Universal Thermodynamic Uncertainty Relation for Quantum $f-$Divergences
Domingos S. P. Salazar
TL;DR
The paper develops a universal framework linking quantum f-divergences to classical χ^2 statistics via the Nussbaum–Szkoła bridge, showing every Petz f-divergence is a nonnegative χ^2_λ mixture. This yields a universal thermodynamic uncertainty relation (TUR) D_f(ρ||σ) ≥ ∫_0^1 w_f(λ) h_λ(x,y,z) dλ, where h_λ is a closed-form fluctuation contrast and w_f(λ) is determined by the Stieltjes representation of f. The result unifies known quantum TURs (e.g., KL, Jeffreys, Hellinger, Rényi) and provides a method to derive new TURs for arbitrary operator-convex generators, with tightness proven via commuting binary states. The framework offers a modular approach to quantify how uncertainty (mean/variance) constrains quantum dissimilarities, with broad implications for quantum thermodynamics and nonequilibrium statistics.
Abstract
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.