Coherent-Classical Estimation for Linear Quantum Systems

We consider a coherent-classical estimation scheme for a class of linear quantum systems, where the estimator is a mixed quantum-classical system that may or may not involve coherent feedback. We show that when the quantum plant or the quantum part of the estimator (coherent controller) is an annihilation operator only system, coherent-classical estimation without coherent feedback can provide no improvement over purely-classical estimation. Otherwise, it can be better for certain homodyne detector angles, although it is inferior for the optimal choice of homydne angle. Moreover, we show that coherentclassical estimation with coherent feedback is as good as classical-only estimation, when both the plant and the controller are annihilation operator only systems. Otherwise, it can be superior, and if so, certainly for the best choice of homodyne angle.