A COMPREHENSIVE GUIDE TO QUANTUM MECHANICS
1 Invitation to Quantum Information Science 1 11 From Classical Information Science to Quantum Information Science 1 12 Further Expansion of Quantum Information Science 4 13 Feedback from Quantum Information Science to Physics 6 14 Toward Realization of Quantum Information Processing 8 15 Organization of This Book 9 References 12 2 Quantum Mechanics for Qubit Systems 13 21 Preliminary 13 22 Preparation 15 221 Conceptual Preparation: Physical System, State, Measurement of Physical Quantity 15 222 Notational Preparation: Dirac Notation 16 23 Qubit Systems 20 231 A Qubit System 21 232 Time Evolution in Qubit System 24 233 Composition of Qubit Systems: n-Qubit Systems 28 References 34 3 Foundations on Quantum Computing 37 31 What is Computation? 37 32 Mathematical Notation for Information Science 40 33 Classical Circuit Model 41 34 Quantum Circuit Model 45 References 53 4 Quantum Algorithms 55 41 Introduction 55 42 Deutsch–Jozsa Algorithm 56 43 Grover’s Algorithm 59 431 Construction of Grover’s Algorithm 59 432 Analysis of Success Probability 62 433 Generalization: Multiple Solutions 64 44 Shor’s Algorithm 66 441 Quantum Algorithm for Period Finding 66 442 Quantum Algorithm for Factorization 70 443 Quantum Algorithm for Discrete Logarithm 72 45 Other Quantum Algorithms 73 References 73 5 Foundations of Quantum Mechanics and Quantum Information Theory 75 51 Introduction 75 511 Postulates and Preconditions 76 512 Hilbert Space and Linear Operators 77 513 Dirac Notation II 78 52 Postulates for Quantum Mechanics 80 521 Quantum States, and Measurements of Physical Quantities 81 522 Time Evolution 88 523 Composite Systems 89 524 Comment on the Measurement Process: State-Changes due to Measurements 91 53 Reformulation of Quantum Mechanics 93 531 General Class of Quantum States 93 532 General Class of Measurements 107 533 General Class of Time Evolutions 115 534 General Class of Measurement Processes 119 References 125 6 Information Quantities in Quantum Systems 127 61 Introduction 127 62 Information Quantities in Classical Systems 128 621 Shannon Entropy 128 622 Entropy and Typical Sequence 130 623 Joint Entropy and Conditional Entropy 134 624 Conditional Probability and Classical Channel 135 625 Divergence 136 626 f -Divergence 138 627 Mutual Information 141 628 Concavity and Subadditivity of the Entropy 143 629 Fano Inequality 144 63 Information Quantities in Quantum Systems 145 631 Von Neumann Entropy 145 632 Quantum Relative Entropy 147 633 Mutual Information in Quantum Systems 149 634 Concavity and Subadditivity of the von Neumann Entropy 152 635 Trace Distance 153 636 Fidelity and Uhlmann’s Theorem 155 637 Properties of Fidelity 158 638 Entanglement Fidelity 162 References 165 7 Quantum Entanglement 167 71 Introduction 167 72 Basic Concepts of Entanglement 168 721 Quantum and Classical Correlation 168 722 Product State and Maximally Entangled State 169 723 Quantum Teleportation 171 724 Superdense Coding 173 73 Quantifying Entanglement 174 731 Local Operations and Classical Communication (LOCC) 174 732 Basic Unit of Entanglement 176 733 Entanglement Concentration 177 734 Quantum Data Compression 180 735 Entanglement Dilution 183 736 Amount of Entanglement 185 74 Multipartite Entanglement 186 741 GHZ and W State 186 742 Stochastic LOCC 187 743 Classification of Multipartite Entanglement 188 75 Mixed-State Entanglement 190 751 Entanglement Criteria 190 752 LOCC on Mixed States 194 753 Entanglement Measure 196 754 Entanglement of Formation 197 755 Relative Entropy of Entanglement 199 756 Relationship among Entanglement Measures 200 757 Entanglement Monotone 202 References 202 8 Classical-Quantum Channel Coding 205 81 Introduction 205 82 Quantum Hypothesis Testing 206 821 Problem of Quantum Hypothesis Testing 206 822 Trace Inequality for Quantum Hypothesis Testing 208 823 Asymptotic Theory of Quantum Hypothesis Testing 210 824 Properties of Relative Rényi Entropy 213 83 Classical-Quantum Channel Coding 216 831 Message Transmission Over Quantum Channels 216 832 Proof of the C-Q Channel Coding Theorem (Converse Part) 219 833 Proof of the C-Q Channel Coding Theorem (Direct Part) 223 References 228 9 Quantum Error Correction and Quantum Cryptography 231 91 Introduction 231 92 Algebraic Error Correction in the Classical System 232 921 Formulation 232 922 Evaluation of Error Probability Under Code Ensemble 236 923 Examples of Code Ensemble 239 924 Asymptotic Theory 241 925 Error Correction with Confidentiality 242 93 Quantum Error Correcting Code 246 931 Pauli Channel 246 932 Stabilizer Code 247 933 CSS Code 250 934 Asymptotic Theory 254 94 Application to Quantum Secret Communication 255 941 Channel to the Environment System 255 942 Leaked Information Without Privacy Amplification 256 943 Leaked Information with Privacy Amplification 260 95 Application to Quantum Cryptography (Quantum Key Distribution) 264 References 268