Wehrl entropy, entropic uncertainty relations and entanglement

TL;DR

This paper explores the Wehrl entropy's role in formulating entropic uncertainty relations and quantifying entanglement in continuous variable systems, introducing a Wehrl mutual information as a measurable entanglement witness.

Contribution

It demonstrates the Wehrl-Lieb inequality's closeness to equality and introduces Wehrl mutual information as a new measurable entanglement witness for pure states.

Findings

Wehrl-Lieb inequality is closer to equality than Białynicki-Birula and Mycielski relation.

Wehrl mutual information can serve as a measurable entanglement witness.

Provides a lower bound on entanglement entropy.

Abstract

The Wehrl entropy is an entropy associated to the Husimi quasi-probability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Bia{\l}ynicki-Birula and Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how a Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.