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Lecture notes-monograph series

Purification of quantum trajectories

Hans Maassen, Burkhard Kümmerer

Published Jan 1, 2006• 28 citations
Publisher

Abstract

<!-- *** 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.
DOI
10.1214/lnms/1196285826

Related papers

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.

Institute of Mathematical Statistics eBooks•Jan 1, 2006•28 citations

Purification of quantum trajectories

Hans Maassen, Burkhard Kümmerer

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.

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The rate of purification of quantum trajectories

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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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Purification of quantum trajectories in infinite dimensions

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In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems ( Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 06(02), 223-243, 2003 and IMS Lectures Notes-Monograph Series , 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.

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The rate of purification of quantum trajectories

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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 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.

Journal of Physics A Mathematical and Theoretical•Sep 8, 2026•0 citations

Asymptotic Purification of Quantum Trajectories under Disordered Generalized Measurements

Owen Ekblad, Eloy Moreno-Nadales, Lubashan Pathirana +1 more

arXiv:2404.03168•Annales Henri Poincaré•Nov 7, 2025•0 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.
In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems ( Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 06(02), 223-243, 2003 and IMS Lectures Notes-Monograph Series , 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.
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.