Alice and Bob Meet Banach. The Interface of Asymptotic Geometric Analysis and Quantum Information Theory

Cover......Page 1
Title page......Page 4
Dedication......Page 6
Contents......Page 8
List of Tables......Page 14
List of Figures......Page 16
Preface......Page 20
Part  1 .  Alice and Bob Mathematical Aspects   of Quantum Information Theory......Page 24
0.2. Euclidean and Hilbert spaces......Page 26
0.3. Bra-ket notation......Page 27
0.5. Complexification......Page 29
0.6. Matrices vs. operators......Page 30
0.10. States, classical and quantum......Page 31
1.1.1. Gauges......Page 34
1.1.2. First examples: ell-p-balls, simplices, polytopes, and convex hulls......Page 35
1.1.3. Extreme points, faces......Page 36
1.1.4. Polarity......Page 38
1.1.5. Polarity and the facial structure......Page 40
1.2. Cones......Page 41
1.2.1. Cone duality......Page 42
1.2.2. Nondegenerate cones and facial structure......Page 44
1.3.1. Majorization......Page 45
1.3.2. Schatten norms......Page 46
1.3.3. Von Neumann and Rényi entropies......Page 50
Section 1.3......Page 52
2.1.1. Pure and mixed states......Page 54
2.1.2. The Bloch ball D(C2)......Page 55
2.1.3. Facial structure......Page 56
2.1.4. Symmetries......Page 57
2.2.1. Partial trace......Page 58
2.2.2. Schmidt decomposition......Page 59
2.2.3. A fundamental dichotomy: Separability vs. entanglement......Page 60
2.2.4. Some examples of bipartite states......Page 62
2.2.6. Partial transposition......Page 64
2.2.7. PPT states......Page 66
2.2.8. Local unitaries and symmetries of Sep......Page 69
2.3.1. The Choi and Jamiołkowski isomorphisms......Page 70
2.3.2. Positive and completely positive maps......Page 71
2.3.3. Quantum channels and Stinespring representation......Page 73
2.3.4. Some examples of channels......Page 75
2.4.1. Cones of operators......Page 78
2.4.2. Cones of superoperators......Page 79
2.4.3. Symmetries of the PSD cone......Page 81
2.4.4. Entanglement witnesses......Page 83
2.4.5. Proofs of Størmer’s theorem......Page 85
Section 2.2......Page 86
Section 2.4......Page 87
3.1. Simple-minded quantum mechanics......Page 90
3.3. Composite systems and quantum marginals: Mixed states......Page 91
3.4. The partial trace: Purification of mixed states......Page 93
3.5. Unitary evolution and quantum operations: The completely positive maps......Page 94
3.6. Other measurement schemes......Page 96
3.7. Local operations......Page 97
Notes and Remarks......Page 98
Part  2 .  Banach and His Spaces Asymptotic Geometric Analysis Miscellany......Page 100
4.1.1. Distances between convex sets......Page 102
4.1.2. Symmetrization......Page 103
4.1.3. Zonotopes and zonoids......Page 104
4.1.4. Projective tensor product......Page 105
4.2.1. Definition and characterization......Page 107
4.2.2. Convex bodies with enough symmetries......Page 112
4.3.1. The Brunn–Minkowski inequality......Page 114
4.3.2. log-concave measures......Page 116
4.3.3. Mean width and the Urysohn inequality......Page 117
4.3.5. Symmetrization inequalities......Page 121
4.4. Volume of central sections and the isotropic position......Page 124
Section 4.1......Page 126
Section 4.3......Page 127
Section 4.4......Page 128
5.1.1. Definitions......Page 130
5.1.2. Nets and packings on the Euclidean sphere......Page 131
5.1.3. Nets and packings in the discrete cube......Page 136
5.1.4. Metric entropy for convex bodies......Page 137
5.1.5. Nets in Grassmann manifolds, orthogonal and unitary groups......Page 139
5.2. Concentration of measure......Page 140
5.2.1. A prime example: concentration on the sphere......Page 142
5.2.2. Gaussian concentration......Page 144
5.2.3. Concentration tricks and treats......Page 147
5.2.4. Geometric and analytic methods. Classical examples......Page 152
5.2.5. Some discrete settings......Page 159
5.2.6. Deviation inequalities for sums of independent random variables......Page 162
Section 5.1......Page 165
Section 5.2......Page 166
6.1. Gaussian processes......Page 172
6.1.1. Key example and basic estimates......Page 173
6.1.2. Comparison inequalities for Gaussian processes......Page 175
6.1.3. Sudakov and dual Sudakov inequalities......Page 177
6.1.4. Dudley’s inequality and the generic chaining......Page 180
6.2. Random matrices......Page 183
6.2.1. infinity-Wasserstein distance......Page 184
6.2.2. The Gaussian Unitary Ensemble (GUE)......Page 185
6.2.3. Wishart matrices......Page 189
6.2.4. Real RMT models and Chevet–Gordon inequalities......Page 196
6.2.5. A quick initiation to free probability......Page 199
Section 6.1......Page 201
Section 6.2......Page 202
7.1.1. ell-norm and ell-position......Page 204
