The $\alpha \to 1$ Limit of the Sharp Quantum Rényi Divergence

Abstract

Fawzi and Fawzi recently defined the sharp Rényi divergence, $D_{\alpha }^{\#}$, for $\alpha \in (1,\infty )$, as an additional quantum Rényi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit $\alpha \to 1$ of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric Rényi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call kringel divergences, and for which we prove various properties including the data-processing inequality.

11 pages, no figures. v3: Fixed a typo in Lemma 3 and simplified the argument in Lemma 11. v2: Restructured the proof of the main result and added clarifications to improve readability