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Preface | p. ix |
Convexity, colours and statistics | p. 1 |
Convex sets | p. 1 |
High-dimensional geometry | p. 8 |
Colour theory | p. 13 |
What is 'distance'? | p. 17 |
Probability and statistics | p. 24 |
Geometry of probability distributions | p. 28 |
Majorization and partial order | p. 28 |
Shannon entropy | p. 35 |
Relative entropy | p. 40 |
Continuous distributions and measures | p. 45 |
Statistical geometry and the Fisher-Rao metric | p. 47 |
Classical ensembles | p. 53 |
Generalized entropies | p. 55 |
Much ado about spheres | p. 62 |
Spheres | p. 62 |
Parallel transport and statistical geometry | p. 67 |
Complex, Hermitian and Kahler manifolds | p. 73 |
Symplectic manifolds | p. 79 |
The Hopf fibration of the 3-sphere | p. 81 |
Fibre bundles and their connections | p. 87 |
The 3-sphere as a group | p. 93 |
Cosets and all that | p. 98 |
Complex projective spaces | p. 102 |
From art to mathematics | p. 102 |
Complex projective geometry | p. 106 |
Complex curves, quadrics and the Segre embedding | p. 109 |
Stars, spinors and complex curves | p. 112 |
The Fubini-Study metric | p. 114 |
CP[superscript n] illustrated | p. 120 |
Symplectic geometry and the Fubini-Study measure | p. 127 |
Fibre bundle aspects | p. 128 |
Grassmannians and flag manifolds | p. 131 |
Outline of quantum mechanics | p. 135 |
Quantum mechanics | p. 135 |
Qubits and Bloch spheres | p. 137 |
The statistical and the Fubini-Study distances | p. 140 |
A real look at quantum dynamics | p. 143 |
Time reversals | p. 147 |
Classical and quantum states: a unified approach | p. 151 |
Coherent states and group actions | p. 156 |
Canonical coherent states | p. 156 |
Quasi-probability distributions on the plane | p. 161 |
Bloch coherent states | p. 169 |
From complex curves to SU (K) coherent states | p. 174 |
SU (3) coherent states | p. 177 |
The stellar representation | p. 182 |
The stellar representation in quantum mechanics | p. 182 |
Orbits and coherent states | p. 184 |
The Husimi function | p. 187 |
Wehrl entropy and the Lieb conjecture | p. 192 |
Generalized Wehrl entropies | p. 195 |
Random pure states | p. 197 |
From the transport problem to the Monge distance | p. 203 |
The space of density matrices | p. 209 |
Hilbert-Schmidt space and positive operators | p. 209 |
The set of mixed states | p. 213 |
Unitary transformations | p. 216 |
The space of density matrices as a convex set | p. 219 |
Stratification | p. 224 |
An algebraic afterthought | p. 229 |
Summary | p. 231 |
Purification of mixed quantum states | p. 233 |
Tensor products and state reduction | p. 234 |
The Schmidt decomposition | p. 236 |
State purification and the Hilbert-Schmidt bundle | p. 239 |
A first look at the Bures metric | p. 242 |
Bures geometry for N = 2 | p. 245 |
Further properties of the Bures metric | p. 247 |
Quantum operations | p. 251 |
Measurements and POVMs | p. 251 |
Positive and completely positive maps | p. 262 |
Environmental representations | p. 268 |
Some spectral properties | p. 270 |
Unital and bistochastic maps | p. 272 |
One qubit maps | p. 275 |
Duality: maps versus states | p. 281 |
Positive and decomposable maps | p. 281 |
Dual cones and super-positive maps | p. 288 |
Jamiolkowski isomorphism | p. 290 |
Quantum maps and quantum states | p. 292 |
Density matrices and entropies | p. 297 |
Ordering operators | p. 297 |
Von Neumann entropy | p. 301 |
Quantum relative entropy | p. 307 |
Other entropies | p. 311 |
Majorization of density matrices | p. 313 |
Entropy dynamics | p. 318 |
Distinguishability measures | p. 323 |
Classical distinguishability measures | p. 323 |
Quantum distinguishability measures | p. 328 |
Fidelity and statistical distance | p. 333 |
Monotone metrics and measures | p. 339 |
Monotone metrics | p. 339 |
Product measures and flag manifolds | p. 344 |
Hilbert-Schmidt measure | p. 347 |
Bures measure | p. 350 |
Induced measures | p. 351 |
Random density matrices | p. 354 |
Random operations | p. 358 |
Quantum entanglement | p. 363 |
Introducing entanglement | p. 363 |
Two qubit pure states: entanglement illustrated | p. 367 |
Pure states of a bipartite system | p. 371 |
Mixed states and separability | p. 380 |
Geometry of the set of separable states | p. 389 |
Entanglement measures | p. 394 |
Two-qubit mixed states | p. 404 |
Epilogue | p. 415 |
Basic notions of differential geometry | p. 417 |
Differential forms | p. 417 |
Riemannian curvature | p. 418 |
A key fact about mappings | p. 419 |
Basic notions of group theory | p. 421 |
Lie groups and Lie algebras | p. 421 |
SU(2) | p. 422 |
SU(N) | p. 422 |
Homomorphisms between low-dimensional groups | p. 423 |
Geometry: do it yourself | p. 424 |
Hints and answers to the exercises | p. 428 |
References | p. 437 |
Index | p. 462 |
Table of Contents provided by Ingram. All Rights Reserved. |
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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 Schrdinger, 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.
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