Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schr?dinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
Author(s): Ingemar Bengtsson, Karol Zyczkowski
Edition: 1
Publisher: Cambridge University Press
Year: 2006
Language: English
Pages: 466
Cover......Page 1
GEOMETRY OF QUANTUM STATES: An Introduction to Quantum Entanglement......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1.1 Convex sets ......Page 14
1.2 High-dimensional geometry ......Page 21
1.3 Colour theory ......Page 26
1.4 What is ‘distance’? ......Page 30
1.5 Probability and statistics ......Page 37
2.1 Majorization and partial order ......Page 41
2.2 Shannon entropy ......Page 48
2.3 Relative entropy ......Page 53
2.4 Continuous distributions and measures ......Page 58
2.5 Statistical geometry and the Fisher–Rao metric ......Page 60
2.6 Classical ensembles ......Page 66
2.7 Generalized entropies ......Page 68
3.1 Spheres ......Page 75
3.2 Parallel transport and statistical geometry ......Page 80
3.3 Complex, Hermitian and Kahler manifolds......Page 86
3.4 Symplectic manifolds ......Page 92
3.5 The Hopf fibration of the 3-sphere ......Page 94
3.6 Fibre bundles and their connections ......Page 100
3.7 The 3-sphere as a group ......Page 106
3.8 Cosets and all that ......Page 111
4.1 From art to mathematics ......Page 115
4.2 Complex projective geometry ......Page 119
4.3 Complex curves, quadrics and the Segre embedding ......Page 122
4.4 Stars, spinors and complex curves ......Page 125
4.5 The Fubini–Study metric ......Page 127
4.6 CP^n illustrated......Page 133
4.7 Symplectic geometry and the Fubini–Study measure ......Page 140
4.8 Fibre bundle aspects ......Page 141
4.9 Grassmannians and flag manifolds ......Page 144
5.1 Quantum mechanics ......Page 148
5.2 Qubits and Bloch spheres ......Page 150
5.3 The statistical and the Fubini–Study distances ......Page 153
5.4 A real look at quantum dynamics ......Page 156
5.5 Time reversals ......Page 160
5.6 Classical and quantum states: a unified approach ......Page 164
6.1 Canonical coherent states ......Page 169
6.2 Quasi-probability distributions on the plane ......Page 174
6.3 Bloch coherent states ......Page 182
6.4 From complex curves to SU(K) coherent states ......Page 187
6.5 SU(3) coherent states ......Page 190
7.1 The stellar representation in quantum mechanics ......Page 195
7.2 Orbits and coherent states ......Page 197
7.3 The Husimi function ......Page 200
7.4 Wehrl entropy and the Lieb conjecture ......Page 205
7.5 Generalized Wehrl entropies ......Page 208
7.6 Random pure states ......Page 210
7.7 From the transport problem to the Monge distance ......Page 216
8.1 Hilbert–Schmidt space and positive operators ......Page 222
8.2 The set of mixed states ......Page 226
8.3 Unitary transformations ......Page 229
8.4 The space of density matrices as a convex set ......Page 232
8.5 Stratification ......Page 237
8.6 An algebraic afterthought ......Page 242
8.7 Summary ......Page 244
9 Purification of mixed quantum states ......Page 246
9.1 Tensor products and state reduction ......Page 247
9.2 The Schmidt decomposition ......Page 249
9.3 State purification and the Hilbert–Schmidt bundle ......Page 252
9.4 A first look at the Bures metric ......Page 255
9.5 Bures geometry for N = 2 ......Page 258
9.6 Further properties of the Bures metric ......Page 260
10.1 Measurements and POVMs ......Page 264
10.3 Positive and completely positive maps ......Page 275
10.4 Environmental representations ......Page 281
10.5 Some spectral properties ......Page 283
10.6 Unital and bistochastic maps ......Page 285
10.7 One qubit maps ......Page 288
11.1 Positive and decomposable maps ......Page 294
11.2 Dual cones and super-positive maps ......Page 301
11.3 Jamiolkowski isomorphism......Page 303
11.4 Quantum maps and quantum states ......Page 305
12.1 Ordering operators ......Page 310
12.2 Von Neumann entropy ......Page 314
12.3 Quantum relative entropy ......Page 320
12.4 Other entropies ......Page 324
12.5 Majorization of density matrices ......Page 326
12.6 Entropy dynamics ......Page 331
13.1 Classical distinguishability measures ......Page 336
13.2 Quantum distinguishability measures ......Page 341
13.3 Fidelity and statistical distance ......Page 346
14.1 Monotone metrics ......Page 352
14.2 Product measures and flag manifolds ......Page 357
14.3 Hilbert–Schmidt measure ......Page 360
14.4 Bures measure ......Page 363
14.5 Induced measures ......Page 364
14.6 Random density matrices ......Page 367
14.7 Random operations ......Page 371
15.1 Introducing entanglement ......Page 376
15.2 Two qubit pure states: entanglement illustrated ......Page 380
15.3 Pure states of a bipartite system ......Page 384
15.4 Mixed states and separability ......Page 393
15.5 Geometry of the set of separable states ......Page 402
15.6 Entanglement measures ......Page 407
15.7 Two-qubit mixed states ......Page 417
Epilogue ......Page 428
A1.1 Differential forms ......Page 430
A1.2 Riemannian curvature ......Page 431
A1.3 A key fact about mappings ......Page 432
A2.1 Lie groups and Lie algebras ......Page 434
A2.3 SU(N) ......Page 435
A2.4 Homomorphisms between low-dimensional groups ......Page 436
Appendix 3 Geometry: do it yourself ......Page 437
Appendix 4 Hints and answers to the exercises ......Page 441
References ......Page 450
Index ......Page 475
