Principles of Quantum Computation and Information: Basic Tools and Special Topics

Contents – Volume II......Page 17
Preface......Page 9
5. Quantum Information Theory......Page 22
5.1 The density matrix......Page 23
5.1.1 The density matrix for a qubit......Page 29
5.1.2 Composite systems......Page 32
5.1.3 The quantum copying machine......Page 36
5.2 The Schmidt decomposition......Page 38
5.3 Purification......Page 41
5.4 The Kraus representation......Page 43
5.5 Measurement of the density matrix for a qubit......Page 49
5.6 Generalized measurements......Page 51
5.6.1 Weak measurements......Page 53
5.6.2 POVM measurements......Page 55
5.7 The Shannon entropy......Page 58
5.8.1 Shannon’s noiseless coding theorem......Page 59
5.8.2 Examples of data compression......Page 61
5.9 The von Neumann entropy......Page 62
5.9.1 Example 1: source of orthogonal pure states......Page 64
5.9.2 Example 2: source of non-orthogonal pure states......Page 65
5.10.1 Schumacher’s quantum noiseless coding theorem......Page 68
5.10.2 Compression of an n-qubit message......Page 69
5.10.3 Example 1: two-qubit messages......Page 71
5.10.4 Example 2: three-qubit messages......Page 73
5.11 Accessible information......Page 76
5.11.2 Example 1: two non-orthogonal pure states......Page 78
5.11.3 Example 2: three non-orthogonal pure states......Page 82
5.12 Entanglement concentration and von Neumann entropy......Page 84
5.13 The Peres separability criterion......Page 88
5.14.1 Thermodynamic entropy......Page 90
5.14.2 Statistical entropy......Page 93
5.14.3 Dynamical Kolmogorov–Sinai entropy......Page 95
5.15 A guide to the bibliography......Page 98
6. Decoherence......Page 100
6.1 Decoherence models for a single qubit......Page 101
6.1.1 The quantum black box......Page 102
6.1.2 Measuring a quantum operation acting on a qubit......Page 104
6.1.3 Quantum circuits simulating noise channels......Page 105
6.1.4 The bit-flip channel......Page 108
6.1.5 The phase-flip channel......Page 109
6.1.6 The bit-phase-flip channel......Page 110
6.1.7 The depolarizing channel......Page 111
6.1.8 Amplitude damping......Page 112
6.1.9 Phase damping......Page 114
6.1.10 De-entanglement......Page 116
6.2 The master equation......Page 119
6.2.1 Derivation of the master equation......Page 120
6.2.2 The master equation and quantum operations......Page 124
6.2.3 The master equation for a single qubit......Page 127
6.3.1 Schrodinger’s cat......Page 130
6.3.2 Decoherence and destruction of cat states......Page 132
6.4 Decoherence and quantum measurements......Page 140
6.5 Quantum chaos......Page 143
6.5.1 Dynamical chaos in classical mechanics......Page 144
6.5.2 Quantum chaos and the correspondence principle......Page 147
6.5.3 Time scales of quantum chaos......Page 150
6.5.4 Quantum chaos and Anderson localization......Page 157
6.5.5 The hydrogen atom in a microwave field......Page 160
6.5.6 Quantum chaos and universal spectral fluctuations......Page 165
6.5.7 The chaos border for the quantum computer hardware......Page 177
6.5.8 The quantum Loschmidt echo......Page 181
6.5.9 Dynamical stability of quantum motion......Page 188
6.5.10 Dynamical chaos and dephasing: the double-slit experiment......Page 190
6.5.11 Entanglement and chaos......Page 195
6.6 Decoherence and quantum computation......Page 199
6.6.1 Decoherence and quantum trajectories......Page 204
6.7 Quantum computation and quantum chaos......Page 214
6.7.1 Quantum versus classical errors......Page 216
6.7.2 Static imperfections versus noisy gates......Page 217
6.8 A guide to the bibliography......Page 222
7. Quantum Error Correction......Page 224
7.1 The three-qubit bit-flip code......Page 226
7.2 The three-qubit phase-flip code......Page 230
7.3 The nine-qubit Shor code......Page 231
7.4 General properties of quantum error correction......Page 236
7.4.1 The quantum Hamming bound......Page 238
7.5 The five-qubit code......Page 19
7.6 Classical linear codes......Page 241
7.6.1 The Hamming codes......Page 243
7.7 CSS codes......Page 246
7.8 Decoherence-free subspaces......Page 249
7.8.1 Conditions for decoherence-free dynamics......Page 251
7.8.2 The spin-boson model......Page 253
7.9 The Zeno effect......Page 255
7.10 Fault-tolerant quantum computation......Page 259
7.10.1 Avoidance of error propagation......Page 260
7.10.3 The noise threshold for quantum computation......Page 262
7.11 Quantum cryptography over noisy channels......Page 265
7.12 Quantum channels with memory......Page 271
7.13 A guide to the bibliography......Page 275
8. First Experimental Implementations......Page 278
8.1 NMR quantum computation......Page 279
8.1.1 The system Hamiltonian......Page 280
8.1.2 The physical apparatus......Page 283
8.1.3 Quantum ensemble computation......Page 284
8.1.4 Refocusing......Page 287
8.1.5 Demonstration of quantum algorithms......Page 288
8.2 Cavity quantum electrodynamics......Page 293
8.2.1 Rabi oscillations......Page 300
8.2.2 Entanglement generation......Page 303
8.2.3 The quantum phase gate......Page 307
8.3.1 The Paul trap......Page 309
8.3.2 Laser pulses......Page 312
8.3.3 Realization of the Cirac–Zoller CNOT gate......Page 319
8.3.4 Entanglement generation......Page 321
8.4.1 Spins in semiconductors......Page 326
8.4.2 Quantumdots......Page 327
8.4.3 Superconducting qubit circuits......Page 331
8.5.1 Linear optics......Page 338
8.5.2 Experimental quantum teleportation......Page 343
8.5.3 Experimental quantum-key distribution......Page 352
8.6 Problems and prospects......Page 356
8.7 A guide to the bibliography......Page 357
Appendix B Solutions to the exercises......Page 360
Bibliography......Page 422
Index......Page 440