We prove that the Gärtner–Ellis generating function of probability distributions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest, including systems that are not translation invariant. The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. The derivation and analysis of the expansions of logarithms of Berezin integrals are the subject of the second part of the present work. Such technical results are also useful, for instance, in the context of quantum information theory, in the computation of relative entropy densities associated with fermionic Gibbs states, and in the theory of quantum normal fluctuations for weakly interacting fermion systems.We prove that the Gärtner–Ellis generating function of probability distributions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest, including systems that are not translation invariant. The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. The derivation and analysis of the expansions of logarithms of Berezin integrals are the subject of the second part of the present work. Such technical results are also useful, for instance, in the context of quantum information theory, in the computation of relative entropy densities associated with fermionic Gibbs states, and in the theory of quantum normal fluctuations for weakly interacting fermion systems.

arXiv:2105.13308Reviews in Mathematical PhysicsJun 11, 20216 citations

Quantum data processing equality

Tai-Hsuan Yang

The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.

quant-pharXiv:2609.33815Sep 29, 20260 citations

Note on Entropies of Quantum Dynamical Systems

Noboru Watanabe

Foundations of PhysicsMay 21, 201015 citations

Universal Thermodynamic Uncertainty Relation for Quantum $f-$ DivergencesUniversal Thermodynamic Uncertainty Relation for Quantum $f-$Divergences

Domingos S. P. Salazar

We show that any Petz $f$ -divergence (where $f$ is operator convex) between quantum states admits a universal $χ^2$ -mixture representation: the distinguishability of $ρ$ from $σ$ is obtained as a positive superposition of quadratic contrasts $χ^2_λ$ , with nonnegative weights $w_f(λ)$ determined explicitly from the Stieltjes representation of the generator $f$ . This identifies $χ^2_λ$ as atomic building blocks for quantum $f$ -divergences and yields closed-form $w_f$ for canonical choices (relative entropy/KL, Hellinger/Bures, R'{e}nyi). By mapping $χ^2_λ$ into a classical Pearson $χ^2$ , we leverage the Chapman-Robbins variational representation and obtain a tight and universal quantum thermodynamic uncertainty relation: any $f$ -divergence is lower bounded by a function of the statistics of quantum observables (mean and variance), reproducing previous and novel results in quantum thermodynamics as applications.We show that any Petz $f$-divergence (where $f$ is operator convex) between quantum states admits a universal $χ^2$-mixture representation: the distinguishability of $ρ$ from $σ$ is obtained as a positive superposition of quadratic contrasts $χ^2_λ$, with nonnegative weights $w_f(λ)$ determined explicitly from the Stieltjes representation of the generator $f$. This identifies $χ^2_λ$ as atomic building blocks for quantum $f$-divergences and yields closed-form $w_f$ for canonical choices (relative entropy/KL, Hellinger/Bures, R'{e}nyi). By mapping $χ^2_λ$ into a classical Pearson $χ^2$, we leverage the Chapman-Robbins variational representation and obtain a tight and universal quantum thermodynamic uncertainty relation: any $f$-divergence is lower bounded by a function of the statistics of quantum observables (mean and variance), reproducing previous and novel results in quantum thermodynamics as applications.

quant-pharXiv:2511.10817Nov 17, 20250 citations

Universal Thermodynamic Uncertainty Relation for Quantum

f-

Divergences

We show that any Petz

f

-divergence (where

f

is operator convex) between quantum states admits a universal

χ^2

-mixture representation: the distinguishability of

ρ

from

σ

is obtained as a positive superposition of quadratic contrasts

χ^2_λ

, with nonnegative weights

w_f(λ)

determined explicitly from the Stieltjes representation of the generator

f

. This identifies

χ^2_λ

as atomic building blocks for quantum

f

-divergences and yields closed-form

w_f

for canonical choices (relative entropy/KL, Hellinger/Bures, R'{e}nyi). By mapping

χ^2_λ

into a classical Pearson

χ^2

, we leverage the Chapman-Robbins variational representation and obtain a tight and universal quantum thermodynamic uncertainty relation: any

f

-divergence is lower bounded by a function of the statistics of quantum observables (mean and variance), reproducing previous and novel results in quantum thermodynamics as applications.