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Peer-reviewedCC BYJournal of Physics A: Mathematical and Theoretical

The rate of purification of quantum trajectories

  • Maël Bompais
  • Nina Amini
  • Juan Garrahan
  • Madalin Guta
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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.

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Cite

BibTeXRIS
BibTeX
@article{bompais2026rate,
  title = {{The rate of purification of quantum trajectories}},
  author = {Maël Bompais and Nina Amini and Juan Garrahan and Madalin Guta},
  journal = {Journal of Physics A: Mathematical and Theoretical},
  year = {2026},
  month = sep,
  doi = {10.1088/1751-8121/aea45b},
  url = {https://hal.science/hal-05752358v1},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The rate of purification of quantum trajectories
AU  - Maël Bompais
AU  - Nina Amini
AU  - Juan Garrahan
AU  - Madalin Guta
PY  - 2026
DA  - 2026/09/08
JO  - Journal of Physics A: Mathematical and Theoretical
DO  - 10.1088/1751-8121/aea45b
UR  - https://hal.science/hal-05752358v1
AB  - 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.
SN  - 1751-8113
SN  - 1751-8121
ER  -