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 Bia{}ynicki-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.