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arXiv · 2601.14023

The rate of purification of quantum trajectories

Maël Bompais·Nina H. Amini·Juan P. Garrahan·Mădălin Guţă
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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, K\"{u}mmerer 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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Maël Bompais, Nina H. Amini, Juan P. Garrahan, Mădălin Guţă. 2026-01-20. The rate of purification of quantum trajectories. https://arxiv.org/abs/2601.14023

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