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

• URL: http://arxiv.org/abs/2102.06576v3
• Date: Mon, 8 Mar 2021 17:11:33 GMT
• Title: The $\alpha \to 1$ Limit of the Sharp Quantum R\'enyi Divergence
• Authors: Bjarne Bergh, Robert Salzmann and Nilanjana Datta
• Abstract summary: Fawzi and Fawzi recently defined the sharp R'enyi divergence, $D_alpha#$, for $alpha in (1, infty)$. 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.
• Score: 6.553031877558699
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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