Asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement

TL;DR

This paper studies the long-term behavior of quantum trajectories with imperfect measurements, providing conditions for convergence and establishing ergodic properties under irreducible quantum channels.

Contribution

It extends previous results by establishing necessary and sufficient conditions for convergence of estimated quantum trajectories with imperfect measurements.

Findings

Convergence of estimated trajectories to true trajectories under irreducible channels

Uniqueness of the invariant measure for quantum trajectories

Demonstration of ergodic convergence in quantum systems

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.