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arXiv:1509.05124 · IEEE Conference on Decision and Control (CDC)

Pole placement design for quantum systems via coherent observers

Zibo Miao, M. R. James, Valery Ugrinovskii

Published Dec 1, 2015• 7 citations
arXivHTMLPDF

Abstract

We previously extended Luenberger's approach for observer design to the quantum case, and developed a class of coherent observers which tracks linear quantum stochastic systems in the sense of mean values. In light of the fact that the Luenberger observer is commonly and successfully applied in classical control, it is interesting to investigate the role of coherent observers in quantum feedback. As the first step in exploring observer-based coherent control, in this paper we study pole-placement techniques for quantum systems using coherent observers, and in such a fashion, poles of a closed-loop quantum system can be relocated at any desired locations. In comparison to classical feedback control design incorporating the Luenberger observer, here direct coupling between a quantum plant and the observer-based controller are allowed to enable a greater degree of freedom for the design of controller parameters. A separation principle is presented, and we show how to design the observer and feedback independently to be consistent with the laws of quantum mechanics. The proposed scheme is applicable to coherent feedback control of quantum systems, especially when the transient dynamic response is of interest, and this issue is illustrated in an example.

DOI
10.1109/cdc.2015.7403128

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We previously extended Luenberger's approach for observer design to the quantum case, and developed a class of coherent observers which tracks linear quantum stochastic systems in the sense of mean values. In light of the fact that the Luenberger observer is commonly and successfully applied in classical control, it is interesting to investigate the role of coherent observers in quantum feedback. As the first step in exploring observer-based coherent control, in this paper we study pole-placement techniques for quantum systems using coherent observers, and in such a fashion, poles of a closed-loop quantum system can be relocated at any desired locations. In comparison to classical feedback control design incorporating the Luenberger observer, here direct coupling between a quantum plant and the observer-based controller are allowed to enable a greater degree of freedom for the design of controller parameters. A separation principle is presented, and we show how to design the observer and feedback independently to be consistent with the laws of quantum mechanics. The proposed scheme is applicable to coherent feedback control of quantum systems, especially when the transient dynamic response is of interest, and this issue is illustrated in an example.
The purpose of this paper is to investigate the extension of the Luenberger observer design approach to linear quantum stochastic systems to obtain coherent quantum observers. We show how a physically realizable quantum observer can be designed, consistent with the laws of quantum mechanics. The quantum observer has the property that the mean values of the observer variables asymptotically track the corresponding mean values of the plant. In addition, we discuss entanglement of the joint plant-observer state, as well as the implications of the no-cloning theorem. Several examples are considered.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
It is becoming apparent that coherent feedback control of quantum systems has significant advantages over measurement-based control, but not much progress has been made on coherent estimators. In our previous work, a class of coherent quantum observers was developed, which tracks linear quantum stochastic systems in the sense of mean values. However, in the majority of quantum technologies, apart from mean value quantities, correlations and variances are critical elements because they are connected with energy and entanglement. Therefore, in this paper, we create a covariance matrix tracking (CMT) coherent observer that enables the asymptotic tracking of both the mean and covariance matrix of a quantum system. The existence of CMT observers is analyzed, and a sufficient and necessary condition is provided to ensure the physical realization of CMT observers to be consistent with the laws of quantum mechanics. The performance of a CMT vs. mean tracking coherent observer is compared in an illustrated example, and we demonstrate for the first time coherent tracking of entanglement. These results shed light on observer-based coherent control design.
This paper is concerned with the coherent quantum linear-quadratic-Gaussian control problem of minimising an infinite-horizon mean square cost for a measurement-free field-mediated interconnection of a quantum plant with a stabilising quantum controller. The plant and the controller are multimode open quantum harmonic oscillators, governed by linear quantum stochastic differential equations and coupled to each other and the external multichannel bosonic fields in the vacuum state. We discuss an interplay between the quantum physical realizability conditions and the Luenberger structure associated with the classical separation principle. This leads to a quadratic constraint on the controller gain matrices, which is formulated in the framework of a swapping transformation for the conjugate positions and momenta in the canonical representation of the controller variables. For the class of coherent quantum controllers with the Luenberger dynamics, we obtain first-order necessary conditions of optimality in the form of algebraic equations, involving a matrix-valued Lagrange multiplier.
In this paper we extend our previous research on coherent observer-based pole placement approach to study the synthesis of robust decoherence-free (DF) modes for linear quantum passive systems, which is aimed at preservation of quantum information. In particular, DF modes can be generated by placing the poles on the imaginary axis via a coherent feedback design scheme, and these modes can further be simultaneously made robust against perturbations to the system parameters by minimizing the condition number associated with imaginary poles. We develop explicit algebraic conditions for the existence of such a coherent quantum controller, with the corresponding deign procedure provided. Examples are given to illustrate the process of tuning the DF modes towards perfect robustness via the proposed pole placement technique.
Coherent feedback control of quantum systems has demonstrable advantages over measurement-based control, but so far there has been little work done on coherent estimators and more specifically coherent observers. Coherent observers are input the coherent output of a specified quantum plant and are designed such that some subset of the observer's and plant's expectation values converge in the asymptotic limit. We previously developed a class of mean tracking (MT) observers for open harmonic oscillators that only converged in mean position and momentum; here we develop a class of covariance matrix tracking (CMT) coherent observers that track both the mean and the covariance matrix of a quantum plant. We derive necessary and sufficient conditions for the existence of a CMT observer and find that there are more restrictions on a CMT observer than there are on a MT observer. We give examples where we demonstrate how to design a CMT observer and show that it can be used to track properties like the entanglement of a plant. As the CMT observer provides more quantum information than a MT observer, we expect it will have greater application in future coherent feedback schemes mediated by coherent observers. Investigation of coherent quantum estimators and observers is important in the ongoing discussion of quantum measurement because they provide an estimation of a system's quantum state without explicit use of the measurement postulate in their derivation.