The $\alpha \to 1$ Limit of the Sharp Quantum R\'enyi Divergence

TL;DR

This paper investigates the limit of the sharp quantum Rényi divergence as alpha approaches 1, establishing its convergence to the Belavkin-Staszewski relative entropy and introducing a new family of divergences called kringel divergences.

Contribution

The paper provides a new expression for the sharp divergence in terms of geometric Rényi divergence minimization, solving an open problem and introducing kringel divergences with proven properties.

Findings

Limit of sharp divergence as alpha approaches 1 equals Belavkin-Staszewski entropy.

Derived a new minimization expression for the divergence.

Introduced kringel divergences with data-processing inequality.

Abstract

Fawzi and Fawzi recently defined the sharp R\'enyi divergence, $D_\alpha^\#$, for $\alpha \in (1, \infty)$, as an additional quantum R\'enyi 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 R\'enyi 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.