Table of Contents
Preface vii
1 Foundations of Probability 1
1.1 Interpretation Problem in Quantum Mechanics and Classical Probability Theory 4
1.2 Kolmogorov Axiomatics of Probability Theory 6
1.2.1 Events as sets and probability as measure on a family of sets representing events 6
1.2.2 The role of countable-additivity (σ-additivity) 8
1.2.3 Probability space 10
1.3 Elementary Properties of Probability Measure 10
1.3.1 Consequences of finite-additivity 11
1.3.2 Bell's inequality in Wigner's form 13
1.3.3 Monotonicity of probability 14
1.4 Random Variables 14
1.5 Conditional Probability; Independence; Repeatability 17
1.6 Formula of Total Probability 18
1.7 Law of Large Numbers 19
1.8 Kolmogorov's Interpretation of Probability 21
1.9 Random Vectors; Existence of Joint Probability Distribution 22
1.9.1 Marginal probability 22
1.9.2 From Boole and Vorob'ev to Bell 23
1.9.3 No-signaling in quantum physics 25
1.9.4 Kolmogorov theorem about existence of stochastic processes 28
1.10 Frequency (von Mises) Theory of Probability 29
1.11 Subjective Interpretation of Probability 36
1.12 Gnedenko's Viewpoint on Subjective Probability and Bayesian Inference 43
1.13 Cournot's Principle 44
2 Randomness 47
2.1 Random Sequences 48
2.1.1 Approach of von Mises: randomness as unpredictability 50
2.1.2 Laplace-Ville-Martin-Löf: randomness as typicality 53
2.2 Kolmogorov: Randomness as Complexity 54
2.3 Kolmogorov-Chaitin Randomness 56
2.4 Randomness: Concluding Remarks 58
3 Supplementary Notes on Measure-theoretic and Frequency Approaches 61
3.1 Extension of Probability Measure 61
3.1.1 Lebesgue measure on the real line 61
3.1.2 Outer and inner probabilities, Lebesgue measurability 62
3.2 Complete Probability 65
3.3 Von Mises Views 67
3.3.1 Problem of verification 67
3.3.2 Jordan measurability 69
3.4 Role of the Axiom of Choice in the Measure Theory 70
3.5 Possible Generalizations of Probability Theory 71
3.5.1 Negative probabilities 72
3.5.2 On generalizations of the frequency theory of probability 74
3.5.3 p-adic probability 75
3.6 Quantum Theory: No Statistical Stabilization for Hidden Variables? 77
4 Introduction to Quantum Formalism 79
4.1 Quantum States 79
4.2 First Steps Towards Quantum Measurement Theory 84
4.2.1 Projection measurements 85
4.2.2 Projection postulate for pure states 87
4.3 Conditional Probabilities 88
4.4 Quantum Logic 91
4.5 Atomic Instruments 92
4.6 Symmetric Informationally Complete Quantum Instruments 93
4.7 Schrödinger and von Neumann Equations 94
4.8 Compound Systems 95
4.9 Dirac's Symbolic Notations 98
4.10 Quantum Bits 99
4.11 Entanglement 100
4.12 General Theory of Quantum Instruments 101
4.12.1 Davis-Levis instruments 102
4.12.2 Complete positivity 104
5 Quantum and Contextual Probability 107
5.1 Probabilistic Structure of Two-Slit Experiment 110
5.2 Quantum versus Classical Interference 113
5.2.1 Quantum waves? 114
5.2.2 Prequantum classical statistical field theory 115
5.3 Formula of Total Probability with Interference Term 120
5.3.1 Context-conditioning 120
5.3.2 Contextual analog of the two-slit experiment 123
5.3.3 Non-Kolmogorovean probability models 125
5.3.4 Trigonometric and hyperbolic interference 126
5.4 Constructive Wave Function Approach 127
5.4.1 Inverse Born's rule problem 127
5.4.2 Quantum-like representation algorithm 130
5.4.3 Double stochasticity 133
5.4.4 Supplementary observables 134
5.4.5 Symmetrically conditioned observables 136
5.4.6 Non-doubly stochastic matrices of transition probabilities 137
