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Australian Control Conference

Coherent quantum observers for n-level quantum systems

Zibo Miao, Luis A. Duffaut Espinosa, Ian R. Petersen, Valery Ugrinovskii, M. R. James

Published Nov 1, 2013• 10 citations
Publisher

Abstract

The purpose of this paper is to find coherent quantum observers for open n-level quantum systems. Recently, a class of linear coherent observers has been developed for quantum harmonic oscillators. However, open n-level quantum systems, which are characterized by bilinear quantum stochastic differential equations, escape the realm of the known theory. Therefore, in this paper we show how a coherent quantum observer is designed to track the corresponding n-level quantum plant asymptotically in the sense of mean values. We also discuss suboptimal quantum observers in the sense of least mean squares estimation.

DOI
10.1109/aucc.2013.6697291

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The purpose of this paper is to find coherent quantum observers for open n-level quantum systems. Recently, a class of linear coherent observers has been developed for quantum harmonic oscillators. However, open n-level quantum systems, which are characterized by bilinear quantum stochastic differential equations, escape the realm of the known theory. Therefore, in this paper we show how a coherent quantum observer is designed to track the corresponding n-level quantum plant asymptotically in the sense of mean values. We also discuss suboptimal quantum observers in the sense of least mean squares estimation.
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.
We previously developed coherent quantum observers for linear and bilinear quantum plants corresponding to open harmonic oscillators and finite level quantum systems respectively. In this paper we consider the case where two XY-coupled qubits are interacting with a common environment, and a class of coherent quantum observers is designed aimed at tracking variables characterising the two qubits including dipole-dipole product terms in the sense of mean values. The quantum observers proposed here have the potential to be employed in coherent feedback control design involving coupling between qubits.
This paper proposes a direct coupling coherent quantum observer for a quantum plant which consists of a two level quantum system. The quantum observer, which is a quantum harmonic oscillator, includes homodyne detection measurements. It is shown that the observer can be designed so that it does not affect the quantum variable of interest in the quantum plant and that measured output converges in a given sense to the plant variable of interest. Also, the plant variable of interest-observer system can be described by a set of linear quantum stochastic differential equations. A minimum variance unbiased estimator form of the Kalman filter is derived for linear quantum systems and applied to the direct coupled coherent quantum observer.
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.
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.
The paper is concerned with a problem of coherent (measurement-free) filtering for physically realizable (PR) linear quantum plants. The state variables of such systems satisfy canonical commutation relations and are governed by linear quantum stochastic differential equations, dynamically equivalent to those of an open quantum harmonic oscillator. The problem is to design another PR quantum system, connected unilaterally to the output of the plant and playing the role of a quantum filter, so as to minimize a mean square discrepancy between the dynamic variables of the plant and the output of the filter. This coherent quantum filtering (CQF) formulation is a simplified feedback-free version of the coherent quantum LQG control problem which remains open despite recent studies. The CQF problem is transformed into a constrained covariance control problem which is treated by using the Frechet differentiation of an appropriate Lagrange function with respect to the matrices of the filter.