Wehrl entropy, Lieb conjecture and entanglement monotones

We propose to quantify the entanglement of pure states of $N \times N$ bipartite quantum system by defining its Husimi distribution with respect to $SU(N)\times SU(N)$ coherent states. The Wehrl entropy is minimal if and only if the pure state analyzed is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized (R{\'e}nyi) subentropies, are proved to be Schur--convex, so they are entanglement monotones and may be used as alternative measures of entanglement.