Quantum Information Theory and Quantum Statistics

by Petz, Denes

• ISBN13: 9783540746348
• ISBN10: 354074634X
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-01-01
• Publisher: SPRINGER

Summary

This concise and readable book addresses primarily readers with a background in classical statistical physics and introduces quantum mechanical notions as required. Conceived as a primer to bridge the gap between statistical physics and quantum information, it emphasizes concepts and thorough discussions of the fundamental notions and prepares the reader for deeper studies, not least through a selection of well chosen exercises.

Table of Contents

Introduction	p. 1
Prerequisites from Quantum Mechanics	p. 3
Postulates of Quantum Mechanics	p. 4
State Transformations	p. 14
Notes	p. 22
Exercises	p. 22
Information and its Measures	p. 25
Shannon's Approach	p. 26
Classical Source Coding	p. 28
von Neumann Entropy	p. 34
Quantum Relative Entropy	p. 37
Renyi Entropy	p. 45
Notes	p. 49
Exercises	p. 50
Entanglement	p. 53
Bipartite Systems	p. 53
Dense Coding and Teleportation	p. 63
Entanglement Measures	p. 67
Notes	p. 69
Exercises	p. 70
More About Information Quantities	p. 73
Shannon's Mutual Information	p. 73
Markov Chains	p. 74
Entropy of Partied Systems	p. 76
Strong Subadditivity of the von Neumann Entropy	p. 78
The Holevo Quantity	p. 79
The Entropy Exchange	p. 80
Notes	p. 81
Exercises	p. 82
Quantum Compression	p. 83
Distances Between States	p. 83
Reliable Compression	p. 85
Universality	p. 88
Notes	p. 90
Exercises	p. 90
Channels and Their Capacity	p. 91
Information Channels	p. 91
The Shannon Capacity	p. 92
Holevo Capacity	p. 95
Classical-quantum Channels	p. 104
Entanglement-assisted Capacity	p. 105
Notes	p. 106
Exercises	p. 106
Hypothesis Testing	p. 109
The Quantum Stein Lemma	p. 110
The Quantum Chernoff Bound	p. 116
Notes	p. 119
Exercises	p. 120
Coarse-grainings	p. 121
Basic Examples	p. 121
Conditional Expectations	p. 123
Commuting Squares	p. 131
Superadditivity	p. 133
Sufficiency	p. 133
Markov States	p. 138
Notes	p. 141
Exercises	p. 142
State Estimation	p. 143
Estimation Schemas	p. 143
Cramer-Rao Inequalities	p. 150
Quantum Fisher Information	p. 154
Contrast Functionals	p. 162
Notes	p. 163
Exercises	p. 164
Appendix: Auxiliary Linear and Convex Analysis	p. 165
Hilbert Spaces and Their Operators	p. 165
Positive Operators and Matrices	p. 167
Functional Calculus for Matrices	p. 170
Distances	p. 175
Majorization	p. 177
Operator Monotone Functions	p. 180
Positive Mappings	p. 189
Matrix Algebras	p. 195
Conjugate Convex Function	p. 198
Some Trace Inequalities	p. 199
Notes	p. 200
Exercises	p. 200
Bibliography	p. 205
Index	p. 211
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