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 generalize Uhlmann’s theorem to $\alpha$-Rényi relative entropies for $\alpha \in [\frac{1}{2},\mathrm{\infty }]$, a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to $\alpha =\frac{1}{2}$, $\alpha =1$, and $\alpha =\mathrm{\infty }$, respectively.