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Higher-order corrections to the quantum Cramér-Rao bound

J. R. Hervas1, A. Z. Goldberg2, A. S. Sanz1, Z. Hradil3, J. Řeháček3, and L. L. Sánchez-Soto1,4,5

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Phys. Rev. A 112, 022426 – Published 20 August, 2025

DOI: https://doi.org/10.1103/ltz3-163t

Abstract

Quantum Fisher information and the associated quantum Cramér-Rao bound (QCRB) are fundamental tools in frequentist quantum metrology, offering both analytical simplicity and practical precision limits for parameter estimation. The QCRB sets a lower bound on the mean square error (MSE) in the idealized limit of infinite measurement trials (ν→∞). Here we perform a systematic expansion in powers of 1/ν to refine MSE estimates in realistic, finite-resource scenarios. These corrections reveal differences between measurements that appear equally optimal under the QCRB. They also help to distinguish among multiple optimal state families for estimating an unknown unitary transformation. Additionally, we explore the Bhattacharyya bound and its quantum counterpart, which constrain these corrections. Our results are relevant for preasymptotic metrology, enabling optimized protocols with limited resources without reliance on numerical simulations.

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Phys. Rev. A 112, 022426– Published 20 August, 2025
  • Received 8 May 2025
  • Accepted 29 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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