Uhlmann’s theorem for relative entropies

Uhlmann's theorem states that for any two quantum states $\rho_{AB}$ and $\sigma_A$, there exists an extension $\sigma_{AB}$ of $\sigma_A$ such that the fidelity between $\rho_{AB}$ and $\sigma_{AB}$ equals the fidelity between their reduced states $\rho_A$ and $\sigma_A$. In this work, we extend Uhlmann's theorem to $\alpha$-R'enyi relative entropies for $\alpha \in [\frac{1}{2},\infty)$, a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to $\alpha=\frac{1}{2}$, $\alpha=1$, and $\alpha\to \infty$, respectively.