Design and Stability of Discrete-Time Quantum Filters with Measurement Imperfections

TL;DR

This paper develops a stable, optimal discrete-time quantum filter that accounts for measurement imperfections and decoherence, combining quantum filtering and classical probability methods for improved quantum state estimation.

Contribution

It introduces a recursive quantum filtering framework that handles measurement errors and decoherence, proving its optimality and stability in quantum feedback systems.

Findings

The filter is proven to be stable with respect to initial conditions.

Recursive expressions are derived combining quantum filtering and classical Bayes' law.

The filter achieves optimal estimation of the quantum state despite measurement imperfections.

Abstract

This work considers the theory underlying a discrete-time quantum filter recently used in a quantum feedback experiment. It proves that this filter taking into account decoherence and measurement errors is optimal and stable. We present the general framework underlying this filter and show that it corresponds to a recursive expression of the least-square optimal estimation of the density operator in the presence of measurement imperfections. By measurement imperfections, we mean in a very general sense unread measurement performed by the environment (decoherence) and active measurement performed by non-ideal detectors. However, we assume to know precisely all the Kraus operators and also the detection error rates. Such recursive expressions combine well known methods from quantum filtering theory and classical probability theory (Bayes' law). We then demonstrate that such a recursive filter is always stable with respect to its initial condition: the fidelity between the optimal filter state (when the initial filter state coincides with the real quantum state) and the filter state (when the initial filter state is arbitrary) is a sub-martingale.