Design and stability of discrete-time quantum filters with measurement imperfections

Abhinav Somaraju (1) , Igor Dotsenko (2, 3) , Clément Sayrin (2) , Pierre Rouchon (4)

1 Brussels Photonics Team
2 LKB (Lhomond) - Laboratoire Kastler Brossel
3 LKB [Collège de France] - Laboratoire Kastler Brossel
4 CAS - Centre Automatique et Systèmes

This work considers the theory underlying a discrete-time quantum filter recently used in a quantum feedback experiment. It proves that this filter taking into account decoherence and measurement errors is optimal and stable. We present the general framework underlying the filter and show that it corresponds to a recursive expression of the least-square optimal estimation of the density operator in the presence of measurement imperfections. By measurement imperfections, we mean in a very general sense unread measurement performed by the environment (decoherence) and active measurement performed by non-ideal detectors. However, we assume to know precisely all the Kraus operators and also the detection error rates. Such recursive expressions combine well known methods from quantum filtering theory and classical probability theory (Bayes' law). We then demonstrate that such a recursive filter is always stable with respect to its initial condition: the fidelity between the optimal filter state (when the initial filter state coincides with the real quantum state) and the filter state (when the initial filter state is arbitrary) is a sub-martingale.

Mots clés: Atomic measurements; Cavity resonators; Equations; Error analysis; Mathematical model; Photonics; Quantum mechanics

Domaines: Automatique / Robotique

Abhinav Somaraju, Igor Dotsenko, Clément Sayrin, Pierre Rouchon. Design and stability of discrete-time quantum filters with measurement imperfections. American Control Conference 2012, Jun 2012, Montreal, Canada. pp.5084 - 5089. ⟨hal-00738687⟩