On asymptotic stability of quantum trajectories and their Cesaro mean

We address the question of asymptotic stability of quantum trajectories, also referred as quantum filters. We determine the limit of the quantum fidelity between the true quantum trajectory and the estimated one. Under a purification and absolute continuity of initial conditions assumptions we show that this limit equals to one meaning that quantum filters are asymptotically stable. In the general case, under an identifiability, a spectral assumption and absolute continuity of initial conditions assumption we show that the limit of the Cesaro mean of the estimated trajectory is the same as the true one. (paper)