Pointwise mutual information bounded by stochastic Fisher information

Pedro B. Melo

arxiv: 2603.12573 · v2 · pith:DY4WNS5Jnew · submitted 2026-03-13 · 🪐 quant-ph · cond-mat.stat-mech

Pointwise mutual information bounded by stochastic Fisher information

Pedro B. Melo This is my paper

Pith reviewed 2026-05-22 10:34 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords pointwise mutual informationstochastic Fisher informationFisher informationmutual informationquantum sensingstochastic dynamicsinformation bounds

0 comments

The pith

Pointwise mutual information is upper-bounded by stochastic Fisher information.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives general upper bounds on pointwise mutual information using stochastic Fisher information. These bounds are shown to average to known bounds on mutual information from Fisher information. The connection holds for general cases in both classical and quantum settings. This is relevant for understanding limits on information extraction from single realizations in stochastic dynamics and quantum sensing.

Core claim

We derive general upper bounds to pointwise mutual information in terms of stochastic Fisher information and show these bounds average to known results in the literature for bounds to mutual information in terms of Fisher information. These results deepen the connection between information-theoretical quantities and are shown to hold in general cases. We test the bounds in classical systems and provide a quantum generalization.

What carries the argument

Stochastic Fisher information, which is used to derive upper bounds on pointwise mutual information.

Load-bearing premise

The stochastic Fisher information is well-defined and finite for the probability distributions or quantum states under consideration, and the averaging recovers the known bounds without additional restrictions.

What would settle it

A numerical example or analytical counterexample in a simple probability distribution where pointwise mutual information exceeds the derived bound from stochastic Fisher information.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper derives pointwise bounds on mutual information using stochastic Fisher information that average to known results.

read the letter

The main takeaway is that the authors have derived general upper bounds to pointwise mutual information in terms of stochastic Fisher information. They show that these bounds average to the known bounds on mutual information in terms of Fisher information from the literature. This pointwise approach is what stands out as new. It builds on the established program of using Fisher information to bound mutual information but moves it to the level of individual points rather than averages. The paper does well by including tests in classical systems and providing a quantum generalization, which helps make the abstract claims more concrete. The focus on applications to stochastic dynamics and quantum sensing is appropriate, since those areas often involve single realizations where ensemble averages are not available. The soft spots are in the details of the derivation. The abstract does not include the explicit steps or the precise definition of stochastic Fisher information, so it is hard to judge if the bound is non-trivial or if it relies on particular assumptions about differentiability or support of the distributions. The averaging identity is presented as a consistency check, which is helpful, but verifying that the stochastic version is defined independently would strengthen the case. No major circularity is apparent from the description, but the full proof would clarify this. The work is aimed at researchers in quantum information and statistical physics interested in fundamental limits for information extraction. A reader familiar with Fisher information bounds would get value from seeing how the pointwise version is constructed and applied. I recommend sending this to peer review for a thorough check of the mathematics and the quantum extension.

Referee Report

0 major / 3 minor

Summary. The manuscript derives general upper bounds to pointwise mutual information in terms of stochastic Fisher information and shows that these bounds average to known results in the literature for bounds to mutual information in terms of Fisher information. The results are claimed to hold in general cases, with tests in classical systems and a quantum generalization provided. Applications to stochastic dynamics and quantum sensing are discussed.

Significance. If the central derivation holds, the work strengthens connections between pointwise information measures and Fisher information by providing bounds applicable to single realizations rather than ensembles. The averaging consistency with established mutual-information/Fisher-information bounds serves as an external anchor, and the quantum extension broadens potential utility in quantum sensing where single-shot limits are relevant.

minor comments (3)

[Introduction / main derivation section] The definition and independence of stochastic Fisher information (relative to the pointwise mutual information) should be stated explicitly early in the manuscript, e.g., in the section introducing the main inequality, to allow direct verification that the bound is non-tautological.
[Averaging / expectation step] The averaging identity that recovers the known mutual-information bounds should include an explicit interchange-of-integral step or dominated-convergence argument if the support or differentiability conditions are non-trivial.
[Numerical tests section] In the classical numerical tests, specify the exact distributions or parameter ranges used so that readers can reproduce the tightness of the pointwise bound.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for recommending minor revision. We appreciate the recognition that our bounds on pointwise mutual information recover known averaged results and that the quantum generalization may be relevant for single-shot quantum sensing. Since the report contains no specific major comments requiring point-by-point replies, we focus on preparing the minor revisions.

