Shannon's inequality for the Rényi entropy and an application to the uncertainty principle

Abstract

We consider Shannon's inequality for the Rényi entropy. The Rényi entropy corresponds to a nonlinear diffusion equation, for example, the porous-medium equation. Shannon's inequality implies the positivity of the relative Lyapunov functional for a nonlinear Fokker–Planck type equation. We derive Shannon's inequality for the Rényi entropy and give the other proof of this positivity. Moreover, by combining the Shannon and the logarithmic Sobolev inequalities for the Rényi entropy, we obtain the Heisenberg type uncertainty principle.