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 $\alpha \in \left [{{\frac {1}{2},\infty }}\right]$ , a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to $\alpha =\frac {1}{2}$ , $\alpha =1$ , and $\alpha =\infty $ , respectively.