Circularity Check

0 steps flagged

No significant circularity; derivation anchored by external literature recovery

full rationale

The paper derives pointwise upper bounds on pointwise mutual information using stochastic Fisher information via standard definitions and inequalities. The central step is showing that these bounds, when averaged, recover known mutual-information/Fisher-information bounds from the literature. This averaging follows directly from the integral representation of mutual information and serves as an external consistency check rather than a definitional reduction. No self-citation is load-bearing for the core inequality, no parameter is fitted and relabeled as a prediction, and the construction does not reduce to its inputs by construction. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are identifiable from the abstract. The derivation presumably relies on standard definitions of mutual information and Fisher information from prior literature.

pith-pipeline@v0.9.0 · 5588 in / 1081 out tokens · 32628 ms · 2026-05-22T10:34:39.041427+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We derive general upper bounds to pointwise mutual information in terms of stochastic Fisher information... i(x, θ) = log p(x|θ)/p(x), ι(x, θ) = (∂θ log p(x|θ))², Λ₂(x, θ) = ι(x, θ)p(θ)² + ṗ(θ)² + 2∂θ log p(x|θ)ṗ(θ)p(θ)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

recovering the average bounds... I(X,Θ) ≤ log(∫ √(F(θ)f(θ)² + ḟ(θ)²) dθ) - ∫ p(θ) log f(θ) dθ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Demkowicz-Dobrza´ nski, W

R. Demkowicz-Dobrza´ nski, W. G´ orecki, and M. Gut ¸˘ a, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical53, 363001 (2020)

work page 2020
[2]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical53, 023001 (2019)

work page 2019
[3]

A. Rath, C. Branciard, A. Minguzzi, and B. Vermersch, Quantum fisher information from randomized measurements, Phys. Rev. Lett.127, 260501 (2021)

work page 2021
[4]

J. L. Beckey, M. Cerezo, A. Sone, and P. J. Coles, Variational quantum algorithm for estimating the quantum fisher information, Phys. Rev. Res.4, 013083 (2022)

work page 2022
[5]

Brunel and J.-P

N. Brunel and J.-P. Nadal, Mutual information, fisher information, and population coding, Neural computation 10, 1731 (1998)

work page 1998
[6]

G´ orecki, X

W. G´ orecki, X. Lu, C. Macchiavello, and L. Maccone, Mutual information bounded by fisher information, Phys. Rev. Res.7, L022013 (2025)

work page 2025
[7]

X. Lu, W. G´ orecki, C. Macchiavello, and L. Maccone, Number of bits returned by a quantum estimation, Phys. Rev. A110, 032405 (2024)

work page 2024
[8]

Wei and A

X.-X. Wei and A. A. Stocker, Mutual information, fisher information, and efficient coding, Neural computation 28, 305 (2016)

work page 2016
[9]

Fogelmark, M

K. Fogelmark, M. A. Lomholt, A. Irb¨ ack, and T. Ambj¨ ornsson, Fitting a function to time-dependent ensemble averaged data, Scientific Reports8, 6984 (2018)

work page 2018
[10]

Radaelli, G

M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, Fisher information of correlated stochastic processes, New Journal of Physics25, 053037 (2023)

work page 2023
[11]

Elouard and M

C. Elouard and M. H. Mohammady, Work, heat and entropy production along quantum trajectories, in Thermodynamics in the quantum regime: Fundamental Aspects and New Directions(Springer, 2019) pp. 363– 393

work page 2019
[12]

Manzano and R

G. Manzano and R. Zambrini, Quantum thermodynamics under continuous monitoring: A general framework, AVS Quantum Science4(2022)

work page 2022
[13]

Ferri-Cort´ es, J

M. Ferri-Cort´ es, J. A. Almanza-Marrero, R. L´ opez, R. Zambrini, and G. Manzano, Conditional fluctuation theorems and entropy production for monitored quantum systems under imperfect detection, Physical Review Research7, 013077 (2025)

work page 2025
[14]

Makkeh, A

A. Makkeh, A. J. Gutknecht, and M. Wibral, Introducing a differentiable measure of pointwise shared information, Physical Review E103, 032149 (2021)

work page 2021
[15]

C. K. Williams, On suspicious coincidences and pointwise mutual information, Neural Computation34, 2037 (2022)

work page 2037
[16]

Czy˙ z, F

P. Czy˙ z, F. Grabowski, J. E. Vogt, N. Beerenwinkel, and A. Marx, On the properties and estimation of pointwise mutual information profiles (2024), arXiv:2310.10240 [stat.ML]

work page arXiv 2024
[17]

P. B. Melo, S. M. Duarte Queir´ os, and W. A. M. Morgado, Stochastic thermodynamics of fisher information, Phys. Rev. E111, 014101 (2025)

work page 2025
[18]

