On the generalization of linear least mean squares estimation to quantum systems with non-commutative outputs

TL;DR

This paper extends linear least mean squares estimation to quantum systems with non-commutative outputs, formulating a quantum filtering problem and identifying conditions for physical realizability.

Contribution

It generalizes classical Kalman filtering to quantum systems and derives necessary and sufficient conditions for physically realizable quantum estimators.

Findings

Formulation of a quantum least mean squares estimation problem.

Derivation of conditions for physical realizability of quantum filters.

Extension of classical filtering theory to non-commutative quantum outputs.

Abstract

The purpose of this paper is to study the problem of generalizing the Belavkin-Kalman filter to the case where the classical measurement signal is replaced by a fully quantum non-commutative output signal. We formulate a least mean squares estimation problem that involves a non-commutative system as the filter processing the non-commutative output signal. We solve this estimation problem within the framework of non-commutative probability. Also, we find the necessary and sufficient conditions which make these non-commutative estimators physically realizable. These conditions are restrictive in practice.