Uhlmann's theorem for measured divergences

Uhlmann's theorem is a cornerstone of quantum information theory, stating that for any quantum state $\rho_{AB}$ and any state $\sigma_A$, there exists an extension $\sigma_{AB}$ of $\sigma_A$ such that the fidelity between $\rho_{AB}$ and $\sigma_{AB}$ equals the fidelity between their marginals $\rho_A$ and $\sigma_A$. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann's theorem to a broad class of measured $f$-divergences, including the measured $\alpha$-R\'enyi divergences for all $\alpha \geq 0$. The well-known Uhlmann's theorem for the fidelity corresponds to the special case $\alpha = \frac{1}{2}$. Since most commonly used quantum R\'enyi divergences, including the Petz and sandwiched R\'enyi divergences, cannot satisfy this property (except for degenerate cases). This fundamentally distinguishes measured $f$-divergences from other quantum divergences and highlights their unique mathematical structure.