Quantum Measurement and Control

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Scope and aims......Page 13
Historical background......Page 14
The structure of this book......Page 16
Acknowledgements......Page 17
1.1.1 Basic concepts......Page 19
1.1.2 Example: binary variables......Page 20
1.1.3 Bayesian inference......Page 22
1.1.5 Most general formulation of classical measurements......Page 24
1.2.1 Probability and quantum mechanics......Page 26
1.2.2 Projective measurements......Page 28
1.2.3 Systems and meters......Page 32
1.2.4 Example: binary measurement......Page 34
1.2.5 Measurement operators and effects......Page 37
1.2.6 Non-selective evolution and choice of basis......Page 43
1.3.1 \'Correlations without correlata\'......Page 45
1.3.2 Measurement in the Heisenberg picture......Page 47
1.3.3 The binary example......Page 49
1.4.1 Operations and effects......Page 50
1.4.2 Classification of measurements......Page 53
1.4.3 Classification exercise......Page 59
1.5 Measuring a single photon......Page 60
1.6 Further reading......Page 67
2.1.1 Introduction......Page 69
2.1.2 The Helstrom–Holevo lower bound......Page 70
2.2.1 Distinguishability and Fisher information......Page 72
2.2.2 Quantum statistical distance......Page 74
2.2.3 Achieving Braunstein–Caves optimality......Page 77
2.3.1 Spatial displacement......Page 80
2.3.2 Spatial displacement of a squeezed state......Page 81
2.3.3 Harmonic oscillator phase......Page 83
2.4.1 The standard quantum limit for interferometry......Page 86
2.4.2 Canonical phase-difference measurements......Page 90
2.4.3 Optimal states for interferometry......Page 92
2.5.1 Constrained measurements......Page 94
2.5.2 Adaptive measurements......Page 97
2.6 Experimental results for adaptive phase estimation......Page 101
2.7 Quantum state discrimination......Page 103
2.7.1 Minimizing the error probability......Page 104
2.7.2 Experimental demonstration of the Helstrom bound......Page 107
2.7.3 Inconclusive state discrimination......Page 108
2.7.4 Experimental demonstration of the IDP bound......Page 110
2.8.1 Quantum tomography......Page 112
2.8.2 Other work......Page 113
3.1 Introduction......Page 115
3.2 The Born–Markov master equation......Page 117
3.3.1 Spontaneous emission......Page 120
3.3.2 Cavity emission......Page 125
3.4.1 The spin–boson model......Page 127
3.4.2 Quantum Brownian motion......Page 130
3.5 Fermionic reservoirs......Page 131
3.6 The Lindblad form and positivity......Page 137
3.7.1 Einselection......Page 139
3.7.2 A more realistic model......Page 141
3.8 Preferred ensembles......Page 142
3.8.1 Quantum steering......Page 144
3.8.3 Examples......Page 145
3.9.1 Theoretical analysis......Page 148
3.9.2 Experimental observation......Page 151
3.10.1 Quantum electromechanical systems......Page 154
3.10.2 A superconducting box......Page 156
3.11 Heisenberg-picture dynamics......Page 159
3.11.1 Quantum Langevin equations......Page 160
3.11.2 Generalization for a non-vacuum bath......Page 163
3.12 Further reading......Page 164
4.1 Introduction......Page 166
4.2.1 Master equations and continuous measurements......Page 167
4.2.2 Stochastic evolution......Page 168
4.2.3 Quantum trajectories for simulations......Page 170
4.3.1 Photon emission and detection......Page 172
4.3.2 Output correlation functions......Page 173
4.3.3 Coherent field input......Page 174
4.4.1 Adding a local oscillator......Page 175
4.4.2 The continuum limit......Page 177
4.4.3 Linear quantum trajectories......Page 180
4.4.4 Output field correlation functions......Page 183
4.5.1 Heterodyne detection......Page 184
4.5.2 Completely general dyne detection......Page 186
4.6.1 Direct photodetection......Page 190
4.6.2 Adaptive detection with a local oscillator......Page 193
4.6.3 Homodyne detection......Page 196
4.6.4 Heterodyne detection......Page 198
4.7.1 Inputs and outputs......Page 199
4.7.2 Direct photodetection......Page 200
4.7.3 Homodyne detection......Page 202
4.7.4 Heterodyne detection......Page 203
4.7.5 Completely general dyne detection......Page 204
4.7.6 Relation to distribution functions......Page 205
4.8.1 Inefficient detection......Page 208
4.8.2 Detection with a white-noise input field......Page 209
4.8.3 Detection with dark noise......Page 212
4.8.4 Detectors with a finite bandwidth......Page 213
4.9.1 Monitoring a single quantum dot......Page 219
4.9.2 Monitoring a double quantum dot......Page 225
4.9.3 Other theoretical approaches to monitoring in mesoscopic systems......Page 231
4.10 Further reading......Page 233
5.1 Introduction......Page 234
5.2.1 Linearized theory of photodetection......Page 235
5.2.2 In-loop \'squeezing\'......Page 238
5.2.3 Stability......Page 240
5.2.4 In-loop and out-of-loop spectra......Page 243
5.2.5 Commutation relations......Page 245
5.2.6 QND measurements of in-loop beams......Page 247
