Covariance matrix tracking coherent observers for linear quantum stochastic systems

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.