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IEEE Conference on Decision and Control (CDC)

On the robustness of feedback stabilization for discrete-time quantum trajectories beyond QND measurements

Maël Bompais, Nina H. Amini

Published Dec 9, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1109/cdc57313.2025.11312429

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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.
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.
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This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.