5.2.7 Applications to noise reduction......Page 248
5.3.1 QND measurements......Page 249
5.3.2 QND-based feedback......Page 251
5.3.3 Parametric down-conversion......Page 252
5.3.4 Feedback, feedforward and robustness......Page 253
5.4 Feedback control of a monitored system......Page 255
5.4.1 General feedback......Page 256
5.4.2 The feedback master equation......Page 257
5.4.3 Application: protecting Schrödinger\'s cat......Page 259
5.4.4 Feedback in the Heisenberg picture......Page 261
5.4.5 Markovian feedback in the Heisenberg picture......Page 263
5.5.1 The homodyne feedback master equation......Page 264
5.5.2 Feedback with white noise......Page 266
5.5.3 Homodyne feedback in the Heisenberg picture......Page 267
5.6.1 The linear system......Page 269
5.6.2 Adding linear feedback......Page 272
5.6.3 Understanding feedback through conditioning......Page 274
5.7.1 Spin-squeezing......Page 277
5.7.2 Measurement and feedback......Page 279
5.7.3 Experimental considerations......Page 282
5.8.1 Coherent quantum feedback......Page 283
5.8.2 Other applications of quantum feedback......Page 284
6.1 Introduction......Page 287
6.2 Freezing a conditional state......Page 288
6.2.1 The system......Page 290
6.2.2 Conditional evolution......Page 292
6.2.4 Experimental results......Page 294
6.3.1 Notation and terminology......Page 296
6.3.2 The Kushner–Stratonovich equation......Page 298
6.3.3 Optimal feedback control......Page 300
6.4 Linear classical systems......Page 301
6.4.1 Unconditional dynamics......Page 302
6.4.2 Conditional dynamics......Page 306
6.4.3 Stabilizing solutions......Page 310
6.4.4 LQG optimal feedback control......Page 313
6.4.5 Markovian feedback......Page 319
6.5.1 Notation and terminology......Page 326
6.5.2 The Belavkin equation......Page 327
6.5.3 Optimal feedback control......Page 329
6.6.1 Quantum phase-space......Page 330
6.6.2 Linear unconditional dynamics......Page 333
6.6.3 Linear conditional dynamics......Page 338
6.6.4 Stabilizing solutions......Page 343
6.6.5 LQG optimal feedback control......Page 346
6.6.6 Optimal unravellings......Page 351
6.6.7 Markovian feedback......Page 353
6.7.1 Approximations to LQG quantum feedback......Page 355
6.7.2 State-based quantum feedback control in finite-dimensional systems......Page 356
6.7.3 Beyond state-based control......Page 357
7.1 Introduction......Page 359
7.2.1 The protocol......Page 361
7.2.2 A criterion for demonstrating qubit teleportation......Page 364
7.3 Quantum teleportation for continuous variables......Page 365
7.3.1 The ideal protocol......Page 366
7.3.2 A more realistic protocol......Page 367
7.3.3 A criterion for demonstrating CV teleportation......Page 369
7.3.4 Experimental demonstration of CV teleportation......Page 370
7.4.1 Types of quantum errors......Page 371
7.4.2 Quantum error correction......Page 374
7.4.3 Detected errors......Page 377
7.4.4 Dynamical decoherence......Page 379
7.5.1 Feedback to correct spontaneous emission jumps......Page 380
7.5.2 Feedback to correct spontaneous-emission diffusion......Page 382
7.5.3 Generalization to spontaneous emission of n qubits......Page 383
7.5.4 Generalization to arbitrary local measurements on n qubits......Page 384
7.5.5 Other generalizations......Page 385
7.6 QEC using continuous feedback......Page 386
7.6.1 Practical considerations for charge qubits......Page 390
7.7 Continuous QEC without measurement......Page 393
7.8.1 Measurement-induced optical nonlinearity......Page 397
7.8.2 Two-qubit gates......Page 400
7.8.3 Teleporting to determinism......Page 403
7.9 Adaptive phase measurement and single-rail LOQC......Page 408
7.9.1 Dyne measurement on single-rail qubits......Page 409
7.9.2 Adaptive phase measurement on single-rail qubits......Page 410
7.9.3 Preparing arbitrary single-rail qubit states......Page 411
7.9.4 Quantum gates using adaptive phase measurements......Page 412
7.10 Further reading......Page 413
A.1.1 Pure states and operators......Page 416
A.1.2 Mixed states......Page 419
A.1.3 Time evolution......Page 420
A.2.1 Multipartite systems......Page 422
A.2.2 Entanglement......Page 423
A.3.1 Position......Page 425
A.3.2 Momentum......Page 426
A.3.3 Minimum-uncertainty states......Page 427
A.4.1 Number states......Page 428
A.4.2 Coherent states......Page 429
A.5 Quasiprobability distributions......Page 432
A.5.2 Antinormal order and the Q function......Page 433
A.5.3 Symmetric order and the Wigner function......Page 434
B.1 Gaussian white noise......Page 436
B.2 It stochastic differential calculus......Page 438
B.3 The Ito–Stratonovich relation......Page 440
B.4.1 The meaning of \'solution\'......Page 441
B.5 The connection to the Fokker–Planck equation......Page 442
B.6 More general noise......Page 443
B.6.1 Multi-dimensional generalization......Page 446
References......Page 448
Index......Page 467