Contents v
5.3.1 Classical source coding 284
5.3.2 Quantum source coding 289
5.3.3 Encoding classical information into quantum
states 294
5.4 Exercises 306
5.5 Bibliographic remarks 308
6 Bipartite entanglement 310
6.1 Separability 310
6.1.1 Separable operators and states 310
6.1.2 Separable maps and the LOCC paradigm 324
6.1.3 Separable and LOCC measurements 332
6.2 Manipulation of entanglement 339
6.2.1 Entanglement transformation 339
6.2.2 Distillable entanglement and entanglement cost 345
6.2.3 Bound entanglement and partial transposition 352
6.3 Phenomena associated with entanglement 358
6.3.1 Teleportation and dense coding 359
6.3.2 Non-classical correlations 371
6.4 Exercises 384
6.5 Bibliographic remarks 386
7 Permutation invariance and unitarily invariant measures 390
7.1 Permutation-invariant vectors and operators 390
7.1.1 The subspace of permutation-invariant vectors 391
7.1.2 The algebra of permutation-invariant operators 400
7.2 Unitarily invariant probability measures 408
7.2.1 Uniform spherical measure and Haar measure 408
7.2.2 Applications of unitarily invariant measures 420
7.3 Measure concentration and it applications 429
7.3.1 L´evy’s lemma and Dvoretzky’s theorem 430
7.3.2 Applications of measure concentration 447
7.4 Exercises 460
7.5 Bibliographic remarks 462
8 Quantum channel capacities 464
8.1 Classical information over quantum channels 464
8.1.1 Classical capacities of quantum channels 465
8.1.2 The Holevo–Schumacher–Westmoreland theorem 476
8.1.3 The entanglement-assisted classical capacity
theorem 493
8.2 Quantum information over quantum channels 512
