The Rényi Entropies Operate in Positive Semifields

Swiveled Rényi entropies

This paper introduces “swiveled Rényi entropies” as an alternative to the Rényi entropic quantities put forward in [Berta et al., Physical Review A 91, 022333 (2015)]. What distinguishes the swiveled Rényi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A conse...

Sharp Bounds Between Two Rényi Entropies of Distinct Positive Orders

Many axiomatic definitions of entropy, such as the Rényi entropy, of a random variable are closely related to the `α-norm of its probability distribution. This study considers probability distributions on finite sets, and examines the sharp bounds of the `β-norm with a fixed `α-norm, α 6= β, for n-dimensional probability vectors with an integer n ≥ 2. From the results, we derive the sharp bound...

Joint Range of Rényi Entropies

The exact range of the joined values of several Rényi entropies is determined. The method is based on topology with special emphasis on the orientation of the objects studied.

Strong converse theorems using Rényi entropies

Rényi entropies and nonlinear diffusion equations

Since their introduction in the early sixties [20], the Rényi entropies have been used in many contexts, ranging from information theory to astrophysics, turbulence phenomena and others. In this note, we enlighten the main connections between Rényi entropies and nonlinear diffusion equations. In particular, it is shown that these relationships allow to prove various functional inequalities in s...