Rigorous bounds for Renyi entropies of spherically symmetric potentials

TL;DR

This paper derives sharp upper bounds for Rènyi and Shannon entropies in quantum systems with spherically symmetric potentials, improving previous bounds by employing a maximization procedure with covariance constraints.

Contribution

It introduces improved bounds for Rènyi and Shannon entropies based on the second moment, explicitly accounting for radial and angular wavefunction parts.

Findings

Derived sharp upper bounds for entropies in spherically symmetric potentials.

Enhanced previous bounds using a covariance-constrained maximization approach.

Explicitly characterized contributions from radial and angular wavefunction components.

Abstract

The R\'enyi and Shannon entropies are information-theoretic measures which have enabled to formulate the position-momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density-functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment $\langle r^2\rangle$ for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the R\'enyi maximization procedure with a covariance constraint due to Costa, Hero and Vignat \cite{CosHer03}. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given.