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  • Letter
  • Open Access

Mutual information bounded by Fisher information

Wojciech Górecki1,*, Xi Lu2,*, Chiara Macchiavello1,3, and Lorenzo Maccone1,3

  • *These authors contributed equally to this work.
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Phys. Rev. Research 7, L022013 – Published 11 April, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L022013

Abstract

We derive a general upper bound to mutual information in terms of the Fisher information. The bound may be further used to derive a lower bound for the Bayesian quadratic cost. These two provide alternatives to other inequalities in the literature (e.g., the van Trees inequality) that are useful also for cases where the latter ones give trivial bounds. We then generalize them to the quantum case, where they bound the Holevo information in terms of the quantum Fisher information. We illustrate the usefulness of our bounds with a case study in quantum phase estimation. Here, they allow us to adapt to mutual information (useful for global strategies where the prior plays an important role), the known and highly nontrivial bounds for the Fisher information in the presence of noise. The results are also useful in the context of quantum communication, both for continuous and discrete alphabets.

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Outline

Information

Phys. Rev. Research 7, L022013– Published 11 April, 2025
  • Received 25 March 2024
  • Revised 17 July 2024
  • Accepted 10 February 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.

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