Wehrl Entropy and Entanglement Complexity of Quantum Spin Systems

Abstract

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 ($2\le N\le 20$) 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., $2^{-N/2}{\sum }_{s_1,s_2,...,s_N=\uparrow ,\downarrow }|s_1,s_2,...,s_N⟩e^{-i{\varphi }_{s_1,s_2,...,s_N}}$, with ${\varphi }_{s_1,s_2,...,s_N}$ 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 $N\to \infty$: 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.

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Cited by in corpus (1)

• Bayesian Interpretation of Husimi Function and Wehrl Entropy