Exponential Selection and Feedback Stabilization of Invariant Subspaces of Quantum Trajectories

We show that quantum trajectories become exponentially fast supported by one of their minimal invariant subspaces. Exponential convergence is shown in expectation using Lyapunov techniques. The proof is based on an in-depth study of the identifiability of the probability distributions generated in the different subspaces. We furthermore introduce a feedback control strategy that allows for the targeted convergence towards a desired subspace. This convergence is also achieved at exponential speed.