On stability of continuous-time quantum-filters

TL;DR

This paper proves that the fidelity between a quantum state and its filter in continuous-time stochastic master equations is a sub-martingale, indicating stability but not guaranteeing convergence.

Contribution

It generalizes the stability result of quantum filters to non-pure states using fidelity as a measure.

Findings

Fidelity is a sub-martingale in the described setting.

Stability of quantum filtering processes is established.

Asymptotic convergence of filters is not guaranteed.

Abstract

We prove that the fidelity between the quantum state governed by a continuous time stochastic master equation driven by a Wiener process and its associated quantum-filter state is a sub-martingale. This result is a generalization to non-pure quantum states where fidelity does not coincide in general with a simple Frobenius inner product. This result implies the stability of such filtering process but does not necessarily ensure the asymptotic convergence of such quantum-filters.