P. B. Melo, F. Iemini, D. O. Soares-Pinto, S. M. D. Queir´ os, and W. A. M. Morgado, Thermodynamic interpretation to stochastic fisher information and single- trajectory speed limits, Phys. Rev. E112, 014126 (2025)

work page 2025
[19]

P. B. Melo, P. V. Paraguass´ u, S. M. D. Queir´ os, F. Iemini, M. Paternostro, and W. A. M. Morgado, Stochastic quantum information geometry and speed limits at the trajectory level (2026), arXiv:2601.12475

work page arXiv 2026
[20]

Albarelli, M

F. Albarelli, M. A. Rossi, D. Tamascelli, and M. G. Genoni, Restoring heisenberg scaling in noisy quantum metrology by monitoring the environment, Quantum2, 110 (2018)

work page 2018
[21]

de Neeve, A

B. de Neeve, A. V. Lebedev, V. Negnevitsky, and J. P. Home, Time-adaptive phase estimation, Phys. Rev. Res. 7, 023070 (2025)

work page 2025
[22]

Elouard, D

C. Elouard, D. A. Herrera-Mart´ ı, M. Clusel, and A. Auff` eves, The role of quantum measurement in stochastic thermodynamics, npj Quantum Information3, 9 (2017)

work page 2017
[23]

Sagawa and M

T. Sagawa and M. Ueda, Generalized jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104, 090602 (2010)

work page 2010
[24]

Sagawa and M

T. Sagawa and M. Ueda, Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics, Physical review letters109, 180602 (2012). 7 Appendix A: Proof of PMI bounds We begin by expressing the PMI using an arbitrary non-negative functionf(θ) such that suppf(θ)⊇suppp(θ). The PMI can be written algebraically as i(x, θ) = log ...

work page 2012
[25]

Proof of Theorem 1 If the parameter is guaranteed to lie in a finite-size set such that suppp(θ)⊆[a, b], we evaluate the integral over this specific domain by choosingf(θ) to be a boxcar function over [a, b], such thatf(θ) = 1 forθ∈[a, b] and 0 elsewhere. To evaluate the generalized bound without squaring the derivative of a step function, we apply the tr...

work page
[26]

Proof of Theorem 2 Theorem 2 follows directly from the generalized bound by choosing the arbitrary function to be the exact prior probability distribution,f(θ) =p(θ). Substitutingf(θ) =p(θ) into our definition of Λ(x, θ) immediately recovers Λ2(x, θ) =ι(x, θ)p(θ) 2 + ˙p(θ)2 + 2∂θ logp(x|θ) ˙p(θ)p(θ).(A9) The second term of the bound,−logf(θ), becomes−logp...

work page

[1] [1]

Demkowicz-Dobrza´ nski, W

R. Demkowicz-Dobrza´ nski, W. G´ orecki, and M. Gut ¸˘ a, Multi-parameter estimation beyond quantum fisher information, Journal of Physics A: Mathematical and Theoretical53, 363001 (2020)

work page 2020

[2] [2]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical53, 023001 (2019)

work page 2019

[3] [3]

A. Rath, C. Branciard, A. Minguzzi, and B. Vermersch, Quantum fisher information from randomized measurements, Phys. Rev. Lett.127, 260501 (2021)

work page 2021

[4] [4]

J. L. Beckey, M. Cerezo, A. Sone, and P. J. Coles, Variational quantum algorithm for estimating the quantum fisher information, Phys. Rev. Res.4, 013083 (2022)

work page 2022

[5] [5]

Brunel and J.-P

N. Brunel and J.-P. Nadal, Mutual information, fisher information, and population coding, Neural computation 10, 1731 (1998)

work page 1998

[6] [6]

G´ orecki, X

W. G´ orecki, X. Lu, C. Macchiavello, and L. Maccone, Mutual information bounded by fisher information, Phys. Rev. Res.7, L022013 (2025)

work page 2025

[7] [7]

X. Lu, W. G´ orecki, C. Macchiavello, and L. Maccone, Number of bits returned by a quantum estimation, Phys. Rev. A110, 032405 (2024)

work page 2024

[8] [8]

Wei and A

X.-X. Wei and A. A. Stocker, Mutual information, fisher information, and efficient coding, Neural computation 28, 305 (2016)

work page 2016

[9] [9]

Fogelmark, M

K. Fogelmark, M. A. Lomholt, A. Irb¨ ack, and T. Ambj¨ ornsson, Fitting a function to time-dependent ensemble averaged data, Scientific Reports8, 6984 (2018)

work page 2018

[10] [10]

