On stability of quantum trajectories and their Cesaro mean

TL;DR

This paper investigates the stability of quantum trajectories and filters, demonstrating conditions under which the estimated quantum state converges to the true state, either in fidelity or in Cesaro mean.

Contribution

It establishes new stability results for quantum filters, including convergence in fidelity under purification and Cesaro mean convergence under spectral and identifiability assumptions.

Findings

Quantum fidelity between true and estimated trajectories converges to one under purification.

Cesaro mean of estimated trajectory converges to the true trajectory under certain spectral assumptions.

Quantum filters exhibit stability properties ensuring reliable state estimation.

Abstract

We address the question of stability of quantum trajectories, also referred as quantum filters}. We determine the limit of the quantum fidelity between the true quantum trajectory and the {estimated one}. Under a purification assumption we show that this limit equals to one meaning that quantum filters are stable. In the general case, under an identifiability and a spectral assumption we show that the limit of the Cesaro mean of the estimated trajectory is the same as the true one.