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

*Correspondence: [email protected] 1Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA 2CNRS, Laboratoire des signaux et systèmes (L2S) CentraleSupélec, 3 rue Joliot-Curie, Gif-sur-Yvette, 91192, France Full list of author information is available at the end of the article 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.