Quantum Information Theory and Quantum Statistics (eBook)

Preface 6
Contents 8
Introduction 11
Prerequisites from Quantum Mechanics 13
2.1 Postulates of Quantum Mechanics 14
2.2 State Transformations 24
2.3 Notes 32
2.4 Exercises 32
Information and its Measures 35
3.1 Shannon’s Approach 36
3.2 Classical Source Coding 38
3.3 von Neumann Entropy 44
3.4 Quantum Relative Entropy 47
3.5 R ´ enyi Entropy 55
3.6 Notes 59
3.7 Exercises 60
Entanglement 62
4.1 Bipartite Systems 62
4.2 Dense Coding and Teleportation 72
4.3 Entanglement Measures 76
4.4 Notes 78
4.5 Exercises 79
More About Information Quantities 81
5.1 Shannon’s Mutual Information 81
5.2 Markov Chains 82
5.3 Entropy of Partied Systems 84
5.4 Strong Subadditivity of the von Neumann Entropy 86
5.5 The Holevo Quantity 87
5.6 The Entropy Exchange 88
5.7 Notes 89
5.8 Exercises 90
Quantum Compression 91
6.1 Distances Between States 91
6.2 Reliable Compression 93
6.3 Universality 96
6.4 Notes 98
6.5 Exercises 98
Channels and Their Capacity 99
7.1 Information Channels 99
7.2 The Shannon Capacity 100
7.3 Holevo Capacity 103
7.4 Classical-quantum Channels 112
7.5 Entanglement-assisted Capacity 113
7.6 Notes 114
7.7 Exercises 114
Hypothesis Testing 116
8.1 The Quantum Stein Lemma 117
8.2 The Quantum Chernoff Bound 123
8.3 Notes 126
8.4 Exercises 127
Coarse-grainings 128
9.1 Basic Examples 128
9.2 Conditional Expectations 130
9.3 Commuting Squares 138
9.4 Superadditivity 140
9.5 Sufficiency 140
9.6 Markov States 145
9.7 Notes 148
9.8 Exercises 149
State Estimation 150
10.1 Estimation Schemas 150
10.2 Cram ´ er–Rao Inequalities 157
10.3 Quantum Fisher Information 161
10.4 Contrast Functionals 169
10.5 Notes 170
10.6 Exercises 171
Appendix: Auxiliary Linear and Convex Analysis 172
11.1 Hilbert Spaces and Their Operators 172
11.2 Positive Operators and Matrices 174
11.3 Functional Calculus for Matrices 177
11.4 Distances 182
11.5 Majorization 184
11.6 Operator Monotone Functions 187
11.7 Positive Mappings 196
11.8 Matrix Algebras 202
11.9 Conjugate Convex Function 205
11.10 Some Trace Inequalities 206
11.11 Notes 207
11.12 Exercises 207
Bibliography 211
Index 216