Quantum Measurement Theory and its Application 1st edition by Kurt Jacobs ISBN 1139985260 9781139985260

1 Quantum measurement theory

1.1 Introduction and overview

1.2 Classical measurement theory

1.2.1 Understanding Bayes’ theorem

1.2.2 Multiple measurements and Gaussian distributions

1.2.3 Prior states-of-knowledge and invariance

1.3 Quantum measurement theory

1.3.1 The measurement postulate

1.3.2 Quantum states-of-knowledge:density matrices

1.3.3 Quantum measurements

1.4 Understanding quantum measurements

1.4.1 Relationship to classical measurements

1.4.2 Measurements of observables and resolving power

1.4.3 A measurement of position

1.4.4 The polar decomposition: bare measurements and feedback

1.5 Describing measurements within unitary evolution

1.6 Inefficient measurements

1.7 Measurements on ensembles of states

2 Useful concepts from information theory

2.1 Quantifying information

2.1.1 The entropy

2.1.2 The mutual information

2.2 Quantifying uncertainty about a quantum system

2.2.1 The von Neumann entropy

2.2.2 Majorization and density matrices

2.2.3 Ensembles corresponding to a density matrix

2.3 Quantum measurements and information

2.3.1 Information-theoretic properties

2.3.2 Quantifying disturbance

2.4 Distinguishing quantum states

2.5 Fidelity of quantum operations

3 Continuous measurement

3.1 Continuous measurements with Gaussian noise

3.1.1 Classical continuous measurements

3.1.2 Gaussian quantum continuous measurements

3.1.3 When the SME is the classical Kalman–Bucy filter

3.1.4 The power spectrum of the measurement record

3.2 Solving for the evolution: the linear form of the SME

3.2.1 The dynamics of measurement: diffusion gradients

3.2.2 Quantum jumps

3.2.3 Distinguishing quantum from classical

3.2.4 Continuous measurements on ensembles of systems

3.3 Measurements that count events: detecting photons

3.4 Homodyning: from counting to Gaussian noise

3.5 Continuous measurements with more exotic noise?

3.6 The Heisenberg picture: inputs, outputs, and spectra

3.7 Heisenberg-picture techniques for linear systems

3.7.1 Equations of motion for Gaussian states

3.7.2 Calculating the power spectrum of the measurement record

3.8 Parameter estimation: the hybrid master equation

3.8.1 An example: distinguishing two quantum states

4 Statistical mechanics, open systems, and measurement

4.1 Statistical mechanics

4.1.1 Thermodynamic entropy and the Boltzmann distribution

4.1.2 Entropy and information: Landauer’s erasure principle

4.1.3 Thermodynamics with measurements: Maxwell’s demon

4.2 Thermalization I: the origin of irreversibility

4.2.1 A new insight: the Boltzmann distribution from typicality

4.2.2 Hamiltonian typicality

4.3 Thermalization II: useful models

4.3.1 Weak damping: the Redfield master equation

4.3.2 Redfield equation for time-dependent or interacting systems

4.3.3 Baths and continuous measurements

4.3.4 Wavefunction “Monte Carlo” simulation methods

4.3.5 Strong damping: master equations and beyond

4.4 The quantum-to-classical transition

4.5 Irreversibility and the quantum measurement problem

5 Quantum feedback control

5.1 Introduction

5.2 Measurements versus coherent interactions

5.3 Explicit implementations of continuous-time feedback

5.3.1 Feedback via continuous measurements

5.3.2 Coherent feedback via unitary interactions

5.3.3 Coherent feedback via one-way fields

5.3.4 Mixing one-way fields with unitary interactions: a coherent version of Markovian feedback

5.4 Feedback control via continuous measurements

5.4.1 Rapid purification protocols

5.4.2 Control via measurement back-action

5.4.3 Near-optimal feedback control for a single qubit?

5.4.4 Summary

5.5 Optimization

5.5.1 Bellman’s equation and the HJB equation

5.5.2 Optimal control for linear quantum systems

5.5.3 Optimal control for nonlinear quantum systems

6 Metrology

6.1 Metrology of single quantities

6.1.1 The Cramér–Rao bound

6.1.2 Optimizing the Cramér–Rao bound

6.1.3 Resources and limits to precision

6.1.4 Adaptive measurements

6.2 Metrology of signals

6.2.1 Quantum-mechanics-free subsystems

6.2.2 Oscillator-mediated force detection

7 Quantum mesoscopic systems I: circuits and measurements

7.1 Superconducting circuits

7.1.1 Procedure for obtaining the circuit Lagrangian (short method)

7.2 Resonance and the rotating-wave approximation

7.3 Superconducting harmonic oscillators

7.4 Superconducting nonlinear oscillators and qubits

7.4.1 The Josephson junction

7.4.2 The Cooper-pair box and the transmon

7.4.3 Coupling qubits to resonators

7.4.4 The RF-SQUID and flux qubits

7.5 Electromechanical systems

7.6 Optomechanical systems

7.7 Measuring mesoscopic systems

7.7.1 Amplifiers and continuous measurements

7.7.2 Translating between experiment and theory

7.7.3 Implementing a continuous measurement

7.7.4 Quantum transducers and nonlinear measurements

8 Quantum mesoscopic systems II: measurement and control

8.1 Open-loop control

8.1.1 Fast state-swapping for oscillators

8.1.2 Preparing non-classical states

8.2 Measurement-based feedback control

8.2.1 Cooling using linear feedback control

8.2.2 Squeezing using linear feedback control

8.3 Coherent feedback control

8.3.1 The “resolved-sideband” cooling method

8.3.2 Resolved-sideband cooling via one-way fields

8.3.3 Optimal cooling and state-preparation

Appendix A The tensor product and partial trace

Appendix B A fast-track introduction for experimentalists

Appendix C A quick introduction to Ito calculus

Appendix D Operators for qubits and modes

Appendix E Dictionary of measurements

Appendix F Input–output theory

F.1 A mode of an optical or electrical cavity

F.2 The traveling-wave fields at x = 0: the input and output signals

F.3 The Heisenberg equations of motion for the system

F.4 A weakly damped oscillator

F.5 Sign conventions for input–output theory

F.6 The quantum noise equations for the system: Ito calculus

F.7 Obtaining the Redfield master equation

F.8 Spectrum of the measurement signal

Appendix G Various formulae and techniques

G.1 The relationship between Hz and s–1, and writing decay rates in Hz

G.2 Position representation of a pure Gaussian state

G.3 The multivariate Gaussian distribution

G.4 The rotating-wave approximation (RWA)

G.5 Suppression of off-resonant transitions

G.6 Recursion relations for time-independent perturbation theory

G.7 Finding operator transformation, reordering, and splitting relations

G.8 The Haar measure

G.9 General form of the Kushner–Stratonovich equation

G.10 Obtaining steady states for linear open systems

Appendix H Some proofs and derivations

H.1 The Schumacher–Westmoreland–Wootters theorem

H.2 The operator-sum representation for quantum evolution

H.3 Derivation of the Wiseman–Milburn Markovian feedback SME

References

Index