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

Nina H. Amini (1, 2, 3) , Zibo Miao (4) , Yu Pan (4) , Matthew R. James (4) , Hideo Mabuchi (5)

1 L2S - Laboratoire des signaux et systèmes
2 CNRS - Centre National de la Recherche Scientifique
3 CentraleSupélec
4 ANU - Australian National University - Department of engineering
5 Ginzton Laboratory [Stanford]

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.

Keywords: linear quantum stochastic differential equations (QSDEs); quantum noises; Kalman filtering; physical realizability; linear least mean squares estimators; non-commutative outputs; coherent observers

Domains: Mathematical Physics [math-ph]; Optimization and Control [math.OC]; Quantum Physics [quant-ph]

Nina H. Amini, Zibo Miao, Yu Pan, Matthew R. James, Hideo Mabuchi. On the generalization of linear least mean squares estimation to quantum systems with non-commutative outputs. European physical journal quantum technology, 2015, 2 (1), http://link.springer.com/article/10.1140%2Fepjqt%2Fs40507-015-0028-y. ⟨10.1140/epjqt/s40507-015-0028-y⟩. ⟨hal-01267109⟩