Introduction to Optical Quantum Information Processing

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
PART I QUANTUM OPTICS AND QUANTUM INFORMATION......Page 15
1.1 The classical electromagnetic field......Page 17
1.2.1 Field quantization......Page 20
1.2.2 Mode functions and mode operators......Page 23
1.2.3 Photons as excitations of the electromagnetic field......Page 25
1.2.4 Quadrature operators......Page 28
1.3.1 Polarization......Page 30
1.3.2 Transverse mode functions......Page 31
Gaussian modes......Page 32
Hermite–Gaussian modes......Page 35
Laguerre–Gaussian modes......Page 37
1.4.1 The Heisenberg equations of motion......Page 39
1.4.2 Time-bin mode operators......Page 42
1.4.3 Mode transformations and optical elements......Page 43
1.4.4 Normal modes......Page 46
1.4.5 Non-photon-number-preserving transformations......Page 48
1.5 Quantum states of the electromagnetic field......Page 51
1.5.1 Coherent states......Page 52
Single-mode squeezing......Page 55
Two-mode squeezing......Page 59
1.6 References and further reading......Page 60
2.1.1 Classical and quantum bits......Page 62
2.1.2 Two-qubit logical operations and gates......Page 66
2.1.3 The stabilizer formalism......Page 68
2.2.1 The no-cloning theorem and cryptography......Page 71
2.2.2 Teleportation......Page 72
2.2.3 Quantum repeaters......Page 74
2.3.1 The circuit model......Page 76
2.3.2 Cluster states and the one-way model......Page 78
2.3.3 Manipulating cluster states......Page 81
2.3.4 The Gottesman–Knill theorem......Page 86
Universality of different classes of clusters......Page 87
2.3.6 Error correction and fault tolerance......Page 89
Stabilizer codes......Page 90
Fault tolerance......Page 92
2.4 Quantum computation with continuous variables......Page 94
2.4.1 Linear and quadratic interaction Hamiltonians......Page 95
2.4.2 Universal quantum computation......Page 99
2.4.3 Cluster states for continuous variables......Page 100
2.4.4 Approximate position eigenstates and error correction......Page 101
2.5 References and further reading......Page 103
3.1 Density operators and superoperators......Page 104
3.1.1 The qubit density operator on the Bloch sphere......Page 107
3.1.2 Superoperators and completely positive maps......Page 110
3.2 The fidelity......Page 114
3.3.1 Partial-transpose criterion......Page 115
3.3.2 Von Neumann entropy......Page 116
3.3.4 Mutual information and the Holevo bound......Page 118
3.4 Correlation functions and interference of light......Page 119
3.5 Photon correlation measurements......Page 122
3.6 References and further reading......Page 124
PART II QUANTUM INFORMATION IN PHOTONS AND ATOMS......Page 125
4.1 A mathematical model of photodetectors......Page 127
4.1.1 Efficiency and dark counts......Page 128
Detection efficiency......Page 130
Detector dark counts......Page 133
Absolute detection efficiencies and dark currents......Page 134
4.2.1 Photomultiplier tubes and avalanche photodiodes......Page 135
4.2.2 Photon number resolution: detector cascading......Page 138
Time multiplexing......Page 140
4.2.3 Superconducting photon-number detectors......Page 141
4.3 Single-photon sources......Page 143
4.3.1 Hanbury Brown and Twiss, and Hong–Ou–Mandel......Page 144
4.3.2 Weak coherent states and post-selection......Page 148
4.3.3 Heralded parametric downconversion......Page 150
4.3.4 Other single-photon sources......Page 152
4.4.1 Post-selected parametric downconversion......Page 153
4.5 Quantum non-demolition photon detectors......Page 156
4.6 References and further reading......Page 158
5.1 Photons as information carriers......Page 159
5.1.1 Spatial dual-rail qubits......Page 160
5.1.2 Polarization qubits......Page 161
5.1.3 Orbital angular-momentum qubits......Page 166
5.1.4 Time-bin encoding......Page 175
5.2.1 Two-photon entanglement......Page 176
5.2.2 Photonic Bell measurements......Page 178
5.2.3 Teleportation of single photons......Page 180
5.2.4 Entanglement distillation......Page 182
5.3 Decoherence-free subspaces for communication......Page 184
5.4 Quantum cryptography......Page 186
5.4.1 Security of Ekert91......Page 187
5.4.2 Security of BB84......Page 189
5.4.3 Multi-photon states and decoy states......Page 190
5.5 References and further reading......Page 191
6.1 Optical N-port interferometers and scalability......Page 193
6.1.1 Optical simulation of a quantum computer......Page 194
6.2.1 A gate based on Kerr nonlinearities......Page 195
6.2.2 A probabilistic gate for single-photon qubits......Page 197
6.2.3 Improving success probabilities of optical gates......Page 201
6.2.4 A gate that consumes entanglement......Page 204
6.3 Building quantum computers with probabilistic gates......Page 206
6.3.1 Fusion gates and four-photon states......Page 207
6.3.2 Optical cluster-state generation......Page 210
6.3.3 Photonic quantum computing in the circuit model......Page 213
6.4.1 The tree encoding......Page 216
6.4.2 The redundant encoding......Page 219
6.5 Threshold theorem for linear-optical quantum computing......Page 221
6.6 References and further reading......Page 223
7.1 Atomic systems as qubits......Page 224
7.1.1 Manipulating atomic states with classical light......Page 226
7.1.2 Floquet theory and the rotating-wave approximation......Page 230
7.1.3 Three-level systems......Page 233
7.2 The Jaynes–Cummings Hamiltonian......Page 236
