Wehrl entropy and entanglement complexity of quantum spin systems

The Wehrl entropy of a quantum state is the Shannon entropy of its coherent-state distribution function, and remains non-zero even for pure states. We investigate the relationship between this entropy and the many-particle quantum entanglement, for N spin-1/2 particles. Explicitly, we numerically calculate the Wehrl entropy of various N-particle ( 2 <= N <= 20) entangled pure states, with respect to the SU(2) circle times N coherent states. Our results show that for the large-N ( N greater than or similar to 10) systems the Wehrl entropy of the highly chaotic entangled states (e.g. 2-N/2 & sum;s1,s2,& mldr;,sN=up arrow,down arrow |s1,s2,& mldr;,sN > e-i phi s1,s2,& mldr;,sN, with phi s1,s2,& mldr;,sN being random angles) are substantially larger than that of the very regular entangled states (e.g. the Greenberger-Horne-Zeilinger state). Therefore, the Wehrl entropy can reflect the complexity of the quantum entanglement of many-body pure states, as proposed by Sugita (2003 J. Phys. A: Math. Gen. 36 9081). In particular, the Wehrl entropy per particle (WEPP) can be used as a quantitative description of this entanglement complexity. Unlike other quantities used to evaluate this complexity (e.g. the degree of entanglement between a subsystem and the other particles), the WEPP does not necessitate the division of the total system into two subsystems. We further demonstrate that many-body pure entangled states can be classified into three types, based on the behavior of the WEPP in the limit N ->infinity: states approaching that of a maximally mixed state, those approaching completely separable pure states, and a third category lying between these two extremes. Each type exhibits fundamentally different entanglement complexity.