Mutual information bounded by Fisher information

Górecki, Wojciech; Lu, Xi; Macchiavello, Chiara; Maccone, Lorenzo

doi:10.1103/PhysRevResearch.7.L022013

Mutual information bounded by Fisher information

Journal Article · Fri Apr 11 00:00:00 EDT 2025 · Physical Review Research

DOI:https://doi.org/10.1103/PhysRevResearch.7.L022013· OSTI ID:2555874

; ; Macchiavello, Chiara;

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.

Published by the American Physical Society 2025

View Journal Article

Research Organization:: Hangzhou, Zhejiang U.; INFN, Pavia; Pavia U.

Sponsoring Organization:: US Department of Energy; USDOE

Grant/Contract Number:: 89243024CSC000002; AC02-07CH11359

OSTI ID:: 2555874

Alternate ID(s):: OSTI ID: 2446950

Report Number(s):: FERMILAB-PUB-24-0515-SQMS-V; L022013

Journal Information:: Physical Review Research, Journal Name: Physical Review Research Journal Issue: 2 Vol. 7; ISSN PPRHAI; ISSN 2643-1564

Publisher:: American Physical SocietyCopyright Statement

Country of Publication:: United States

Language:: English

References (46)

Uncertainty relations for information entropy in wave mechanics Białynicki-Birula, Iwo; Mycielski, Jerzy Communications in Mathematical Physics, Vol. 44, Issue 2 https://doi.org/10.1007/BF01608825	journal	June 1975
Comparison Between the Cramer-Rao and the Mini-max Approaches in Quantum Channel Estimation Hayashi, Masahito Communications in Mathematical Physics, Vol. 304, Issue 3 https://doi.org/10.1007/s00220-011-1239-4	journal	April 2011
Jeffreys' prior is asymptotically least favorable under entropy risk Clarke, Bertrand S.; Barron, Andrew R. Journal of Statistical Planning and Inference, Vol. 41, Issue 1 https://doi.org/10.1016/0378-3758(94)90153-8	journal	August 1994
Squeezed states of light and their applications in laser interferometers Schnabel, Roman Physics Reports, Vol. 684 https://doi.org/10.1016/j.physrep.2017.04.001	journal	April 2017
The elusive Heisenberg limit in quantum-enhanced metrology Demkowicz-Dobrzański, Rafał; Kołodyński, Jan; Guţă, Mădălin Nature Communications, Vol. 3, Issue 1 https://doi.org/10.1038/ncomms2067	journal	January 2012
Advances in quantum metrology Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo Nature Photonics, Vol. 5, Issue 4 https://doi.org/10.1038/nphoton.2011.35	journal	March 2011
General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology Escher, B. M.; de Matos Filho, R. L.; Davidovich, L. Nature Physics, Vol. 7, Issue 5 https://doi.org/10.1038/nphys1958	journal	March 2011
Advances in photonic quantum sensing Pirandola, S.; Bardhan, B. R.; Gehring, T. Nature Photonics, Vol. 12, Issue 12 https://doi.org/10.1038/s41566-018-0301-6	journal	November 2018
Phase Resolution and Coherent Phase States Hall, Michael J. W. Journal of Modern Optics, Vol. 40, Issue 5 https://doi.org/10.1080/09500349314550841	journal	May 1993
Demonstrating Heisenberg-limited unambiguous phase estimation without adaptive measurements Higgins, B. L.; Berry, D. W.; Bartlett, S. D. New Journal of Physics, Vol. 11, Issue 7 https://doi.org/10.1088/1367-2630/11/7/073023	journal	July 2009
Efficient tools for quantum metrology with uncorrelated noise Kołodyński, Jan; Demkowicz-Dobrzański, Rafał New Journal of Physics, Vol. 15, Issue 7 https://doi.org/10.1088/1367-2630/15/7/073043	journal	July 2013
Super-additivity in communication of classical information through quantum channels from a quantum parameter estimation perspective Czajkowski, Jan; Jarzyna, Marcin; Demkowicz-Dobrzański, Rafał New Journal of Physics, Vol. 19, Issue 7 https://doi.org/10.1088/1367-2630/aa7a7b	journal	July 2017
A fibre bundle over manifolds of quantum channels and its application to quantum statistics Fujiwara, Akio; Imai, Hiroshi Journal of Physics A: Mathematical and Theoretical, Vol. 41, Issue 25 https://doi.org/10.1088/1751-8113/41/25/255304	journal	May 2008
