The Estimation Lie Algebra Associated with Quantum Filters

TL;DR

This paper introduces a Lie algebra framework for quantum filters derived from Stratonovich calculus, exploring conditions for finite-dimensional quantum filters analogous to classical cases, especially under homodyne measurement.

Contribution

It develops a Lie algebra of super-operators for quantum filters and investigates conditions for finite-dimensional filters in quantum systems, extending classical filtering theory.

Findings

Lie algebra of super-operators isomorphic to system operators under homodyne measurement

Framework suggests criteria for the existence of finite-dimensional quantum filters

Extends classical nonlinear filtering geometric theory to quantum context

Abstract

We introduce the Lie algebra of super-operators associated with a quantum filter, specifically emerging from the Stratonovich calculus. In classical filtering, the analogue algebra leads to a geometric theory of nonlinear filtering which leads to well-known results by Brockett and by Mitter characterizing potential models where the curse-of-dimensionality may be avoided, and finite dimensional filters obtained. We discuss the quantum analogue to these results. In particular, we see that, in the case where all outputs are subjected to homodyne measurement, the Lie algebra of super-operators is isomorphic to a Lie algebra of system operators from which one may approach the question of the existence of finite-dimensional filters.