Uhlmann's theorem for measured divergences

Abstract

Uhlmann's theorem is a cornerstone of quantum information theory, stating that for any quantum state $\ddot{I}{}_{AB}$ and any state $\ddot{I}{ƒ}_A$, there exists an extension $\ddot{I}{ƒ}_{AB}$ of $\ddot{I}{ƒ}_A$ such that the fidelity between $\ddot{I}{}_{AB}$ and $\ddot{I}{ƒ}_{AB}$ equals the fidelity between their marginals $\ddot{I}{}_A$ and $\ddot{I}{ƒ}_A$. 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 $\hat{I}\pm$-Rényi divergences for all $\hat{I}\pm \ge 0$. The well-known Uhlmann's theorem for the fidelity corresponds to the special case $\hat{I}\pm =\frac{1}{2}$. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi 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.

v2: close to published version