5.5 Contextual Probabilistic Description of Measurements 137
5.5.1 Contexts, observables, and measurements 137
5.5.2 Contextual probabilistic model 139
5.5.3 Probabilistic compatibility (noncontextuality) 142
5.6 Quantum Formula of Total Probability 146
5.6.1 Interference of von Neumann-Lüders observables 147
5.6.2 Interference of positive operator valued measures 150
6 Interpretations of Quantum Mechanics and Probability 155
6.1 Classification of Interpretations 155
6.1.1 Realism and reality 155
6.1.2 Epistemic and ontic description 158
6.1.3 Individual and statistical interpretations 160
6.1.4 Subquantum models and models with hidden variables 161
6.1.5 Nonlocality 162
6.2 Interpretations of Probability and Quantum State 162
6.3 Orthodox Copenhagen Interpretation 163
6.4 Von Neumann's Interpretation 167
6.5 Zollinger-Br ukner Information Interpretation 168
6.6 Copcnhagen-Göttingen Interpretation: From Bohr and Pauli to Plotnitsky 174
6.7 Quantum Bayesianism - QBism 177
6.7.1 QBism childhood in Växjö 177
6.7.2 Quantum theory is about evaluation of expectations for the content of personal experience 179
6.7.3 QBism as a probability update machinery 180
6.7.4 Agents constrained by Born's rule 185
6.7.5 QBism challenge: Born rule or Hilbert space formalism? 187
6.7.6 QBism and Copenhagen interpretation? 188
6.8 Interpretations in the Spirit of Einstein 189
6.9 Växjö Interpretation 191
6.10 Projection Postulate: von Neumann and Luders Versions 194
7 Randomness: Quantum Versus Classical 199
7.1 Irreducible Quantum Randomness 199
7.2 Lawless Universe? Digital Philosophy? 202
7.3 Unpredictability and Indetermmism 205
8 Probabilistic Structure of Bell's Argument 211
8.1 CHSH-inequality in Kolmogorov Probability Theory 214
8.2 Bell-test: Conditional Compatibility of Observables 215
8.2.1 Random choice of settings 217
8.2.2 Construction of Kolmogorov probability space 218
8.2.3 Validity of CHSH-inequality for correlations taking into account randomness of selection of experimental settings 220
8.2.4 Quantum correlations as conditional classical correlations 221
8.2.5 Violation of the CHSH-inequality for classical conditional correlations 223
8.3 Statistics: Data from Incompatible Contexts 224
8.3.1 Medical studies 224
8.3.2 Cognition and psychology 225
8.3.3 Consistent histories 226
8.3.4 Hidden variables 226
8.4 Contextuality of Bell's Test from the Viewpoint of Quantum Measurement Theory 227
8.5 Inter-relation of Observations on a Compound System and its Subsystems 229
8.5.1 Averages 229
8.5.2 Correlations 230
8.5.3 Towards proper quantum formalization of Bell's experiment 231
8.6 Quantum Conditional Correlations 232
8.7 Classical Probabilistic Realization of "Random Numbers Certified by Bell's Theorem" 234
9 Quantum Probability Outside of Physics: from Molecular Biology to Cognition 237
9.1 Quantum Information Biology 238
9.2 Inter-relation of Quantum Bio-physics and Information Biology 240
9.3 From Information Physics to Information Biology 242
9.3.1 Operational approach 242
9.3.2 Free will problem 242
9.3.3 Bohmian mechanics on information spaces and mental phenomena 243
9.3.4 Information interpretation is biology friendly 245
9.4 Nonclassical Probability? Yes! But, Why Quantum? 245
Appendix A Växjö Interpret at ion-2002 249
A.1 Contextual Statistical Realist Interpretation of Physical Theories 249
A.2 Citation with Comments 257
A.3 On Romantic Interpretation of Quantum Mechanics 258
Appendix B Analogy between non-Kolmogorovian Probability and non-Euclidean Geometry 261
Bibliography 263
Index 279
