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IEEE Conference on Decision and Control (CDC)

Asymptotic Stability of Non-Demolition Quantum Trajectories with Measurement Imperfections

Maël Bompais, Nina H. Amini

Published Dec 13, 2023• 3 citations
Publisher

Abstract

We consider the question of asymptotic stability of quantum trajectories undergoing quantum non-demolition imperfect measurement, that is to say the convergence of the estimated trajectory towards the true trajectory whose parameters and initial state are not necessarily known. We give conditions on the estimated initial state and regions of validity for the estimated parameters so that this convergence is ensured. We illustrate these results through numerical simulations on the physical example [1] and discuss the asymptotic stability for a more realistic general case where decoherence acts on the system. In this case, the evolution is described by new Kraus operators which do not satisfy the quantum non-demolition property.

DOI
10.1109/cdc49753.2023.10384090

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We consider the question of asymptotic stability of quantum trajectories undergoing quantum non-demolition imperfect measurement, that is to say the convergence of the estimated trajectory towards the true trajectory whose parameters and initial state are not necessarily known. We give conditions on the estimated initial state and regions of validity for the estimated parameters so that this convergence is ensured. We illustrate these results through numerical simulations on the physical example [1] and discuss the asymptotic stability for a more realistic general case where decoherence acts on the system. In this case, the evolution is described by new Kraus operators which do not satisfy the quantum non-demolition property.
We consider the question of asymptotic stability of quantum trajectories undergoing quantum non-demolition imperfect measurement, that is to say the convergence of the estimated trajectory towards the true trajectory whose parameters and initial state are not necessarily known. We give conditions on the estimated initial state and regions of validity for the estimated parameters so that this convergence is ensured. We illustrate these results through numerical simulations on the physical example [1] and discuss the asymptotic stability for a more realistic general case where decoherence acts on the system. In this case, the evolution is described by new Kraus operators which do not satisfy the quantum non-demolition property.
We investigate the asymptotic stability and ergodic properties of quantum trajectories under imperfect measurement, extending previous results established for the ideal case of perfect measurement. We establish a necessary and sufficient condition ensuring the convergence of the estimated trajectory, initialized from an estimated state, to the true trajectory. This result is obtained assuming that the associated quantum channel is irreducible. Building on this, we prove the uniqueness of the invariant measure and demonstrate convergence toward this measure.
Quantum trajectories describe the evolution of quantum systems undergoing indirect measurement. In the discrete-time setting, the state of the system is updated by applying Kraus operators according to the measurement results. From an experimental perspective, these Kraus operators can depend on unknown physical parameters p. An interesting and powerful method has been proposed in [1] to estimate a parameter in a finite set; however, complete results of convergence were lacking. This article fills this gap by rigorously showing the consistency of the method, whereas there was only numerical evidence so far. When the parameter belongs to a continuous set, we propose an algorithm to approach its value and show simulation results.
Abstract We address the question of asymptotic stability of quantum trajectories, also referred as quantum filters. We determine the limit of the quantum fidelity between the true quantum trajectory and the estimated one. Under a purification and absolute continuity of initial conditions assumptions we show that this limit equals to one meaning that quantum filters are asymptotically stable. In the general case, under an identifiability, a spectral assumption and absolute continuity of initial conditions assumption we show that the limit of the Cesaro mean of the estimated trajectory is the same as the true one.
In this paper, we present several refinements of the results in [1] concerning feedback stabilization of invariant subspaces of quantum trajectories. First, we consider a more realistic dynamics that accounts for potential measurement imperfections. Second, the strong controllability assumption can be replaced by a simpler and more natural condition whenever the channel is primitive on its minimal invariant subspaces. Finally, we show that the control strategy can be based on the estimated trajectory (starting from an estimated initial state) while still ensuring exponential stabilization towards the target subspace at the same rate, thereby demonstrating the robustness of the approach.
In this paper, we consider stochastic master equations describing the evolution of quantum spin-1/2 systems interacting with electromagnetic fields undergoing continuous-time measurements. We suppose that the initial states and the exact values of the physical parameters are unknown. We prove that the feedback stabilization strategy considered in [1] is robust to these imperfections. This is shown by studying the asymptotic behavior of the coupled stochastic master equations describing the evolutions of the actual state and the estimated one under appropriate assumptions on the feedback controller. We provide sufficient conditions on the feedback controller and a valid domain of estimated parameters which ensure exponential stabilization of the coupled system. Furthermore, our results allow us to answer positively to [2], [Conjecture 4.4] in the case of spin-1/2 systems with unknown initial states, even in presence of imprecisely known physical parameters.