Quantum Systems, Channels, Information (eBook)

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Alexander S. Holevo, Steklov Mathematical Institute, Moscow, Russia.

Preface 5
I Basic structures 15
1 Vectors and operators 17
1.1 Hilbert space 17
1.2 Operators 18
1.3 Positivity 19
1.4 Trace and duality 20
1.5 Convexity 22
1.6 Notes and references 23
2 States, observables, statistics 24
2.1 Structure of statistical theories 24
2.1.1 Classical systems 24
2.1.2 Axioms of statistical description 25
2.2 Quantum states 28
2.3 Quantum observables 30
2.3.1 Quantum observables from the axioms 30
2.3.2 Compatibility and complementarity 32
2.3.3 The uncertainty relation 35
2.3.4 Convex structure of observables 36
2.4 Statistical discrimination between quantum states 39
2.4.1 Formulation of the problem 39
2.4.2 Optimal observables 39
2.5 Notes and references 45
3 Composite systems and entanglement 48
3.1 Composite systems 48
3.1.1 Tensor products 48
3.1.2 Naimark’s dilation 50
3.1.3 Schmidt decomposition and purification 52
3.2 Quantum entanglement vs “local realism” 55
3.2.1 Paradox of Einstein-Podolski-Rosen and Bell’s inequalities 55
3.2.2 Mermin-Peres game 59
3.3 Quantum systems as information carriers 61
3.3.1 Transmission of classical information 61
3.3.2 Entanglement and local operations 62
3.3.3 Superdense coding 63
3.3.4 Quantum teleportation 64
3.4 Notes and references 66
II The primary coding theorems 69
4 Classical entropy and information 71
4.1 Entropy of a random variable and data compression 71
4.2 Conditional entropy and the Shannon information 73
4.3 The Shannon capacity of the classical noisy channel 76
4.4 The channel coding theorem 78
4.5 Wiretap channel 83
4.6 Gaussian channel 85
4.7 Notes and references 86
5 The classical-quantum channel 88
5.1 Codes and achievable rates 88
5.2 Formulation of the coding theorem 89
5.3 The upper bound 92
5.4 Proof of the weak converse 97
5.5 Typical projectors 101
5.6 Proof of the Direct Coding Theorem 106
5.7 The reliability function for pure-state channel 109
5.8 Notes and references 112
III Channels and entropies 115
6 Quantum evolutions and channels 117
6.1 Quantum evolutions 117
6.2 Completely positive maps 120
6.3 Definition of the channel 126
6.4 Entanglement-breaking and PPT channels 128
6.5 Quantum measurement processes 131
6.6 Complementary channels 133
6.7 Covariant channels 138
6.8 Qubit channels 141
6.9 Notes and references 143
7 Quantum entropy and information quantities 146
7.1 Quantum relative entropy 146
7.2 Monotonicity of the relative entropy 147
7.3 Strong subadditivity of the quantum entropy 152
7.4 Continuity properties 154
7.5 Information correlation, entanglement of formation and conditional entropy 156
7.6 Entropy exchange 161
7.7 Quantum mutual information 163
7.8 Notes and references 165
IV Basic channel capacities 167
8 The classical capacity of quantum channel 169
8.1 The coding theorem 169
8.2 The . - capacity 171
8.3 The additivity problem 174
8.3.1 The effect of entanglement in encoding and decoding 174
8.3.2 A hierarchy of additivity properties 178
8.3.3 Some entropy inequalities 180
8.3.4 Additivity for complementary channels 183
8.3.5 Nonadditivity of quantum entropy quantities 185
8.4 Notes and references 192
9 Entanglement-assisted classical communication 194
9.1 The gain of entanglement assistance 194
9.2 The classical capacities of quantum observables 198
9.3 Proof of the Converse Coding Theorem 202
9.4 Proof of the Direct Coding Theorem 204
9.5 Notes and references 208
10 Transmission of quantum information 209
10.1 Quantum error-correcting codes 209
10.1.1 Error correction by repetition 209
10.1.2 General formulation 211
10.1.3 Necessary and sufficient conditions for error correction 212
10.1.4 Coherent information and perfect error correction 214
10.2 Fidelities for quantum information 217
10.2.1 Fidelities for pure states 217
10.2.2 Relations between the fidelity measures 219
10.2.3 Fidelity and the Bures distance 222
10.3 The quantum capacity 224
10.3.1 Achievable rates 224
10.3.2 The quantum capacity and the coherent information 229
10.3.3 Degradable channels 231
10.4 The private classical capacity and the quantum capacity 234
10.4.1 The quantum wiretap channel 234
10.4.2 Proof of the Private Capacity Theorem 237
10.4.3 Large deviations for random operators 243
10.4.4 The Direct Coding Theorem for the quantum capacity 246
10.5 Notes and references 251
V Infinite systems 255
11 Channels with constrained inputs 257
11.1 Convergence of density operators 257
11.2 Quantum entropy and relative entropy 261
11.3 Constrained c-q channel 263
11.4 Classical-quantum channel with continuous alphabet 266
11.5 Constrained quantum channel 268
11.6 Entanglement-assisted capacity of constrained channels 271
11.7 Entanglement-breaking channels in infinite dimensions 273
11.8 Notes and references 278
12 Gaussian systems 280
12.1 Preliminary material 280
12.1.1 Spectral decomposition and Stone’s Theorem 280
12.1.2 Operators associated with the Heisenberg commutation relation 283
12.1.3 Classical signal plus quantum noise 286
12.1.4 The classical-quantum Gaussian channel 289
12.2 Canonical commutation relations 290
12.2.1 Weyl-Segal CCR 290
12.2.2 The symplectic space 293
12.2.3 Dynamics, quadratic operators and gauge transformations 295
12.3 Gaussian states 298
12.3.1 Characteristic function 298
12.3.2 Definition and properties of Gaussian states 299
12.3.3 The density operator of Gaussian state 303
12.3.4 Entropy of a Gaussian state 304
12.3.5 Separability and purification 307
12.4 Gaussian channels 310
12.4.1 Open bosonic systems 310
12.4.2 Gaussian channels: basic properties 314
12.4.3 Gaussian observables 315
12.4.4 Gaussian entanglement-breaking channels 317
12.5 The capacities of Gaussian channels 321
12.5.1 Maximization of the mutual information 321
12.5.2 Gauge-covariant channels 322
12.5.3 Maximization of the coherent information 324
12.5.4 The classical capacity: conjectures 325
12.6 The case of one mode 328
12.6.1 Classification of Gaussian channels 328
12.6.2 Entanglement-breaking channels 334
12.6.3 Attenuation/amplification/classical noise channel 335
12.6.4 Estimating the quantum capacity 339
12.7 Notes and references 343
Bibliography 347
Index 360

lt;P>"Written in a mathematically rigorous way, the book gives an accessible and self-contained introduction to quantum theory of information and presents fundamental results in the subject." Mathematical Reviews

"The book is suited for researchers, and lecturers who find the respective matter worked out and well ordered, as well as for students who may check their grasp, since any step is accompanied by task-oriented exercises. […] This book mediates the origins and the development up to the present state of art in a very instructive manner. It can be best recommended." Zentralblatt für Mathematik