The twenty-first century is seeing the emergence of the first truly quantum technologies; that is, technologies that rely on the counter-intuitive properties of individual quantum systems and can often outperform any conventional technology. Examples include quantum computing, which promises to be much faster than conventional computing for certain problems, and quantum metrology, which promises much more sensitive parameter estimation than that offered by conventional techniques. To realize these promises, it is necessary to understand the measurement and control of quantum systems. This book serves as an introduction to quantum measurement and control, including some of the latest developments in both theory and experiment.

Preface page xi

1 Quantum measurement theory 1

      1.1Classical measurement theory 1

      1.2 Quantum measurement theory 8

      1.3 Representing outcomes as operators 27

      1.4 Most general formulation of quantum measurements 32

      1.5 Measuring a single photon 42

      1.6 Further reading 49

2 Quantum parameter estimation 51

      2.1 Quantum limits to parameter estimation 51

      2.2 Optimality using Fisher information 54

      2.3 Examples of BC-optimal parameter estimation 62

      2.4 Interferometry 一 other optimality conditions 68

      2.5 Interferometry 一 adaptive parameter estimation 76

      2.6 Experimental results for adaptive phase estimation 83

      2.7 Quantum state discrimination 85

      2.8 Further reading 94

3 Open quantum systems 97

      3.1 Introduction 97

      3.2 The Born-Markov master equation 99

      3.3 The radiative-damping master equation 102

      3.4 Irreversibility without the rotating-wave approximation 109

      3.5 Fermionic reservoirs 113

      3.6 The Lindblad form and positivity 119

      3.7 Decoherence and the pointer basis 121

      3.8 Preferred ensembles 124

      3.9 Decoherence in a quantum optical system 130

      3.10 Other examples of decoherence 136

      3.11 Heisenberg-picture dynamics 141

      3.12 Further reading 146

4 Quantum trajectories 148

      4.1 Introduction 148

      4.2 Quantum jumps 149

      4.3 Photodetection 154

      4.4 Homodyne detection 157

      4.5 Heterodyne detection and beyond 166

      4.6 Illustration on the Bloch sphere 172

      4.7 Monitoring in the Heisenberg picture 181

      4.8 Imperfect detection 190

      4.9 Continuous measurement in mesoscopic electronics 201

      4.10 Further reading 215

5 Quantum feedback control 216

      5.1 Introduction 216

      5.2 Feedback with optical beams using linear optics 217

      5.3 Feedback with optical beams using nonlinear optics 231

      5.4 Feedback control of a monitored system 237

      5.5 Homodyne-mediated feedback control 246

      5.6 Markovian feedback in a linear system 251

      5.7 Deterministic spin-squeezing 259

      5.8 Further reading 265

6 State-based quantum feedback control 269

      6.1 Introduction 269

      6.2 Freezing a conditional state 270

      6.3 General classical systems 278

      6.4 Linear classical systems 283

      6.5 General quantum systems 308

      6.6 Linear quantum systems 312

      6.7 Further reading 337

7 Applications to quantum information processing 341

      7.1 Introduction 341

      7.2 Quantum teleportation of a qubit 343

      7.3 Quantum teleportation for continuous variables 347

      7.4 Errors and error correction 353

      7.5 Feedback to correct continuously detected errors 362

      7.6 QEC using continuous feedback 368

      7.7 Continuous QEC without measurement 375

      7.8 Linear optical quantum computation 379

      7.9 Adaptive phase measurement and single-rail LOQC 390

      7.10 Further reading 395

Appendix A: Quantum mechanics and phase-space 398

      A.1 Fundamentals of quantum mechanics 398

      A.2 Multipartite systems and entanglement 404

      A.3 Position and momentum 407

      A.4 The harmonic oscillator 410

      A.5 Quasiprobability distributions 414

Appendix B: Stochastic differential equations 418

      B.1 Gaussian white noise 418

      B.2 Ito stochastic differential calculus 420

      B.3 The ltd-Stratonovich relation 422

      B.4 Solutions to SDEs 423

      B.5 The connection to the Fokker-Planck equation 424

      B.6 More general noise 425

References 430

Index 449

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