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arXiv:quant-ph/9912107 · Physical Review A

Quantum feedback control and classical control theory

Andrew C. Doherty, Salman Habib, Kurt A. Jacobs, Hideo Mabuchi, Sze Meng Tan

Published Jun 13, 2000• 369 citations
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Abstract

We introduce and discuss the problem of quantum feedback control in the context of established formulations of classical control theory, examining conceptual analogies and essential differences. We describe the application of state-observer-based control laws, familiar in classical control theory, to quantum systems and apply our methods to the particular case of switching the state of a particle in a double-well potential.

DOI
10.1103/physreva.62.012105

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We introduce and discuss the problem of quantum feedback control in the context of established formulations of classical control theory, examining conceptual analogies and essential differences. We describe the application of state-observer-based control laws, familiar in classical control theory, to quantum systems and apply our methods to the particular case of switching the state of a particle in a double-well potential.
We give an introduction to feedback control in quantum systems, as well as an overview of the variety of applications which have been explored to date. This introductory review is aimed primarily at control theorists unfamiliar with quantum mechanics, but should also be useful to quantum physicists interested in applications of feedback control. We explain how feedback in quantum systems differs from that in traditional classical systems, and how in certain cases the results from modern optimal control theory can be applied directly to quantum systems. In addition to noise reduction and stabilization, an important application of feedback in quantum systems is adaptive measurement, and we discuss the various applications of adaptive measurements. We finish by describing specific examples of the application of feedback control to cooling and state-preparation in nano-electro-mechanical systems and single trapped atoms.
Introduction In the preceding chapter we introduced quantum feedback control, devoting most space to the continuous feedback control of a localized quantum system. That is, we considered feeding back the current resulting from the monitoring of that system to control a parameter in the system Hamiltonian. We described feedback both in terms of Heisenberg-picture operator equations and in terms of the stochastic evolution of the conditional state. The former formulation was analytically solvable for linear systems. However, the latter could also be solved analytically for simple linear systems, and had the advantage of giving an explanation for how well the feedback could perform. In this chapter we develop further the theory of quantum feedback control using the conditional state. The state can be used not only as a basis for understanding feedback, but also as the basis for the feedback itself. This is a simple but elegant idea. The conditional state is, by definition, the observer's knowledge about the system. In order to control the system optimally, the observer should use this knowledge. Of course a very similar idea was discussed in Section 2.5 in the context of adaptive measurements. There, one's joint knowledge of a quantum system and a classical parameter was used to choose future measurements so as to increase one's knowledge of the classical parameter. The distinction is that in this chapter we consider state-based feedback to control the quantum system itself. This chapter is structured as follows. Section 6.2 introduces the idea of state-based feedback by discussing the first experimental implementation of a state-based feedback protocol to control a quantum state.
This review concerns quantum control results and methods that, over the years, have been used in the various operations involving quantum systems. Most of these methods have been originally developed outside the context of quantum physics, and, then, adapted to take into account the specificities of the various quantum physical platforms. Quantum control consists in designing adequate control signals required to efficiently manipulate systems conforming the laws of quantum mechanics in order to ensure the associated desired behaviours and performances. This work attempts to provide a thorough and self-contained introduction and review of the various quantum control theories and their applications. It encompasses issues spanning quantum control modelling, problem formulation, concepts of controllability, as well as a selection of the main control theories. Given the vastness of the field, we tried our best to be as concise as possible, and, for the details, the reader is pointed out to a profusion of references. The contents of the review are organized in the three major classes of control problems - open-loop control, closed-loop learning control, and feedback control - and, for each one of them, we present the main developments in quantum control theory. Finally, concerning the importance of attaining robustness and reliability due to inherent fragility of quantum systems, methods for quantum robust control are also surveyed.
Abstract. No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
Feedback plays an essential role in classical control theory, which has been investigated in depth. However, in quantum control theory, the study of feedback is still in its infancy. The paper continues to explore the fundamental characteristics of quantum feedback control for a typical coherent model based on the results of [3] by highlighting the significant differences between quantum feedback control and classical feedback control due to quantum measurement. The paper compares the control performance of quantum feedback control and quantum open-loop control in the cases with and without environment-induced decoherence. It concludes that in some sense, the quantum feedback control is still superior to the quantum open-loop control despite the complicated tradeoff in the feedback case, and further points out that how the quantum measurement-induced-collapse can play a positive role in the quantum feedback control.