Uhlmann's theorem for measured divergences

Abstract

Uhlmann's theorem is a cornerstone of quantum information theory, stating that for any quantum state ${\rho }_{AB}$ and any state ${\sigma }_A$, there exists an extension ${\sigma }_{AB}$ of ${\sigma }_A$ such that the fidelity between ${\rho }_{AB}$ and ${\sigma }_{AB}$ equals the fidelity between their marginals ${\rho }_A$ and ${\sigma }_A$. 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 $\alpha$-Rényi divergences for all $\alpha \ge 0$. The well-known Uhlmann's theorem for the fidelity corresponds to the special case $\alpha =\frac{1}{2}$. 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

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