Gauge freedoms in unravelled quantum dynamics: When do different continuous measurements yield identical quantum trajectories?

Calum A. Brown; Katarzyna Macieszczak; Robert L. Jack

doi:10.22331/q-2025-07-09-1787

Gauge freedoms in unravelled quantum dynamics: When do different continuous measurements yield identical quantum trajectories?

Calum A. Brown¹, Katarzyna Macieszczak², and Robert L. Jack^1,3

¹Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
²Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
³Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Published:	2025-07-09, volume 9, page 1787
Editor:	Thomas Elliott
Eprint:	arXiv:2503.09261v3
Doi:	https://doi.org/10.22331/q-2025-07-09-1787
Citation:	Quantum 9, 1787 (2025).

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Abstract

Quantum trajectories of a Markovian open quantum system arise from the back-action of measurements performed in the environment with which the system interacts. In this work, we consider counting measurements of quantum jumps, corresponding to different representations of the same quantum master equation. We derive necessary and sufficient conditions under which these different measurements give rise to the same unravelled quantum master equation, which governs the dynamics of the probability distribution over pure conditional states of the system. Since that equation uniquely determines the stochastic dynamics of a conditional state, we also obtain necessary and sufficient conditions under which different measurements result in identical quantum trajectories. We then consider the joint stochastic dynamics for the conditional state and the measurement record. We formulate this in terms of labelled quantum trajectories, and derive necessary and sufficient conditions under which different representations lead to equivalent labelled quantum trajectories, up to permutations of labels. As those conditions are generally stricter, we finish by constructing coarse-grained measurement records, such that equivalence of the corresponding partially-labelled trajectories is guaranteed by equivalence of the trajectories alone. These general results are illustrated by two examples that demonstrate permutation of labels, and equivalence of different quantum trajectories.

Popular summary

Quantum systems are highly sensitive to their surroundings and always interact with their environments to some extent. By making continuous measurements of the random outputs emitted by a system into its environment, one may reconstruct corresponding quantum trajectories of the system, which have an intrinsic dependence on the measurement protocol. However, there are some situations where different choices of measurement (or monitoring) protocol can give exactly the same ensemble of quantum trajectories. We have characterised when this happens, by means of conditions that are both necessary and sufficient.

We also consider labelled quantum trajectories, which include additional information about the measurement outcomes. For two measurement protocols to yield the same ensemble of labelled quantum trajectories, a more restrictive set of conditions is required, compared with the unlabelled case. However, the original conditions may be recovered if one makes an appropriate coarse-graining of the labels.

These results have implications for symmetry properties of quantum trajectories. The conditions that we derive also reinforce the special status of measurements that always reset the system to the same specific conditional state.

► BibTeX data

@article{Brown2025gaugefreedomsin,
  doi = {10.22331/q-2025-07-09-1787},
  url = {https://doi.org/10.22331/q-2025-07-09-1787},
  title = {Gauge freedoms in unravelled quantum dynamics: {W}hen do different continuous measurements yield identical quantum trajectories?},
  author = {Brown, Calum A. and Macieszczak, Katarzyna and Jack, Robert L.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1787},
  month = jul,
  year = {2025}
}

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Cited by

[1] Calum A Brown, Robert L Jack, and Katarzyna Macieszczak, "Weak unitary symmetries of open quantum dynamics: beyond quantum master equations", New Journal of Physics 28 2, 024505 (2026).

[2] Finn Schmolke, "Asymptotic Fate of Continuously Monitored Quantum Systems", arXiv:2506.10873, (2025).

[3] Calum A. Brown, Robert L. Jack, and Katarzyna Macieszczak, "Weak unitary symmetries of open quantum dynamics: beyond quantum master equations", arXiv:2506.19814, (2025).

[4] Eloy Piñol, Piotr Sierant, Dustin Keys, Romain Veyron, Miguel Angel García-March, Tanner Reese, Morgan W. Mitchell, Jan Wehr, and Maciej Lewenstein, "Distinguishing synthetic unravelings on quantum computers", arXiv:2601.19889, (2026).

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