A.M.Chebotarev LECTURES ON QUANTUM PROBABILITY

SOCIEDAD MATEMATICA MEXICANA 2000 Contents

1 Introduction 1 1.1 Classical and quantum events 1 1.2 Violation of Bell's inequality 4 1.3 Violation of the triangle inequality 7 1.4 Violation of partial order by a convex map 8 1.5 Parallelism between quantum and classical probability theories 8 2 Probabilities and mean values in QP 17 2.1 States and observables 17 2.2 Probability measures generated by states 22 2.3 Tensor product and partial trace 25 2.4 Creation and annihilation operators in l2 27 2.5 Canonical unitary isomorphism 29 2.6 Example 32 2.7 Exercises 36 3 Evolution equations 39 3.1 Evolution of states and observables 39 3.2 Equivalent characterizations of CP-maps 42 3.3 Master Markov equation 48 3.4 Examples of solvable master equations 56 3.5 Perron-Frobenius theorem for CP-maps 60 3.6 Appendix 63 4 Conditional complete positivity 67 4.1 Characterization of CCP-maps 67 4.2 Nonuniqueness of coefficients of a CCP-map 70 4.3 Integral form of MME 72 4.4 Equation XG + G*X + λX = B 76 4.5 Exercises 80 5 Minimal solution of MME 81 5.1 Domain assumptions 81 5.2 Continuity properties 83 5.3 A priori bounds and continuity property 86 5.4 Semigroup property of the minimal solution 91 5.5 Domain of the minimal infinitesimal map 92 5.6 Exercises 101 6 Nonexplosion conditions for MME 103 6.1 Nonexplosion conditions for a minimal QDS 103 6.2 Range of the minimal resolvent 110 6.3 Conservation rules for QDS 115 6.4 Conservative extensions of QDS 122 6.5 Conditions sufficient for explosion 125

6.6 Exercises and open problems 127 7 Sufficient nonexplosion conditions 129 7.1 Jensen inequalities for CP-maps 129 7.2 Regularization of self- adjoint operators 134 7.3 Nonexplosion criteria 137 7.4 Resolvent analysis of conservativity 139 7.5 Appendix 145 7.6 Exercises 147 8 Applications to Markov processes 149 8.1 Jensen inequalities for expectations 149 8.2 Regular Markov jump processes 151 8.3 Normal distribution of jumps 155 8.4 Cauchy distribution of jumps 157 8.5 Representation of Feynman integrals 159 8.6 Regularity of the Azema-Emery process 162 8.7 Nonexplosion criteria for diffusion 164 8.8 Exercises 167 9 Evolution in operator algebras 169 9.1 Characterization of self-adjointness 169 9.2 Two-level atoms interacting with field 177 9.3 A model for heavy ion collision 180 9.4 Pauli processes in l2 182 9.5 Exercises and open problems 182 10 Quantum stochastic processes 187 10.1 Diffusion process and random shifts 187 10.2 Fock space arid annihilation operators 190 10.3 Quantum extension of the Wiener process 196 10.4 Number process in Fock space 198 10.5 Noncommutative Ito multiplication table 201 10.6 Adapted operators and definition of QSDE 205 11 Quantum stochastic differential equations 209 11.1 Example of a solvable QSDE 209 11.2 Example of the strong resolvent limit 217 11.3 QSDE in ω-representation 219 11.4 Ito and Stratonovich forms of QSDE 225 11.5 QSDE as a strong resolvent limit 234 11.6 Exercises and open problems 242 12 Symmetric quantum boundary value problem 243 12.1 Fock building 243 12.2 Symmetric boundary value problem 246 12.3 Localization of solutions 256 12.4 Nonexplosion conditions 260 12.5 Derivation of QSDE and MME 265

12.6 Construction of self-adjoint extensions 268 12.7 Open problems 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

Exp

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