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arXiv:2601.14023

The rate of purification of quantum trajectories

Maël Bompais, Nina H. Amini, Juan P. Garrahan, Mădălin Guţă

quant-phmath-phSubmitted Jan 20, 2026• Announced Jan 21, 2026• 0 citations• Revised Jul 21, 2026 (v2)
v1 · Jan 20, 2026v2 · Jul 21, 2026
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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 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.

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The rate of purification of quantum trajectories

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Purification of quantum trajectories

Hans Maassen, Burkhard Kümmerer

<!-- *** Custom HTML *** --> We prove that the quantum trajectory of repeated perfect measurement on a finite quantum system either asymptotically purifies, or hits upon a family of 'dark' subspaces, where the time evolution is unitary.

Lecture notes-monograph series•Jan 1, 2006•28 citations
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.
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.
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.
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.