Universal Thermodynamic Uncertainty Relation for Quantum $f-$Divergences

Abstract

We show that any Petz $f$-divergence (where $f$ is operator convex) between quantum states admits a universal $\chi^2$-mixture representation: the distinguishability of $\rho$ from $\sigma$ is obtained as a positive superposition of quadratic contrasts $\chi^2_\lambda$, with nonnegative weights $w_f(\lambda)$ determined explicitly from the Stieltjes representation of the generator $f$. This identifies $\chi^2_\lambda$ 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 $\chi^2_\lambda$ into a classical Pearson $\chi^2$, 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.