Wehrl entropy, entropic uncertainty relations, and entanglement

Wehrl entropy is an entropy associated with the Husimi quasiprobability 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 Bialynicki-Birula-Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how 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.

Automated summary

The Wehrl entropy is discussed in relation to entropic uncertainty relations and quantification of entanglement in continuous variables. It is shown that the Wehrl-Lieb inequality is closer to equality than the usual Bialynicki-Birula-Mycielski entropic uncertainty relation almost everywhere. Additionally, the Wehrl mutual information is demonstrated to be useful in obtaining a measurable perfect witness for pure state bipartite entanglement and providing a lower bound on the entanglement entropy.