发表机构
University of California, Irvine（加州大学尔湾分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出平面k-纯度作为多体量子系统的代价函数，通过四副本计算得到其精确哈尔统计，验证了相关公式，为AME存在区域外的多体纠缠提供几何感知基准。
AI 中文摘要
平面k-均匀态在环的每一个k位点区间上具有最大混度的约化态。我们引入平面k-纯度——即这些区间的平均纯度——作为该同时约束的忠实代价函数。对于局域维度为p的n个粒子的哈尔随机纯态，两个区间的联合纯度仅取决于它们的重叠。因此，详尽的四副本计算得到了完整的循环协方差核，以及对所有1≤k≤⌊n/2⌋的闭合方差。在平衡量子比特情形下，k=⌊n/2⌋且N=2ⁿ，当n为偶数时，方差渐近于20/(3nN²)；当n为奇数时，方差渐近于6/(nN²)。位点分解的置换表示给出了所有原始矩，并允许对偏度进行精确的有限求和评估。固定种子模拟验证了非平衡量子比特和qutrit（三态量子位）情形，而n=10的平衡量子比特模拟验证了奇偶性公式和有限尺寸偏度。平衡平面纯度与绝对平衡纯度具有相同的哈尔均值，但波动不同；在n=10时，平面方差约大2.83倍。因此，子系统关联虽然对每个单切割边缘分布不可见，但控制着几何相关切割的集体波动。结合普遍可达性和平衡时线性数量的区间约束，这为AME（绝对最大纠缠）存在区域之外的多体纠缠提供了一种几何感知基准。
英文摘要
Planar $k$-uniform states have maximally mixed reductions on every $k$-site interval of a ring. We introduce planar $k$-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of $n$ parties with local dimension $p$, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every $1\le k\le\lfloor n/2\rfloor$. In the balanced qubit case, $k=\lfloor n/2\rfloor$ and $N=2^n$, the variance is asymptotic to $20/(3nN^2)$ for even $n$ and $6/(nN^2)$ for odd $n$. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through $n=10$ validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately $2.83$ times larger at $n=10$. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.
