  This paper is concerned with quantum harmonic oscillators consisting of a
quantum plant and a directly coupled coherent quantum observer. We employ
discounted quadratic performance criteria in the form of exponentially weighted
time averages of second-order moments of the system variables.
Small-gain-theorem bounds are obtained for the back-action of the observer on
the covariance dynamics of the plant in terms of the plant-observer coupling. A
coherent quantum filtering (CQF) problem is formulated as the minimization of
the discounted mean square of an estimation error, with which the dynamic
variables of the observer approximate those of the plant. The cost functional
also involves a quadratic penalty on the plant-observer coupling matrix in
order to mitigate the back-action effect. For the discounted mean square
optimal CQF problem with penalized back-action, we establish first-order
necessary conditions of optimality in the form of algebraic matrix equations.
By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic
techniques, this set of equations is represented in a more explicit form for
equally dimensioned plant and observer. For a class of such observers with
autonomous estimation error dynamics, we obtain a solution of the CQF problem
and outline a homotopy method. The computation of the performance criteria and
the observer synthesis are illustrated by numerical examples.
