Asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement
Nina H. Amini, Tristan Benoist, Maël Bompais, Clément Pellegrini
TL;DR
The paper extends quantum-trajectory theory from ideal perfect measurements to imperfect measurement scenarios modeled by Kraus maps, proving that irreducibility together with a contractivity condition (Cont) yields asymptotic stability of the true and estimated trajectories. It shows that Cont is necessary and sufficient for fidelity convergence to 1, and uses this stability to establish the existence and uniqueness of a stationary invariant measure for the trajectory process, along with convergence in distribution and an ergodic theorem for continuous observables. The results connect to known purification and ergodic theorems in the perfect-measurement case, while also providing a framework for handling detector imperfections and more general quantum instruments. The work thereby advances the understanding of long-time behavior and filtering properties of quantum systems under realistic measurement imperfections, without providing explicit convergence rates.
Abstract
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.