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 ($2\leq N\leq 20$) entangled pure states, with respect to the SU(2)$^{\otimes N}$ coherent states. Our results show that for the large-$N$ ($N\gtrsim 10$) systems the Wehrl entropy of the highly chaotic entangled states (e.g., $2^{-N/2}\sum_{s_1,s_2,...,s_N=\uparrow,\downarrow}|s_1,s_2,...,s_N\rangle e^{-i\phi_{s_1,s_2,...,s_N}}$, with $\phi_{s_1,s_2,...,s_N}$ 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 A. Sugita (Jour. Phys. A 36, 9081 (2003)). 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 $N \rightarrow \infty$: 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.