7.1.2. K-convexity and the MM*-estimate......Page 205
7.2.1. Dvoretzky’s theorem for Lipschitz functions......Page 209
7.2.2. The Dvoretzky dimension......Page 212
7.2.3. The Figiel–Lindenstrauss–Milman inequality......Page 216
7.2.4. The Dvoretzky dimension of standard spaces......Page 218
7.2.5. Dvoretzky’s theorem for general convex bodies......Page 223
7.2.6. Related results......Page 224
7.2.7. Constructivity......Page 228
Section 7.1......Page 230
Section 7.2......Page 231
Part  3 .  The Meeting: AGA and QIT......Page 234
8.1. Entangled subspaces: Qualitative approach......Page 236
8.2.1. Quantifying entanglement for pure states......Page 238
8.2.3. Minimal output entropy and additivity problems......Page 239
8.2.4. On the 1 to p norm of quantum channels......Page 240
8.3.1. Counterexamples to the multiplicativity problem......Page 241
8.3.2. Almost randomizing channels......Page 243
8.4.1. The basic concentration argument......Page 245
8.4.3. Extremely entangled subspaces......Page 247
8.4.4. Counterexamples to the additivity problem......Page 251
8.5.1. Geometric measure of entanglement......Page 252
8.5.2. The case of many qubits......Page 253
8.5.3. Multipartite entanglement in real Hilbert spaces......Page 254
Section 8.3......Page 255
Section 8.5......Page 256
Chapter 9. Geometry of the set of mixed states......Page 258
9.1.2. The set of all quantum states......Page 259
9.1.3. The set of separable states (the bipartite case)......Page 261
9.1.4. The set of block-positive matrices......Page 263
9.1.5. The set of separable states (multipartite case)......Page 265
9.1.6. The set of PPT states......Page 267
9.2. Distance estimates......Page 268
9.2.1. The Gurvits–Barnum theorem......Page 269
9.2.2. Robustness in the bipartite case......Page 270
9.2.3. Distances involving the set of PPT states......Page 271
9.2.4. Distance estimates in the multipartite case......Page 272
9.3. The super-picture: Classes of maps......Page 273
9.4.1. Approximating the set of all quantum states......Page 275
9.4.2. Approximating the set of separable states......Page 279
9.4.3. Exponentially many entanglement witnesses are necessary......Page 281
Section 9.2......Page 283
Section 9.4......Page 284
10.1.1. Majorization inequalities......Page 286
10.1.2. Spectra and norms of unitarily invariant random matrices......Page 287
10.1.3. Gaussian approximation to induced states......Page 289
10.1.4. Concentration for gauges of induced states......Page 290
10.2.1. Almost sure entanglement for low-dimensional environments......Page 291
10.2.2. The threshold theorem......Page 292
10.3.1. Entanglement of formation......Page 294
Notes and Remarks......Page 295
11.1. Isometrically Euclidean subspaces via Clifford algebras......Page 298
11.2. Local vs. quantum correlations......Page 299
11.2.1. Correlation matrices......Page 300
11.2.2. Bell correlation inequalities and the Grothendieck constant......Page 303
11.3. Boxes and games......Page 306
11.3.1. Bell inequalities as games......Page 307
11.3.2. Boxes and the nonsignaling principle......Page 308
11.3.3. Bell violations......Page 312
Section 11.1......Page 317
Section 11.2......Page 318
Section 11.3......Page 319
12.1.1. Quantum state discrimination......Page 322
12.1.3. Sparsification of POVMs......Page 323
12.2.1. State manipulation via LOCC channels......Page 324
12.2.3. The case of two qubits......Page 325
12.2.4. Some reformulations of distillability......Page 327
Section 12.1......Page 328
Section 12.2......Page 329
A.1. Gaussian random variables......Page 330
A.2. Gaussian vectors......Page 331
Notes and Remarks......Page 332
B.1. The unit sphere S-(n-1) or S-Cd......Page 334
B.3. The orthogonal and unitary groups O(n),U(n)......Page 335
B.4. The Grassmann manifolds Gr(k,Rn),Gr(k,Cn)......Page 337
B.5. The Lorentz group O(1,n-1)......Page 341
Notes and Remarks......Page 342
Appendix C. Extreme maps between Lorentz cones and the S-lemma......Page 344
Notes and Remarks......Page 347
Appendix D. Polarity and the Santaló point via duality of cones......Page 348
Chapter 1......Page 352
Chapter 2......Page 357
Chapter 4......Page 362
Chapter 5......Page 367
Chapter 6......Page 374
Chapter 7......Page 380
Chapter 8......Page 384
Chapter 9......Page 385
Chapter 10......Page 387
Chapter 11......Page 388
Appendix B......Page 393
Appendix D......Page 396
Convex geometry......Page 398
Linear algebra......Page 399
Probability......Page 400
Geometry and asymptotic geometric analysis......Page 401
Quantum information theory......Page 402
Bibliography......Page 404
Websites......Page 431
Index......Page 432
Back Cover......Page 439