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arXiv:2409.18655

Dark Subspaces and Invariant Measures of Quantum Trajectories

Tristan Benoist, Clément Pellegrini, Anna Szczepanek

Published Sep 27, 2024• 0 citations
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Abstract

Quantum trajectories are Markov processes describing the evolution of a quantum system subject to indirect measurements. They can be viewed as place dependent iterated function systems or the result of products of dependent and non identically distributed random matrices. In this article, we establish a complete classification of their invariant measures. The classification is done in two steps. First, we prove a Markov process on some linear subspaces called dark subspaces, defined in (Maassen, Kümmerer 2006), admits a unique invariant measure. Second, we study the process inside the dark subspaces. Using a notion of minimal family of isometries from a reference space to dark subspaces, we prove a set of measures indexed by orbits of a unitary group is the set of ergodic measures of quantum trajectories.

DOI
10.48550/arxiv.2409.18655

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Quantum trajectories are Markov processes describing the evolution of a quantum system subject to indirect measurements. They can be viewed as place dependent iterated function systems or the result of products of dependent and non identically distributed random matrices. In this article, we establish a complete classification of their invariant measures. The classification is done in two steps. First, we prove a Markov process on some linear subspaces called dark subspaces, defined in (Maassen, Kümmerer 2006), admits a unique invariant measure. Second, we study the process inside the dark subspaces. Using a notion of minimal family of isometries from a reference space to dark subspaces, we prove a set of measures indexed by orbits of a unitary group is the set of ergodic measures of quantum trajectories.
Quantum trajectories are Markov chains modeling quantum systems subjected to repeated indirect measurements. Their stationary regime depends on what observables are measured on the probes used to indirectly measure the system. In this article we explore the properties of quantum trajectories when the choice of probe observable is randomized. The randomization induces some regularization of the quantum trajectories. We show that non-singular randomization ensures that quantum trajectories purify and therefore accept a unique invariant probability measure. We furthermore study the regularity of that invariant measure. In that endeavour, we introduce a new notion of ergodicity for quantum channels, which we call multiplicative primitivity. It is a priory stronger than primitivity but weaker than positivity improving. Finally, we compute some invariant measures for canonical quantum channels and explore the limits of our assumptions with several examples.
Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called "Stochastic Schrödinger Equations", which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a "purification" condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast towards this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.
We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process.
Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Under purification and irreducibility assumptions, these Markov processes admit a unique invariant measure - see Benoist et al. Probab. Theory Relat. Fields 2019. In this article we prove, finer limit theorems such as Law of Large Numbers (LLN), Functional Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation Principle. The proof of the LLN is based on Birkhoff's ergodic theorem and an analysis of harmonic functions. The other theorems are proved using martingale approximation of empirical sums.
Abstract We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen showed that such trajectories almost surely purify in the long run. An alternative, simple proof of this result is first provided using Lyapunov methods. The result is then strengthened by showing that purification occurs at an exponential rate in expectation, again via a Lyapunov approach. The quantum filter stability problem is then considered: two trajectories are propagated under the same measurement record—one from the true initial state, the other from an arbitrary initial guess—and it is shown that the estimated trajectory converges exponentially fast to the true one. This quantifies the rate at which information about the current state is progressively revealed through the measurement process.
Abstract We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCREs) with dynamically-defined transition probabilities; we call these disordered quantum trajectories . Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of Kümmerer and Maassen to the present disordered framework.