Quantum metrology from a quantum information science perspective Tóth, Géza; Apellaniz, Iagoba Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 42 https://doi.org/10.1088/1751-8113/47/42/424006	journal	October 2014
Bayesian parameter estimation using Gaussian states and measurements Morelli, Simon; Usui, Ayaka; Agudelo, Elizabeth Quantum Science and Technology, Vol. 6, Issue 2 https://doi.org/10.1088/2058-9565/abd83d	journal	March 2021
Asymptotic Theory of Quantum Channel Estimation Zhou, Sisi; Jiang, Liang PRX Quantum, Vol. 2, Issue 1 https://doi.org/10.1103/PRXQuantum.2.010343	journal	March 2021
Physics-inspired forms of the Bayesian Cramér-Rao bound Tsang, Mankei Physical Review A, Vol. 102, Issue 6 https://doi.org/10.1103/PhysRevA.102.062217	journal	December 2020
Protocol for global multiphase estimation Chesi, Giovanni; Riccardi, Alberto; Rubboli, Roberto Physical Review A, Vol. 108, Issue 1 https://doi.org/10.1103/PhysRevA.108.012613	journal	July 2023
Number of bits returned by a quantum estimation Lu, Xi; Górecki, Wojciech; Macchiavello, Chiara Physical Review A, Vol. 110, Issue 3 https://doi.org/10.1103/PhysRevA.110.032405	journal	September 2024
Phase estimation without a priori phase knowledge in the presence of loss Kołodyński, Jan; Demkowicz-Dobrzański, Rafał Physical Review A, Vol. 82, Issue 5 https://doi.org/10.1103/PhysRevA.82.053804	journal	November 2010
Universality of the Heisenberg limit for estimates of random phase shifts Hall, Michael J. W.; Berry, Dominic W.; Zwierz, Marcin Physical Review A, Vol. 85, Issue 4 https://doi.org/10.1103/PhysRevA.85.041802	journal	April 2012
Discrete-to-continuous transition in quantum phase estimation Rządkowski, Wojciech; Demkowicz-Dobrzański, Rafał Physical Review A, Vol. 96, Issue 3 https://doi.org/10.1103/PhysRevA.96.032319	journal	September 2017
Bayesian Quantum Multiphase Estimation Algorithm Gebhart, Valentin; Smerzi, Augusto; Pezzè, Luca Physical Review Applied, Vol. 16, Issue 1 https://doi.org/10.1103/PhysRevApplied.16.014035	journal	July 2021
Bayesian estimation for quantum sensing in the absence of single-shot detection Dinani, Hossein T.; Berry, Dominic W.; Gonzalez, Raul Physical Review B, Vol. 99, Issue 12 https://doi.org/10.1103/PhysRevB.99.125413	journal	March 2019
Using Entanglement Against Noise in Quantum Metrology Demkowicz-Dobrzański, Rafal; Maccone, Lorenzo Physical Review Letters, Vol. 113, Issue 25 https://doi.org/10.1103/PhysRevLett.113.250801	journal	December 2014
Digital Quantum Estimation Hassani, Majid; Macchiavello, Chiara; Maccone, Lorenzo Physical Review Letters, Vol. 119, Issue 20 https://doi.org/10.1103/PhysRevLett.119.200502	journal	November 2017
Ancilla-Free Quantum Error Correction Codes for Quantum Metrology Layden, David; Zhou, Sisi; Cappellaro, Paola Physical Review Letters, Vol. 122, Issue 4 https://doi.org/10.1103/PhysRevLett.122.040502	journal	January 2019
π -Corrected Heisenberg Limit Górecki, Wojciech; Demkowicz-Dobrzański, Rafał; Wiseman, Howard M. Physical Review Letters, Vol. 124, Issue 3 https://doi.org/10.1103/PhysRevLett.124.030501	journal	January 2020
Multiple-Phase Quantum Interferometry: Real and Apparent Gains of Measuring All the Phases Simultaneously Górecki, Wojciech; Demkowicz-Dobrzański, Rafał Physical Review Letters, Vol. 128, Issue 4 https://doi.org/10.1103/PhysRevLett.128.040504	journal	January 2022
Information Geometry under Hierarchical Quantum Measurement Chen, Hongzhen; Chen, Yu; Yuan, Haidong Physical Review Letters, Vol. 128, Issue 25 https://doi.org/10.1103/PhysRevLett.128.250502	journal	June 2022
