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We consider the question of asymptotic stability of quantum trajectoriesundergoing quantum non-demolition imperfect measurement, that is to say theconvergence of the estimated trajectory towards the true trajectory whoseparameters and initial state are not necessarily known. We give conditions onthe estimated initial state and regions of validity for the estimatedparameters so that this convergence is ensured. We illustrate these resultsthrough numerical simulations on the physical example [1] and discuss theasymptotic stability for a more realistic general case where decoherence actson the system. In this case, the evolution is described by new Kraus operatorswhich do not satisfy the quantum non-demolition property.

On Asymptotic Stability of Non-Demolition Quantum Trajectories with Measurement Imperfections