Uhlmann's theorem for relative entropies

TL;DR

This paper generalizes Uhlmann's theorem to a family of quantum divergences called $oldsymbol{ ext{α-Rényi relative entropies}}$, establishing a fundamental relation between extended states and their divergences for a range of $oldsymbol{ ext{α}}$ values.

Contribution

The work extends Uhlmann's theorem to $oldsymbol{ ext{α-Rényi relative entropies}}$ for $oldsymbol{ ext{α} ext{ in } [rac{1}{2}, ext{infinity}]}$, unifying several important quantum divergences.

Findings

Uhlmann's theorem is generalized to $ ext{α-Rényi relative entropies}$.

The generalization covers $ ext{fidelity}$, $ ext{relative entropy}$, and $ ext{max-relative entropy}$.

The results provide new insights into quantum state extensions and divergences.

Abstract

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 $\alpha$-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 $\alpha=\infty$, respectively.