Introduction to Quantum Information Science

This book offers a concise and up-to-date introduction to the popular field of quantum information. It has originated in a series of invited lecture courses at various universities in different countries. This is reflected in its informal style of exposition and presentation of key results inthe subject. In addition to treating quantum communication, entanglement and algorithms in great depth, this book also addresses a number of interesting miscellaneous topics, such as Maxwell's demon, Landauer's erasure, the Bekenstein bound and Caratheodory's treatment of the Second law ofthermodyanmics. All mathematical derivations are based on clear physical pictures which make even the most involved results - such as the Holevo bound - look comprehensible and transparent. The book is ideal as a first introduction to the subject, but may also appeal to the specialist due to itsunique presentation.

Vlatko Vedral
Centenary Professor of Quantum Information
School of Physics and Astronomy
University of Leeds
Leeds LS2 9JT
Vlatko Vedral studied his undergraduate degree and PhD at Imperial College (1992-1998). After graduating from his PhD in 1998, he took up a junior research fellowship at Merton College in Oxford where he stayed for two years (1998-2000). He returned to Imperial College in 2000 as a governors' lecturer and was promoted to reader in 2003. In October 2004 he moved to Leeds University as the centenary professor of Quantum Information Science. He has taught at many different universities and held visiting professorships at Oxford, Vienna, Singapore and Perimeter Institute in Canada.
Vlatko Vedral is an active researcher in quantum information and quantum mechanics, having published over 100 papers in these fields. He enjoys explaining science to the media and has been interviewed on a number of occasions regarding his work and the state of the field. He has contributed to several introductory books on quantum computing as well as written a textbook on Quantum Optics.

Classical and 3d Quantum Information
Classical information	p. 3
Information and physics	p. 3
Quantifying information	p. 4
Data compression	p. 7
Related measures of information	p. 8
Relative entropy	p. 9
Joint entropy	p. 10
Conditional entropy	p. 10
Mutual information	p. 10
Capacity of a noisy channel	p. 11
Summary	p. 12
Quantum mechanics	p. 14
Dirac notation	p. 14
The qubit, higher dimensions, and the inner product	p. 16
Hilbert spaces	p. 17
Projective measurements and operations	p. 19
Unitary operations	p. 20
Eigenvectors and eigenvalues	p. 21
Spectral decomposition	p. 22
Applications of the spectral theorem	p. 23
Dirac notation shorthands	p. 24
The Mach-Zehnder interferometer	p. 25
The postulates of quantum mechanics	p. 27
Mixed states	p. 28
Entanglement	p. 29
Summary	p. 30
Quantum information-the basics	p. 31
No cloning of quantum bits	p. 31
Quantum cryptography	p. 33
The trace and partial-trace operations	p. 35
Hilbert space extension	p. 37
The Schmidt decomposition	p. 38
Generalized measurements	p. 40
CP-maps and positive operator-valued measurements	p. 41
The postulates of quantum mechanics revisited	p. 42
Summary	p. 42
Quantum communication with entanglement	p. 44
Pure state entanglement and Pauli matrices	p. 44
Dense coding	p. 45
Teleportation	p. 46
Entanglement swapping	p. 48
No instantaneous transfer of information	p. 49
The extended-Hilbert-space view	p. 50
Summary	p. 50
Quantum information I	p. 52
Fidelity	p. 53
Helstrom's discrimination	p. 54
Quantum data compression	p. 55
Entropy of observation	p. 58
Conditional entropy and mutual information	p. 59
Relative entropy	p. 61
Statistical interpretation of relative entropy	p. 62
Summary	p. 66
Quantum information II	p. 68
Equalities and inequalities related to entropy	p. 68
The Holevo bound	p. 71
Capacity of a bosonic channel	p. 73
Information gained through measurements	p. 75
Relative entropy and thermodynamics	p. 76
Entropy increase due to erasure	p. 77
Landauer's erasure and data compression	p. 78
Summary	p. 78
Quantum Entanglement
Quantum entanglement-introduction	p. 81
The historical background of entanglement	p. 81
Bell's inequalities	p. 83
Separable states	p. 85
Pure states and Bell's inequalities	p. 86
Mixed states and Bell's inequalities	p. 87
Entanglement in second quantization	p. 87
Summary	p. 91
Witnessing quantum entanglement	p. 92
Entanglement witnesses	p. 93
The Jamiolkowski isomorphism	p. 95
The Peres-Horodecki criterion	p. 97
More examples of entanglement witnesses	p. 99
Summary	p. 100
Quantum entanglement in practice	p. 102
Measurements with a Mach-Zehnder interferometer	p. 102
Interferometric implementation of Peres-Horodecki criterion	p. 104
Measuring tr <$$$>²?	p. 104
Generalization to tr <$$$>^k	p. 105
Measuring tr (<$$$>^T₂)^k	p. 106
Measuring the fidelity between <$$$> and ¿	p. 106
Summary	p. 107
Measures of entanglement	p. 108
Distillation of multiple copies of a pure state	p. 108
Analogy with the Carnot Cycle	p. 110
Properties of entanglement measures	p. 111
Entanglement of pure states	p. 113
Entanglement of mixed states	p. 113
Measures of entanglement derived from relative entropy	p. 117
Classical information and entanglement	p. 121
Entanglement and thermodynamics	p. 123
Summary	p. 128
Quantum Computation
Quantum algorithms	p. 131
Computational complexity	p. 131
Deutsch's algorithm	p. 133
Deutsch's algorithm and the Holevo bound	p. 135
Oracles	p. 136
Grover's search algorithm	p. 137
Quantum factorization	p. 140
Factorization	p. 141
The quantum Fourier transform	p. 142
Phase estimation	p. 144
Summary	p. 145
Entanglement, computation and quantum measurements	p. 146
Optimization of searches using entanglement	p. 147
Model for quantum measurement	p. 149
Correlations and quantum measurement	p. 151
The ultimate limits of computation: the Bekenstein bound	p. 157
Summary	p. 158
Quantum error correction	p. 160
Introduction	p. 160
A simple example	p. 160
General conditions	p. 162
Reliable quantum cumputation	p. 165
Quantum error correction considered as a Maxwell's demon	p. 167
Pure states	p. 171
Mixed states	p. 172
Summary	p. 173
Outlook	p. 175
Bibliography	p. 179
Index	p. 181
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Introduction to Quantum Information Science

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Table of Contents

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