On the generalization of linear least mean squares estimation to quantum   systems with non-commutative outputs
  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. 
Nina H. Amini, Zibo Miao, Yu Pan, Matthew R. James, and Hideo Mabuchi
math-ph math.MP math.OC quant-ph