The $alpha to 1$ Limit of the Sharp Quantum Renyi Divergence

Fawzi and Fawzi recently defined the sharp Renyi divergence, $D_alpha^#$, for $alpha in (1, infty)$, as an additional quantum Renyi 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 Renyi 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.