  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.
