Fundamentals of Quantum Information

This book provides an introduction to the basic ideas and concepts of quantum computation and information for both undergraduate and graduate students. The book starts with the quantum bits and the entangled states which turn out to bring revolutionary ideas in information theory. This book is self-contained and unified in its description of the cross-disciplinary nature of this field. The book aims to provide intuitive and transparent ideas of the subjects, and is not strictly mathematical. Quantum mechanics and mathematical tools (especially, number theory) are explained with many examples and illustrations. The students can obtain practical problem-solving ability by solving the exercises at the end of each chapter. Detailed solutions to all problems are provided at the end of the book.

Preface	p. v
Introduction	p. 1
Mathematical Realization of Quantum Mechanics	p. 7
State vector	p. 7
Operator	p. 11
Time evolution of wave function	p. 13
Eigenvalue and eigenstate of Hamiltonian	p. 14
Problems	p. 17
Basic Concepts of Quantum Mechanics	p. 19
Wave function and probability interpretation	p. 19
Commutation relation and uncertainty relation	p. 20
Quantum states of spin ½ particles	p. 23
Quantum bit	p. 28
Angular momentum, spin and rotation	p. 29
Problems	p. 35
EPR Pair and Measurement	p. 37
EPR pair	p. 37
Transmission of quantum states	p. 39
Einstein's locality principle in quantum mechanics	p. 41
Measurement of two correlated particles and hidden variable theorem	p. 46
CHSH inequality	p. 46
Classical correlation vs. quantal correlation: a case of nuclear fission	p. 51
EPR experiment by photon pairs	p. 54
Four-photon GHZ states	p. 59
Problems	p. 63
Classical Computers	p. 65
Turing machine	p. 65
von Neumann computer	p. 70
Possibility and complexity of computation	p. 71
Arithmetic operations	p. 72
Factorization and decision of prime numbers	p. 73
Combination problem	p. 74
Computational complexity and amount of computation	p. 78
Logic gates	p. 78
Logic circuits	p. 80
Sequential circuit and memory	p. 83
Problem	p. 88
Quantum Gates	p. 89
Quantum gates	p. 90
Controlled-operation gates	p. 95
Quantum Turing machine	p. 101
Quantum Fourier transformation (3 bit case)	p. 102
Problems	p. 105
Theory of Information and Communication	p. 107
Shannon entropy	p. 107
Amount of information	p. 107
Entropy	p. 108
Coding information	p. 110
Von Neumann entropy	p. 113
Transmission error in communication	p. 113
Errors and channel capacity	p. 113
Shannon's theorem	p. 116
Problem	p. 119
Quantum Computation	p. 121
Quantum bit and quantum register	p. 122
Deutsch-Jozsa algorithm	p. 125
Shor's factorization algorithm	p. 127
Quantum Fourier transformation with n bits	p. 130
Quantum phase estimation and order finding algorithm	p. 133
Calculation of modular exponentials	p. 141
Problems	p. 143
Quantum Cryptography	p. 145
Secret key cryptography	p. 145
"One-time pad" cryptography	p. 147
Public key cryptography	p. 148
Quantum key distribution	p. 153
Non-cloning theorem	p. 154
BB84 protocol	p. 155
B92 protocol	p. 161
E91 protocol	p. 162
Problems	p. 167
Quantum Search Algorithm	p. 169
Oracle function	p. 170
Quantum oracle	p. 170
Problem	p. 180
Physical Devices of Quantum Computers	p. 181
Nuclear magnetic resonance (NMR) computers	p. 182
Principles of NMR computers	p. 182
NMR and spin rotations	p. 184
Statistical treatment	p. 188
Experiment of quantum factorization	p. 191
Ion trap computers	p. 196
Basic principles	p. 196
Ion trap	p. 196
Computation	p. 203
Preparation of the initial states	p. 204
Reading results	p. 205
Example of quantum gate	p. 206
Quantum dot computer	p. 208
Principle	p. 208
Problem	p. 215
Basic Theory of Numbers	p. 217
Congruence	p. 217
Euler's theorem (Fermat's little theorem)	p. 221
Euclid's algorithm	p. 224
Diophantus equation (Indeterminate equation)	p. 225
Chinese remainder theorem	p. 227
Continued fractions	p. 230
Solutions of Problems	p. 235
Bibliography	p. 249
Index	p. 251
Table of Contents provided by Ingram. All Rights Reserved.

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Fundamentals of Quantum Information

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