Fidelity is a sub-martingale for discrete-time quantum filters

TL;DR

This paper proves that the fidelity between a quantum state and its filter estimate generally increases over time in a discrete setting, extending known results to mixed states and showing a sub-martingale property.

Contribution

It establishes that fidelity is a sub-martingale for discrete-time quantum filters, generalizing previous results to include mixed states and broader quantum filtering scenarios.

Findings

Fidelity between the true state and the filter estimate is a sub-martingale.

The result holds for both pure and mixed states.

Fidelity increases on average through the quantum filtering process.

Abstract

Fidelity is known to increase through any Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, fidelity is also increasing for any discrete-time quantum filter: fidelity between the density matrix of the underlying Markov chain and the density matrix of its associated quantum filter is a sub-martingale. This result is not restricted to pure states. It also holds true for mixed states.