量子自旋系统的Wehrl熵与纠缠复杂度
Wehrl Entropy and Entanglement Complexity of Quantum Spin Systems
AI总结：
研究N个自旋1/2粒子系统中Wehrl熵与多体量子纠缠的关系，发现每粒子Wehrl熵（WEPP）能定量反映纠缠复杂度且无需系统分割，并据此将多体纯纠缠态分为三类。
摘要：
量子态的Wehrl熵是其相干态分布函数的Shannon熵，即使对于纯态也保持非零。我们研究了N个spin-1/2粒子系统中该熵与多粒子量子纠缠之间的关系。具体而言，我们针对SU(2)$^{\otimes N}$相干态，数值计算了各种N粒子（$2\leq N\leq 20$）纠缠纯态的Wehrl熵。结果表明，对于大N（$N\gtrsim 10$）系统，高度混沌纠缠态（例如$2^{-N/2}\sum_{s_1,s_2,...,s_N=\uparrow,\downarrow}|s_1,s_2,...,s_N\rangle e^{-iϕ_{s_1,s_2,...,s_N}}$，其中$ϕ_{s_1,s_2,...,s_N}$为随机角度）的Wehrl熵显著大于非常规则纠缠态（例如Greenberger-Horne-Zeilinger态）的Wehrl熵。因此，正如A. Sugita（Jour. Phys. A 36, 9081 (2003)）所提出的，Wehrl熵能够反映多体纯态的量子纠缠复杂度。特别地，每粒子Wehrl熵（WEPP）可作为这种纠缠复杂度的定量描述。与用于评估该复杂度的其他量（例如子系统与其他粒子之间的纠缠度）不同，WEPP不需要将总系统划分为两个子系统。我们进一步证明，基于WEPP在$N \rightarrow \infty$极限下的行为，多体纯纠缠态可分为三类：趋近于最大混合态的状态、趋近于完全可分纯态的状态，以及介于这两个极端之间的第三类。每种类型表现出根本不同的纠缠复杂度。
英文摘要：
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., $2^{-N/2}\sum_{s_1,s_2,...,s_N=\uparrow,\downarrow}|s_1,s_2,...,s_N\rangle e^{-iϕ_{s_1,s_2,...,s_N}}$, with $ϕ_{s_1,s_2,...,s_N}$ 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.
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