Uhlmann's theorem for measured divergences

Abstract

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

Theorems & Definitions (14)

• Proposition 1
• proof
• Theorem 2
• proof : Proof of Theorem \ref{['thm: POVM equals PVM']}
• Theorem 3: Uhlmann's theorem for measured $f$-divergences
• proof
• Corollary 4: Uhlmann's theorem for measured Rényi divergences
• proof
• Corollary 5
• proof