Coherently tracking the covariance matrix of an open quantum system

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.

Figure 1

A quantum plant and the corresponding coherent quantum observer in a cascade arrangement.

Figure 2

Plot of the behavior of a CMT observer with the solid line and the dash-dotted line corresponding to $\left|\right|e_{\mathrm{\Sigma }}\left(t\right){\left|\right|}_F$ and ${\nu }_{-}\left(t\right)$ (see Eq. (40) in [27]), respectively. The joint system is initialized in a Gaussian separable state.

Figure 3

Plot of the fidelity $F\left(t\right)$ (see Eq. (7) in [36]) as a function of time with the solid line and the dash-dotted line corresponding to a CMT observer and a MT observer, respectively.

Figure 4

Plot of an entanglement measure (see Eq. (40) in [27]) of the observer ${\nu }_{-}^o\left(t\right)$ (solid line) and the plant ${\nu }_{-}^p\left(t\right)$ (dash-dotted line) as a function of time.