Limit Theorems for Disordered Quantum Trajectories

arXiv:2609.40096v1 Announce Type: cross Abstract: We study discrete-time quantum trajectories of a finite-dimensional system in a disordered environment, where the instrument applied at each step is determined by an invertible measure-preserving dynamical system. For general quantum instruments on a standard Borel outcome space, we construct the quenched probability law on sequences of measurement outcomes for a fixed realization of the disorder and a given initial state. Under a quenched Doeblin condition with a uniform deterministic minorization constant and an environment-dependent minorizing probability measure, we establish the existence and uniqueness of an equivariant family of random posterior laws and identify its barycenter as the unique dynamically stationary state for the associated non-selective channel cocycle. If the environment is ergodic, we prove a pathwise ergodic theorem valid for every measurable choice of initial state: for almost every realization of the disorder, time averages of posterior states converge almost surely under the corresponding quenched law to the disorder average of the unique dynamically stationary state. For an ergodic environment, we establish quenched variance asymptotics and, when the asymptotic variance is positive, a central limit theorem and standardized Berry--Esseen bounds for additive functionals generated by bounded measurable real-valued functions of the environment and posterior state. These results hold for every measurable initial state and almost every disorder realization. We then show that the stationary annealed joint process of instruments and posterior states inherits the $\alpha$-mixing of the instrument process, up to an exponentially decaying term. Finally, we provide classes of examples involving perfect and imperfect measurements, including models with continuous or finite outcome spaces, that satisfy the standing Doeblin condition.

Abstract:We study discrete-time quantum trajectories of a finite-dimensional system in a disordered environment, where the instrument applied at each step is determined by an invertible measure-preserving dynamical system. For general quantum instruments on a standard Borel outcome space, we construct the quenched probability law on sequences of measurement outcomes for a fixed realization of the disorder and a given initial state. Under a quenched Doeblin condition with a uniform deterministic minorization constant and an environment-dependent minorizing probability measure, we establish the existence and uniqueness of an equivariant family of random posterior laws and identify its barycenter as the unique dynamically stationary state for the associated non-selective channel cocycle. If the environment is ergodic, we prove a pathwise ergodic theorem valid for every measurable choice of initial state: for almost every realization of the disorder, time averages of posterior states converge almost surely under the corresponding quenched law to the disorder average of the unique dynamically stationary state. For an ergodic environment, we establish quenched variance asymptotics and, when the asymptotic variance is positive, a central limit theorem and standardized Berry--Esseen bounds for additive functionals generated by bounded measurable real-valued functions of the environment and posterior state. These results hold for every measurable initial state and almost every disorder realization. We then show that the stationary annealed joint process of instruments and posterior states inherits the $\alpha$-mixing of the instrument process, up to an exponentially decaying term. Finally, we provide classes of examples involving perfect and imperfect measurements, including models with continuous or finite outcome spaces, that satisfy the standing Doeblin condition.

Probability (math.PR); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

arXiv:2609.40096 [math.PR]

(or arXiv:2609.40096v1 [math.PR] for this version)

https://doi.org/10.48550/arXiv.2609.40096 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Limit Theorems for Disordered Quantum Trajectories

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