Explorations in Quantum Measurement and Control

Abstract

This thesis presents the theory of LQG (linear-quadratic-Gaussian) and Markovian feedback control for quantum systems. We devote Part I to a review of both classical and quantum feedback but with an emphasis on LQG and linear systems control. The language of classical stochastic control is that of probability theory and stochastic differential equations. Thus we first introduce the essential mathematics (within the context of measurement and control) such as the Kushner– Stratonovich and Langevin equations. We then specialize to linear classical systems and introduce traditional engineering principles such as stabilizing solutions of Riccati equations, controllability, certainty equivalence, and the like. The classical Kushner–Stratonovich and Langevin equations have well-known quantum analogues — the stochastic master equation and the quantum Langevin equation. These, and other relevant tools for doing quantum feedback control are reviewed. Subsequently the concepts of stabilizing solutions of Riccati equations, controllability, separation theorem, amongst other related notions are shown to be adaptable for quantum systems. An essential difference between quantum and classical feedback lies in the in-loop measurement. Quantum measurements induce quantum backaction, non-existent in classsical measurements. This means the measurement strength and strategy in a quantum feedback loop should be optimized for a given control objective. We illustrate each of these points with examples in LQG control. Part II is devoted to the theory of diffusive quantum measurements (measurements that have Gaussian distributed outcomes) and Markovian feedback control for nonlinear systems. A new and general representation of diffusive quantum measurements is derived. This representation is compared with an old representation (which we introduce in Part I) and shown to possess advantages over the old representation. We also propose a quantum optical scheme as a universal method for physically realizing diffusive quantum measurements. Our new representation of diffusive measurements is then applied to build a general theory of multiple-input multiple-output Markovian quantum feedback. Previously known results of Markovian feedback are reproduced as special cases of this more general framework.This work has not previously been submitted for a degree or diploma in any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made in the thesis itself.