Asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement

We investigate the asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement, extending previous results established for the ideal case of perfect measurement. We establish a necessary and sufficient condition ensuring the convergence of the estimated trajectory, initialized from an estimated state, to the true trajectory. This result is obtained assuming that the associated quantum channel is irreducible. Building on this, we prove the uniqueness of the invariant measure and demonstrate convergence toward this measure.