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 ${\chi }_{\lambda }^2$, with non-negative weights $w_f\left(\lambda \right)$ determined explicitly from the Stieltjes representation of the generator $f$. This identifies ${\chi }_{\lambda }^2$ as atomic building blocks for quantum $f$ divergences and yields closed-form $w_f$ for canonical choices (relative entropy/Kullback-Leibler, Hellinger/Bures, Rényi). By mapping ${\chi }_{\lambda }^2$ 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 results in quantum thermodynamics as applications.

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• Entropy production
• Quantum thermodynamics