Coherently tracking the covariance matrix of an open quantum system

TL;DR

This paper introduces covariance matrix tracking (CMT) coherent observers for quantum systems, enabling simultaneous estimation of mean and covariance, which enhances quantum feedback control and information extraction.

Contribution

It develops a new class of CMT observers, deriving conditions for their existence and demonstrating their ability to track quantum properties like entanglement.

Findings

CMT observers can track both mean and covariance of quantum systems.

More restrictions are needed for CMT compared to mean tracking observers.

Design examples show CMT's potential in quantum property estimation.

Abstract

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 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 covariance matrix of a quantum plant. We derive necessary and sufficient conditions for the existence of a CMT observer, and find 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 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; As they provide estimation of a system's quantum state without explicit use of the measurement postulate in their derivation.