On Renyi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems

TL;DR

This paper explores the use of Renyi entropies to characterize the shape and extension of quantum wave functions in disordered systems, focusing on their application to phase space representations like the Husimi distribution.

Contribution

It introduces a method to analyze the shape and extension of continuous quantum distributions using Renyi entropies, even when these measures diverge.

Findings

Renyi entropy differences are finite and useful for distribution shape analysis.

Husimi distribution localization can be reconstructed from position and momentum marginals.

Numerical simulations support the theoretical predictions in disordered quantum systems.

Abstract

We discuss some properties of the generalized entropies, called Renyi entropies and their application to the case of continuous distributions. In particular it is shown that these measures of complexity can be divergent, however, their differences are free from these divergences thus enabling them to be good candidates for the description of the extension and the shape of continuous distributions. We apply this formalism to the projection of wave functions onto the coherent state basis, i.e. to the Husimi representation. We also show how the localization properties of the Husimi distribution on average can be reconstructed from its marginal distributions that are calculated in position and momentum space in the case when the phase space has no structure, i.e. no classical limit can be defined. Numerical simulations on a one dimensional disordered system corroborate our expectations.