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Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

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

Abhinav Somaraju, I. Dotsenko, C. Sayrin, Pierre Rouchon

Published Jun 1, 2012• 11 citations
Publisher

Abstract

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.

DOI
10.1109/acc.2012.6315442

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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.
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 this 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.
Fidelity is known to increase through a Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, the square of the fidelity is also increasing for a quantum filter: the square of the fidelity between the density matrix of the underlying Markov chain and the density matrix of its associated quantum filter is a super-martingale. Thus discrete-time quantum filters are stable processes and tend to forget their initial conditions.
The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.
The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.
The dynamics of many open quantum systems are described by stochastic master equations. In the discrete-time case, we recall the structure of the derived quantum filter governing the evolution of the density operator conditioned to the measurement outcomes. We then describe the structure of the corresponding particle quantum filters for estimating constant parameter and we prove their stability. In the continuous-time (diffusive) case, we propose a new formulation of these particle quantum filters. The interest of this new formulation is first to prove stability, and also to provide an efficient algorithm preserving, for any discretization step-size, positivity of the quantum states and parameter classical probabilities. This algorithm is tested on experimental data to estimate the detection efficiency for a superconducting qubit whose fluorescence field is measured using a heterodyne detector.
Fidelity is known to increase through any Kraus map: the fidelity between two density matrices is less than the fidelity between their images via a Kraus map. We prove here that, in average, fidelity is also increasing for discrete-time quantum filters attached to an arbitrary Kraus map: fidelity between the density matrix of the underlying Markov chain and the density matrix of the associated quantum filter is a sub-martingale. This result is not restricted to pure states. It also holds true for mixed states.