Optimal Unravellings for Feedback Control in Linear Quantum Systems

For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case, such as: given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state-based) control problems (the stationary Linear-Quadratic-Gaussian class), this question is a semi-definite program. Moreover, the answer also applies to Markovian (current-based) feedback.