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Australian Control Conference

Quantum observers for dissipative interacting qubits in a common environment

Zibo Miao, Yu Pan, M. R. James

Published Dec 28, 2015• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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.
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.
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.