7.3 The optical master equation and quantum jumps......Page 241
7.3.1 The optical master equation......Page 245
7.3.2 Monitored decay in the quantum-jump approach......Page 248
7.4.1 The weak driving limit......Page 250
7.4.2 The double heralding parity gate......Page 252
7.4.3 The broker–client protocol......Page 255
7.4.4 Repeat-until-success CX gate......Page 256
7.5.1 Intra-cavity entangling operations......Page 259
7.5.2 The Zeno effect as a two-photon interaction......Page 262
7.6 References and further reading......Page 265
PART III QUANTUM INFORMATION IN MANY-BODY SYSTEMS......Page 267
8.1 Phase space in quantum optics......Page 269
8.1.1 The Wigner function......Page 270
Translations in phase space......Page 273
Squeezing in phase space......Page 275
8.1.3 Vacuum, coherent, and squeezed coherent states......Page 277
8.1.4 Multi-mode Wigner functions......Page 279
8.1.5 Homodyne detection......Page 280
8.2 Continuous-variable entanglement......Page 281
8.3.1 Quantum teleportation......Page 286
8.3.2 Entanglement swapping......Page 291
8.4 Entanglement distillation......Page 294
8.5.1 Transmission of squeezed coherent states......Page 295
8.5.2 Sharing approximate EPR pairs......Page 301
Establishing encoded Bell pairs......Page 302
Reduction to practical QKD......Page 304
8.6 References and further reading......Page 307
9.1.1 Initializing the qunat states......Page 308
9.1.2 The Heisenberg–Weyl operators......Page 310
9.1.3 The Fourier transform......Page 311
9.1.4 The phase gate......Page 312
9.2 Two-mode Gaussian qunat operations......Page 313
9.3 The Gottesman–Knill theorem for qunats......Page 317
9.4 Nonlinear optical qunat gates......Page 321
9.5 The one-way model for qunats......Page 323
9.5.1 Qunat cluster states......Page 324
9.5.2 Information propagation and processing......Page 325
9.6.1 State preparation of GKP codes......Page 332
9.6.2 Qunat error correction with linear optics and squeezing......Page 334
9.6.3 Propagation errors in cluster states......Page 338
9.7 References and further reading......Page 340
10.1 An ensemble of identical two-level atoms......Page 341
10.1.1 The interaction of atomic ensembles with quantized fields......Page 347
10.1.2 Phase shift of a single mode......Page 350
10.2.1 Three-level systems......Page 351
10.2.2 Four-level systems......Page 355
10.3.1 Quantum memories......Page 358
10.3.2 Quantum repeaters......Page 362
10.4 The atomic ensemble as a single qubit......Page 366
Quantum computing with atomic ensembles......Page 368
10.5 Photon–photon interactions via atomic ensembles......Page 369
10.5.1 Weak Kerr nonlinearities......Page 370
10.5.2 Intensity coupling to matter qubits......Page 373
10.6 References and further reading......Page 374
11.1 Basic concepts of solid-state systems......Page 375
11.1.1 Bloch’s theorem......Page 376
11.1.2 Wannier functions......Page 377
11.1.3 Conduction and valence bands......Page 379
11.1.4 Spin......Page 381
11.1.5 Heterostructures and the envelope-function approximation......Page 383
11.1.6 Optical selection rules......Page 386
11.1.7 Selection rules in quantum dots......Page 388
11.2.1 Exciton qubits......Page 389
11.2.2 Electron spin......Page 390
11.2.3 Crystal defects......Page 393
11.3.1 Exciton–exciton interactions......Page 395
Static dipole–dipole interaction......Page 396
11.3.2 Spin interactions......Page 397
11.4.1 Optical gates with exciton qubits......Page 398
11.4.2 Optical gates with spin qubits......Page 401
11.4.3 Passivating spin–spin interactions optically......Page 404
11.4.4 Optical control of spin–spin exchange interactions......Page 406
11.5.2 Global control......Page 407
11.6 References and further reading......Page 409
12.1 Phonons......Page 411
12.2 Electron–phonon coupling......Page 414
12.2.1 Deformation potential coupling......Page 416
12.3 The master equation for electrons and phonons......Page 417
12.4.1 Exploiting the spectral-density function......Page 420
12.4.2 Phonon decoherence for optically controlled spin qubits......Page 422
12.4.3 Adiabatic gating......Page 424
12.5 Strong coupling effects......Page 426
12.5.1 An exact model: independent bosons......Page 427
12.5.2 The optical spectrum......Page 428
12.5.3 Dephasing......Page 431
12.6 References and further reading......Page 433
13.1 Parameter estimation and Fisher information......Page 435
13.1.1 Bounds on precision measurements......Page 436
13.2.1 Distance between classical probability distributions......Page 439
13.2.2 Distance between density operators......Page 443
13.3 The dynamical evolution of states......Page 447
13.4 Entanglement-assisted parameter estimation......Page 451
13.5 Optical quantum metrology......Page 454
13.5.1 The Jordan–Schwinger representation......Page 455
13.5.2 The creation of YMCK and NOON states......Page 460
13.5.3 Squeezed-state metrology......Page 462
13.5.4 The Mach–Zehnder interferometer revisited......Page 465
13.6 References and further reading......Page 466
Appendix A: Baker–Campbell–Hausdorff relations......Page 468
Appendix B: The Knill–Laflamme–Milburn protocol......Page 471
Appendix C: Cross-Kerr nonlinearities for single photons......Page 476
References......Page 479
Index......Page 491
Qubit operators......Page 503
Advanced quantum mechanical relations......Page 504
Operators in quantum optics......Page 505
Continuous variable operators......Page 506