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Introduction to Quantum Information Science - Vlatko Vedral
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Contents PART I CLASSICAL AND QUANTUM INFORMATION 1 Classical information 3 1.1 Information and physics 3 1.2 Quantifying information 4 1.3 Data compression 7 1.4 Related measures of information 8 1.4.1 Relative entropy 9 1.4.2 Joint entropy 10 1.4.3 Conditional entropy 10 1.4.4 Mutual information 10 1.5 Capacity of a noisy channel 11 1.6 Summary 12 2 Quantum mechanics 14 2.1 Dirac notation 14 2.2 The qubit, higher dimensions, and the inner product 16 2.3 Hilbert spaces 17 2.4 Projective measurements and operations 19 2.5 Unitary operations 20 2.6 Eigenvectors and eigenvalues 21 2.7 Spectral decomposition 22 2.8 Applications of the spectral theorem 23 2.9 Dirac notation shorthands 24 2.10 The Mach–Zehnder interferometer 25 2.11 The postulates of quantum mechanics 27 2.12 Mixed states 28 2.13 Entanglement 29 2.14 Summary 30 3 Quantum information—the basics 31 3.1 No cloning of quantum bits 31 3.2 Quantum cryptography 33 3.3 The trace and partial-trace operations 35 3.4 Hilbert space extension 37 3.5 The Schmidt decomposition 38 viii CONTENTS 3.6 Generalized measurements 40 3.7 CP-maps and positive operator-valued measurements 41 3.8 The postulates of quantum mechanics revisited 42 3.9 Summary 42 4 Quantum communication with entanglement 44 4.1 Pure state entanglement and Pauli matrices 44 4.2 Dense coding 45 4.3 Teleportation 46 4.4 Entanglement swapping 48 4.5 No instantaneous transfer of information 49 4.6 The extended–Hilbert–space view 50 4.7 Summary 50 5 Quantum information I 52 5.1 Fidelity 53 5.2 Helstrom’s discrimination 54 5.3 Quantum data compression 55 5.4 Entropy of observation 58 5.5 Conditional entropy and mutual information 59 5.6 Relative entropy 61 5.7 Statistical interpretation of relative entropy 62 5.8 Summary 66 6 Quantum information II 68 6.1 Equalities and inequalities related to entropy 68 6.2 The Holevo bound 71 6.3 Capacity of a bosonic channel 73 6.4 Information gained through measurements 75 6.5 Relative entropy and thermodynamics 76 6.6 Entropy increase due to erasure 77 6.7 Landauer’s erasure and data compression 78 6.8 Summary 78 PART II QUANTUM ENTANGLEMENT 7 Quantum entanglement—introduction 81 7.1 The historical background of entanglement 81 7.2 Bell’s inequalities 83 7.3 Separable states 85 7.4 Pure states and Bell’s inequalities 86 7.5 Mixed states and Bell’s inequalities 87 7.6 Entanglement in second quantization 87 7.7 Summary 91 8 Witnessing quantum entanglement 92 8.1 Entanglement witnesses 93 8.2 The Jamiolkowski isomorphism 95 CONTENTS ix 8.3 The Peres–Horodecki criterion 97 8.4 More examples of entanglement witnesses 99 8.5 Summary 100 9 Quantum entanglement in practice 102 9.1 Measurements with a Mach–Zehnder interferometer 102 9.2 Interferometric implementation of Peres–Horodeckicriterion 104 9.2.1 Measuring tr ? 2 ? 104 9.2.2 Generalization to tr ? k 105 9.2.3 Measuring tr (? T 2 ) k 106 9.3 Measuring the fidelity between ? and σ 106 9.4 Summary 107 10 Measures of entanglement 108 10.1 Distillation of multiple copies of a pure state 108 10.2 Analogy with the Carnot Cycle 110 10.3 Properties of entanglement measures 111 10.4 Entanglement of pure states 113 10.5 Entanglement of mixed states 113 10.6 Measures of entanglement derived from relative entropy 117 10.7 Classical information and entanglement 121 10.8 Entanglement and thermodynamics 123 10.9 Summary 128 PART III QUANTUM COMPUTATION 11 Quantum algorithms 131 11.1 Computational complexity 131 11.2 Deutsch’s algorithm 133 11.2.1 Deutsch’s algorithm and the Holevo bound 135 11.3 Oracles 136 11.4 Grover’s search algorithm 137 11.5 Quantum factorization 140 11.5.1 Factorization 141 11.5.2 The quantum Fourier transform 142 11.5.3 Phase estimation 144 11.6 Summary 145 12 Entanglement, computation and quantum measurements 146 12.1 Optimization of searches using entanglement 147 12.2 Model for quantum measurement 149 12.3 Correlations and quantum measurement 151 12.4 The ultimate limits of computation: the Bekenstein bound 157 12.5 Summary 158 13 Quantum error correction 160 13.1 Introduction 160 13.2 A simple example 160 x CONTENTS 13.3 General conditions 162 13.4 Reliable quantum computation 165 13.5 Quantum error correction considered as a Maxwell’s demon 167 13.5.1 Pure states 171 13.5.2 Mixed states 172 13.6 Summary 173 14 Outlook 175 Bibliography 179 Index 181 |
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