Radaelli, G

M. Radaelli, G. T. Landi, K. Modi, and F. C. Binder, Fisher information of correlated stochastic processes, New Journal of Physics25, 053037 (2023)

work page 2023

[11] [11]

Elouard and M

C. Elouard and M. H. Mohammady, Work, heat and entropy production along quantum trajectories, in Thermodynamics in the quantum regime: Fundamental Aspects and New Directions(Springer, 2019) pp. 363– 393

work page 2019

[12] [12]

Manzano and R

G. Manzano and R. Zambrini, Quantum thermodynamics under continuous monitoring: A general framework, AVS Quantum Science4(2022)

work page 2022

[13] [13]

Ferri-Cort´ es, J

M. Ferri-Cort´ es, J. A. Almanza-Marrero, R. L´ opez, R. Zambrini, and G. Manzano, Conditional fluctuation theorems and entropy production for monitored quantum systems under imperfect detection, Physical Review Research7, 013077 (2025)

work page 2025

[14] [14]

Makkeh, A

A. Makkeh, A. J. Gutknecht, and M. Wibral, Introducing a differentiable measure of pointwise shared information, Physical Review E103, 032149 (2021)

work page 2021

[15] [15]

C. K. Williams, On suspicious coincidences and pointwise mutual information, Neural Computation34, 2037 (2022)

work page 2037

[16] [16]

Czy˙ z, F

P. Czy˙ z, F. Grabowski, J. E. Vogt, N. Beerenwinkel, and A. Marx, On the properties and estimation of pointwise mutual information profiles (2024), arXiv:2310.10240 [stat.ML]

work page arXiv 2024

[17] [17]

P. B. Melo, S. M. Duarte Queir´ os, and W. A. M. Morgado, Stochastic thermodynamics of fisher information, Phys. Rev. E111, 014101 (2025)

work page 2025

[18] [18]

P. B. Melo, F. Iemini, D. O. Soares-Pinto, S. M. D. Queir´ os, and W. A. M. Morgado, Thermodynamic interpretation to stochastic fisher information and single- trajectory speed limits, Phys. Rev. E112, 014126 (2025)

work page 2025

[19] [19]

P. B. Melo, P. V. Paraguass´ u, S. M. D. Queir´ os, F. Iemini, M. Paternostro, and W. A. M. Morgado, Stochastic quantum information geometry and speed limits at the trajectory level (2026), arXiv:2601.12475

work page arXiv 2026

[20] [20]

Albarelli, M

F. Albarelli, M. A. Rossi, D. Tamascelli, and M. G. Genoni, Restoring heisenberg scaling in noisy quantum metrology by monitoring the environment, Quantum2, 110 (2018)

work page 2018

[21] [21]

de Neeve, A

B. de Neeve, A. V. Lebedev, V. Negnevitsky, and J. P. Home, Time-adaptive phase estimation, Phys. Rev. Res. 7, 023070 (2025)

work page 2025

[22] [22]

Elouard, D

C. Elouard, D. A. Herrera-Mart´ ı, M. Clusel, and A. Auff` eves, The role of quantum measurement in stochastic thermodynamics, npj Quantum Information3, 9 (2017)

work page 2017

[23] [23]

Sagawa and M

T. Sagawa and M. Ueda, Generalized jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104, 090602 (2010)

work page 2010

[24] [24]

Sagawa and M

T. Sagawa and M. Ueda, Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics, Physical review letters109, 180602 (2012). 7 Appendix A: Proof of PMI bounds We begin by expressing the PMI using an arbitrary non-negative functionf(θ) such that suppf(θ)⊇suppp(θ). The PMI can be written algebraically as i(x, θ) = log ...

work page 2012

[25] [25]

Proof of Theorem 1 If the parameter is guaranteed to lie in a finite-size set such that suppp(θ)⊆[a, b], we evaluate the integral over this specific domain by choosingf(θ) to be a boxcar function over [a, b], such thatf(θ) = 1 forθ∈[a, b] and 0 elsewhere. To evaluate the generalized bound without squaring the derivative of a step function, we apply the tr...

work page

[26] [26]

Proof of Theorem 2 Theorem 2 follows directly from the generalized bound by choosing the arbitrary function to be the exact prior probability distribution,f(θ) =p(θ). Substitutingf(θ) =p(θ) into our definition of Λ(x, θ) immediately recovers Λ2(x, θ) =ι(x, θ)p(θ) 2 + ˙p(θ)2 + 2∂θ logp(x|θ) ˙p(θ)p(θ).(A9) The second term of the bound,−logf(θ), becomes−logp...

work page