* topology, 22