Exponentially fast selection of sectors for quantum trajectories beyond non demolition measurements

Abstract: We show that, in long time, quantum trajectories select an invariant subspace of the Hilbert space of the system being indirectly measured. This selection is shown to be exponentially fast in an almost sure sense and in average. This result generalizes a known result for non demolition measurements to arbitrary repeated indirect measurements. Our proofs are based on the introduction of a deformation of the original instrument to an equivalent one with a unique invariant state.