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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.
We prove decomposition rules for quantum Renyi mutual information, generalising the relation $I(A:B) = H(A) - H(A|B)$ to inequalities between Renyi mutual information and Renyi entropy of different orders. The proof uses Beigis generalisation of Reis
We derive an inequality for the linear entropy, that gives sharp bounds for all finite dimensional systems. The derivation is based on generalised Bloch decompositions and provides a strict improvement for the possible distribution of purities for al
We evaluate the variance of coefficients of the characteristic polynomial of the quantum evolution operator for chaotic 4-regular quantum graphs (networks) via periodic orbits without taking the semiclassical limit. The variance of the n-th coefficie
This paper studies forward and reverse projections for the R{e}nyi divergence of order $alpha in (0, infty)$ on $alpha$-convex sets. The forward projection on such a set is motivated by some works of Tsallis {em et al.} in statistical physics, and th
We study the entropy production of the sandwiched Renyi divergence under the primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Renyi divergence of any o