Uhlmann's theorem for relative entropies

Giulia Mazzola; David Sutter; Renato Renner

doi:10.1109/TIT.2025.3591775

paper

Uhlmann's theorem for relative entropies

arXiv:2502.01749 · doi:10.1109/TIT.2025.3591775

Abstract

Uhlmann's theorem states that, for any two quantum states $I ¨ _{A B}$ and $I ¨ _{A}$ , there exists an extension $I ¨ _{A B}$ of $I ¨ _{A}$ such that the fidelity between $I ¨ _{A B}$ and $I ¨ _{A B}$ equals the fidelity between their reduced states $I ¨ _{A}$ and $I ¨ _{A}$ . In this work, we generalize Uhlmann's theorem to $I^\pm$ -RÃ©nyi relative entropies for $I^\pm \in [\frac{1}{2}, \infty]$ , a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to $I^\pm = \frac{1}{2}$ , $I^\pm = 1$ , and $I^\pm = \infty$ , respectively.

25 pages, added Appendix A about the relation between sufficiency and Uhlmann property