MEASUREMENT THEORY QUANTUM AND ITS APPLICATIONS KURT JACOBS. University of Massachusetts at Boston. fg Cambridge WW UNIVERSITY PRESS


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1 QUANTUM MEASUREMENT THEORY AND ITS APPLICATIONS KURT JACOBS University of Massachusetts at Boston fg Cambridge WW UNIVERSITY PRESS
2 Contents Preface page xi 1 Quantum measurement theory Introduction and overview Classical measurement theory Understanding Bayes'theorem Multiple measurements and Gaussian distributions Prior statesofknowledge and invariance Quantum measurement theory The measurement postulate Quantum statesofknowledge: density matrices Quantum measurements Understanding quantum measurements Relationship to classical measurements Measurements of observables and resolving power A measurement of position The polar decomposition: bare measurements and feedback Describing measurements within unitary evolution Inefficient measurements Measurements on ensembles of states 40 2 Useful concepts from information theory Quantifying information The entropy The mutual information Quantifying uncertainty about a quantum system The von Neumann entropy Majorization and density matrices Ensembles corresponding to a density matrix Quantum measurements and information Informationtheoretic properties Quantifying disturbance 72 vii
3 viii Contents 2.4 Distinguishing quantum states Fidelity of quantum operations 82 3 Continuous measurement Continuous measurements with Gaussian noise Classical continuous measurements Gaussian quantum continuous measurements When the SME is the classical KalmanBucy filter The power spectrum of the measurement record Solving for the evolution: the linear form of the SME The dynamics of measurement: diffusion gradients Quantum jumps Distinguishing quantum from classical Continuous measurements on ensembles of systems Measurements that count events: detecting photons Homodyning: from counting to Gaussian noise Continuous measurements with more exotic noise? The Heisenberg picture: inputs, outputs, and spectra Heisenbergpicture techniques for linear systems Equations of motion for Gaussian states Calculating the power spectrum of the measurement record Parameter estimation: the hybrid master equation An example: distinguishing two quantum states Statistical mechanics, open systems, and measurement Statistical mechanics Thermodynamic entropy and the Boltzmann distribution Entropy and information: Landauer's erasure principle Thermodynamics with measurements: Maxwell's demon Thermalization I: the origin of irreversibility A new insight: the Boltzmann distribution from typicality Hamiltonian typicality Thermalization II: useful models Weak damping: the Redfield master equation Redfield equation for timedependent or interacting systems Baths and continuous measurements Wavefunction "Monte Carlo" simulation methods Strong damping: master equations and beyond The quantumtoclassical transition Irreversibility and the quantum measurement problem Quantum feedback control 5.1 Introduction Measurements versus coherent interactions Explicit implementations of continuoustime feedback 239
4 Contents ix Feedback via continuous measurements Coherent feedback via unitary interactions Coherent feedback via oneway fields Mixing oneway fields with unitary interactions: a coherent version of Markovian feedback Feedback control via continuous measurements Rapid purification protocols Control via measurement backaction Nearoptimal feedback control for a single qubit? Summary Optimization Bellman's equation and the HJB equation Optimal control for linear quantum systems Optimal control for nonlinear quantum systems Metrology Metrology of single quantities The CramerRao bound Optimizing the CramerRao bound Resources and limits to precision Adaptive measurements Metrology of signals Quantummechanicsfree subsystems Oscillatormediated force detection Quantum mesoscopic systems I: circuits and measurements Superconducting circuits Procedure for obtaining the circuit Lagrangian (short method) Resonance and the rotatingwave approximation Superconducting harmonic oscillators Superconducting nonlinear oscillators and qubits The Josephson junction The Cooperpair box and the transmon Coupling qubits to resonators The RFSQUID and flux qubits Electromechanical systems Optomechanical systems Measuring mesoscopic systems Amplifiers and continuous measurements Translating between experiment and theory Implementing a continuous measurement Quantum transducers and nonlinear measurements Quantum mesoscopic systems II: measurement and control Openloop control 383
5 X Contents Fast stateswapping for oscillators Preparing nonclassical states Measurementbased feedback control Cooling using linear feedback control Squeezing using linear feedback control Coherent feedback control The "resolvedsideband" cooling method Resolvedsideband cooling via oneway fields Optimal cooling and statepreparation 416 Appendix A The tensor product and partial trace 432 Appendix B A fasttrack introduction for experimentalists 441 Appendix C A quick introduction to Ito calculus 448 Appendix D Operators for qubits and modes 451 Appendix E Dictionary ofmeasurements 456 Appendix F Inputoutput theory 458 FA A mode of an optical or electrical cavity 458 F.2 The travelingwave fields atx = 0: the input and output signals 462 F.3 The Heisenberg equations of motion for the system 463 FA A weakly damped oscillator 467 F.5 Sign conventionsfor inputoutput theory 467 F.6 The quantum noise equations for the system: Ito calculus 468 F. 7 Obtaining the Redfield master equation 469 F.8 Spectrum of the measurement signal 470 Appendix G Various formulae and techniques 475 G. I The relationship between Hz and s~], and writing decay rates in Hz 475 G.2 Position representation of a pure Gaussian state 475 G.3 The multivariate Gaussian distribution 476 G.4 The rotatingwave approximation (RWA) 476 G.5 Suppression of offresonant transitions Ml G.6 Recursion relationsfor timeindependent perturbation theory 478 G. 7 Finding operator transformation, reordering, and splitting relations 479 G.8 The Haar measure 484 G.9 Generalform of the KushnerStratonovich equation 485 G. 10 Obtaining steady states for linear open systems 486 Appendix H Some proofs and derivations 490 //./ The SchumacherWestmorelandWootters theorem 490 H.2 The operatorsum representation for quantum evolution 492 H.3 Derivation of the WisemanMilburn Markovian feedback SME 494 References 498 Index 539
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