Bures geodesics and quantum metrology

Dominique Spehner

doi:10.22331/q-2025-04-18-1715

Bures geodesics and quantum metrology

Dominique Spehner

Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile
Univ. Grenoble Alpes, CNRS, Institut Fourier and LPMMC, F-38000 Grenoble, France

Published:	2025-04-18, volume 9, page 1715
Editor:	Tommaso Macrì
Eprint:	arXiv:2308.08706v3
Doi:	https://doi.org/10.22331/q-2025-04-18-1715
Citation:	Quantum 9, 1715 (2025).

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Abstract

We study the geodesics on the manifold of mixed quantum states for the Bures metric. It is shown that these geodesics correspond to physical non-Markovian evolutions of the system coupled to an ancilla. Furthermore, we argue that geodesics lead to optimal precision in single-parameter estimation in quantum metrology. More precisely, if the unknown parameter $x$ is a phase shift proportional to the time parametrizing the geodesic, the estimation error obtained by processing the data of measurements on the system is equal to the smallest error that can be achieved from joint detections on the system and ancilla, meaning that there is no information loss on this parameter in the ancilla. This error can saturate the Heisenberg bound. Reciprocally, assuming that the system-ancilla output and input states are related by a unitary $e^{-i x H}$ with $H$ a $x$-independent Hamiltonian, we show that if the error obtained from measurements on the system is equal to the minimal error obtained from joint measurements on the system and ancilla then the system evolution is given by a geodesic. In such a case, the measurement on the system bringing most information on $x$ is $x$-independent and can be determined in terms of the intersections of the geodesic with the boundary of quantum states. These results show that geodesic evolutions are of interest for high-precision detections in systems coupled to an ancilla in the absence of measurements on the ancilla.

Featured image: Quantum circuit implementing a geodesic transformation for the estimation of a phase shift $x$, with an error reaching the Heisenberg scaling.

Collection 2024 – T1 – Quantum many-body systems out-of-equilibrium

Popular summary

Geodesics play a prominent role in classical mechanics and general relativity as they describe the trajectories of free particles and light. In this work, we study geodesics on the manifold of mixed states in quantum mechanics. We first show that these geodesics can be realized as non-Markovian quantum evolutions resulting from a coupling of the system with its environment. Then, we show that geodesics are of interest in quantum metrology. Quantum metrology aims at devising schemes that extract as precise as possible estimates of the parameters associated to a quantum system (probe). We prove that geodesic evolutions lead to the highest possible precision for the estimation of an unknown parameter (phase shift) in quantum probes coupled to their environment when measurements on the environment are not possible.

► BibTeX data

@article{Spehner2025buresgeodesics,
  doi = {10.22331/q-2025-04-18-1715},
  url = {https://doi.org/10.22331/q-2025-04-18-1715},
  title = {Bures geodesics and quantum metrology},
  author = {Spehner, Dominique},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1715},
  month = apr,
  year = {2025}
}

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