The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement

Cover......Page 1
About the Book and the Authors......Page 2
THE MATHEMATICAL LANGUAGE OF QUANTUM THEORY: From Uncertainty to Entanglement......Page 4
ISBN 978-0-521-19583-6 Hardback......Page 5
Contents......Page 6
Preface......Page 10
Introduction......Page 12
1.1.1 Finite- and infinite-dimensional Hilbert spaces......Page 14
1.1.2 Basis expansion......Page 20
1.1.3 Example: L^2(\\Omega)......Page 22
1.2.1 The C*-algebra of bounded operators......Page 24
1.2.2 Partially ordered vector space of selfadjoint operators......Page 30
1.2.3 Orthocomplemented lattice of projections......Page 33
1.2.4 Group of unitary operators......Page 38
1.2.5 Ideal of trace class operators......Page 43
1.3.1 Weak operator topology......Page 47
1.3.2 Dirac notation and rank-1 operators......Page 49
1.3.3 Spectral and singular-value decompositions......Page 51
1.3.4 Linear functionals and dual spaces......Page 53
1.3.5 Tensor product......Page 55
2.1.1 Basic statistical framework......Page 58
2.1.2 State space......Page 63
2.1.3 State space for a finite-dimensional system......Page 73
2.1.4 From states to effects......Page 81
2.1.5 From effects to states......Page 85
2.1.6 Dispersion-free states and Gleason’s theorem......Page 89
2.2.1 Superposition of two pure states......Page 94
2.2.2 Interference......Page 96
2.3 Symmetry......Page 99
2.3.1 Unitary and antiunitary transformations......Page 100
2.3.2 State automorphisms......Page 105
2.3.3 Pure state automorphisms and Wigner’s theorem......Page 110
2.4 Composite systems......Page 111
2.4.1 System versus subsystems......Page 112
2.4.2 State purification......Page 116
3.1 Observables as positive operator-valued measures......Page 118
3.1.1 Definition and basic properties of observables......Page 119
3.1.2 Observables and statistical maps......Page 124
3.1.3 Discrete observables......Page 126
3.1.4 Real observables......Page 128
3.1.5 Mixtures of observables......Page 129
3.1.6 Coexistence of effects......Page 134
3.2.1 Projection-valued measures......Page 139
3.2.2 Sharp observables and selfadjoint operators......Page 142
3.2.3 Complementary observables......Page 147
3.3.1 Informational completeness......Page 151
3.3.2 Symmetric informationally complete observables......Page 157
3.3.3 State estimation......Page 159
3.4 Testing quantum systems......Page 161
3.4.1 Complete versus incomplete information......Page 162
3.4.2 Unambiguous discrimination of states......Page 163
3.4.3 How distinct are two states?......Page 172
3.5.1 State distinction and state determination......Page 175
3.5.2 Coarse-graining......Page 177
3.6.1 Single-mode electromagnetic field......Page 181
3.6.2 Nonideal photon-counting observables......Page 182
4.1 Transforming quantum systems......Page 186
4.1.1 Operations and complete positivity......Page 187
4.1.2 Schrödinger versus Heisenberg picture......Page 192
4.2.1 Isolated versus open systems......Page 194
4.2.2 Stinespring’s dilation theorem......Page 198
4.2.3 Operator-sum form of channels......Page 201
4.3.1 Mixtures of channels......Page 204
4.3.2 Concatenating channels......Page 207
4.3.3 Disturbance and noise......Page 211
4.3.4 Conjugate channels......Page 214
4.4 Parametrizations of quantum channels......Page 216
4.4.1 Matrix representation......Page 217
4.4.2 The \\Chi-matrix representation......Page 218
4.4.3 Choi–Jamiolkowski isomorphism......Page 220
4.5.1 Strictly contractive channels......Page 223
4.5.2 Random unitary channels......Page 225
4.5.3 Phase-damping channels......Page 227
4.6 Example: qubit channels......Page 229
5.1 Three levels of description of measurements......Page 235
5.1.1 Measurement models......Page 236
5.1.2 Instruments......Page 239
5.1.3 Compatibility of the three descriptions......Page 241
5.2.1 Conditional output states......Page 243
5.2.2 No information without disturbance......Page 245
5.2.3 Disturbance in a rank-1 measurement......Page 248
5.2.4 Example: BB84 quantum key distribution......Page 249
5.3.1 Von Neumann’s measurement model......Page 254
5.3.2 Lüders instrument for a discrete observable......Page 256
5.3.3 Lüders’ theorem......Page 257
5.3.4 Example: mean king’s problem......Page 258
5.4 Repeatable measurements......Page 260
5.4.1 Repeatability......Page 261
5.4.2 Wigner–Araki–Yanase theorem......Page 263
5.4.3 Approximate repeatability......Page 265
5.5.1 Programming of observables and channels......Page 267
5.5.2 Universal processor for channels......Page 268
5.5.3 Probabilistic programming......Page 271
6.1 Entangled bipartite systems......Page 274
6.1.1 Entangled vectors......Page 275
6.1.2 Entangled positive operators......Page 280
6.1.3 Nonlocal channels......Page 283
6.2.1 LOCC ordering and separable states......Page 290
6.2.2 Maximally entangled states......Page 292
6.2.3 Majorization criterion for LOCC......Page 298
6.3 Entanglement detection......Page 299
6.3.1 Entanglement witnesses......Page 300
6.3.2 Quantum nonlocal realism......Page 305
6.3.3 Positive but not completely positive maps......Page 308
6.3.4 Negative partial transpose (NPT) criterion......Page 312
6.3.5 Range criterion......Page 313
6.4.1 Entanglement teleportation and distillation......Page 315
6.4.2 Multipartite entanglement......Page 320
6.4.3 Evolution of quantum entanglement......Page 323
6.5 Example: Werner states......Page 326
Sets of channels and of operations......Page 330
References......Page 331
Index......Page 338