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arXiv:0709.1682 · IEEE Transactions on Automatic Control

Quantum Risk-Sensitive Estimation and Robustness

Naoki Yamamoto, Luc Bouten

Published Jan 1, 2009• 66 citations
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Abstract

This paper studies a quantum risk-sensitive estimation problem and investigates robustness properties of the filter. This is a direct extension to the quantum case of analogous classical results. All investigations are based on a discrete approximation model of the quantum system under consideration. This allows us to study the problem in a simple mathematical setting. We close the paper with some examples that demonstrate the robustness of the risk-sensitive estimator.

DOI
10.1109/tac.2008.2009571

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The use of control theory in a LQG form for quantum mechanical situations has revealed a new conceptualization of knowledge, measurement and feedback. The benefits of state estimation by the Kalman filter has shown up in theoretical calculations for cooling and confining of an atom in a harmonic potential, surpassing previous proposals of direct feedback. The goal of this thesis is to explore the possible application of a new form of filter/controller, the risk-sensitive LEQG filter, which has been shown to enjoy a sense of robustness against model uncertainties. An investigation is launched into some simple models, and a computer simulation has highlighted circumstances where the LEQG filter does provide better control.
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In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is in general unrealistic. Therefore, in this paper we consider a class of linear quantum systems subject to time-varying norm-bounded parametric uncertainty and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although the proposed observer is different from the optimal filter in the sense of the least mean square error, it is demonstrated in a typical quantum control problem that the observer is fairly robust against a parametric uncertainty even when the other estimators, the optimal Kalman filter and the risk-sensitive observer, fail in the estimation due to the uncertain perturbation
The importance of feedback control is being increasingly appreciated in quantum physics and applications. This paper describes the use of optimal control methods in the design of quantum feedback control systems, and in particular the paper formulates and solves a risk-sensitive optimal control problem. The resulting risk-sensitive optimal control is given in terms of an unnormalized conditional state, whose dynamics include the cost function used to specify the performance objective. The risk-sensitive conditional dynamic equation describes the evolution of our knowledge of the quantum system tempered by our purpose for the controlled quantum system. Robustness properties of risk-sensitive controllers are discussed and an example is provided.