Using Adaptiveness and Causal Superpositions Against Noise in Quantum Metrology Kurdziałek, Stanisław; Górecki, Wojciech; Albarelli, Francesco Physical Review Letters, Vol. 131, Issue 9 https://doi.org/10.1103/PhysRevLett.131.090801	journal	August 2023
Quantum Metrology Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo Physical Review Letters, Vol. 96, Issue 1 https://doi.org/10.1103/PhysRevLett.96.010401	journal	January 2006
Does Nonlinear Metrology Offer Improved Resolution? Answers from Quantum Information Theory Hall, Michael J. W.; Wiseman, Howard M. Physical Review X, Vol. 2, Issue 4 https://doi.org/10.1103/PhysRevX.2.041006	journal	October 2012
Adaptive Quantum Metrology under General Markovian Noise Demkowicz-Dobrzański, Rafał; Czajkowski, Jan; Sekatski, Pavel Physical Review X, Vol. 7, Issue 4 https://doi.org/10.1103/PhysRevX.7.041009	journal	October 2017
Entropic uncertainty relations and their applications Coles, Patrick J.; Berta, Mario; Tomamichel, Marco Reviews of Modern Physics, Vol. 89, Issue 1 https://doi.org/10.1103/RevModPhys.89.015002	journal	February 2017
Quantum sensing Degen, C. L.; Reinhard, F.; Cappellaro, P. Reviews of Modern Physics, Vol. 89, Issue 3 https://doi.org/10.1103/RevModPhys.89.035002	journal	July 2017
Quantum metrology with nonclassical states of atomic ensembles Pezzè, Luca; Smerzi, Augusto; Oberthaler, Markus K. Reviews of Modern Physics, Vol. 90, Issue 3 https://doi.org/10.1103/RevModPhys.90.035005	journal	September 2018
Quantum Estimation for Quantum Technology Paris, Matteo G. A. International Journal of Quantum Information, Vol. 07, Issue supp01 https://doi.org/10.1142/S0219749909004839	journal	January 2009
Optimal Short-Term Population Coding: When Fisher Information Fails Bethge, M.; Rotermund, D.; Pawelzik, K. Neural Computation, Vol. 14, Issue 10 https://doi.org/10.1162/08997660260293247	journal	October 2002
Mutual Information, Fisher Information, and Population Coding Brunel, Nicolas; Nadal, Jean-Pierre Neural Computation, Vol. 10, Issue 7 https://doi.org/10.1162/089976698300017115	journal	October 1998
Fisher and Shannon Information in Finite Neural Populations Yarrow, Stuart; Challis, Edward; Seriès, Peggy Neural Computation, Vol. 24, Issue 7 https://doi.org/10.1162/NECO_a_00292	journal	July 2012
Mutual Information, Fisher Information, and Efficient Coding Wei, Xue-Xin; Stocker, Alan A. Neural Computation, Vol. 28, Issue 2 https://doi.org/10.1162/NECO_a_00804	journal	February 2016
Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data Butucea, Cristina; Guţă, Mădălin; Artiles, Luis The Annals of Statistics, Vol. 35, Issue 2 https://doi.org/10.1214/009053606000001488	journal	April 2007
Some Classes of Global Cramer-Rao Bounds Bobrovsky, B. Z.; Mayer-Wolf, E.; Zakai, M. The Annals of Statistics, Vol. 15, Issue 4 https://doi.org/10.1214/aos/1176350602	journal	December 1987
Squeezing metrology: a unified framework Maccone, Lorenzo; Riccardi, Alberto Quantum, Vol. 4 https://doi.org/10.22331/q-2020-07-09-292	journal	July 2020
Applications of the van Trees Inequality: A Bayesian Cramér-Rao Bound Gill, Richard D.; Levit, Boris Y. Bernoulli, Vol. 1, Issue 1/2 https://doi.org/10.2307/3318681	journal	March 1995

Similar Records

Photon-Counting Interferometry to Detect Geontropic Space-Time Fluctuations with GQuEST

Journal Article · Thu Feb 13 19:00:00 EST 2025 · Physical Review. X · OSTI ID:2516765

Entanglement Witness for Indistinguishable Electrons Using Solid-State Spectroscopy

Journal Article · Tue Mar 11 20:00:00 EDT 2025 · Physical Review. X · OSTI ID:2530671

Related Subjects

97 MATHEMATICS AND COMPUTING
Quantum information theory
Quantum metrology
Quantum parameter estimation

Mutual information bounded by Fisher information

Citation Formats

References (46)

Similar Records

Related Subjects