发表机构
The George Washington University; Joint Center for Quantum Information and Computer Science and Joint Quantum Institute, NIST/University of Maryland; Microsoft Quantum（乔治华盛顿大学; 量子信息与计算联合中心及联合量子研究所，美国国家标准与技术研究院/马里兰大学; 微软量子）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对特殊三色断点图开发循环细化生成函数，结合拓扑解释得到卡特兰加权求值等结果，还将其应用于线性光学随机高斯态纠缠的雷尼熵展开。
AI 中文摘要
断点图起源于比较基因组学，其交替循环编码基因组之间的关系。我们研究一类与置换相关的受限三色断点图，并为两个极值族开发了循环细化的生成函数。这些族具有自然的拓扑解释：它们的典范曲面分别为球面和射影平面。球面族以非交叉构型为特征，而射影平面族则允许不同的分解，涉及一类特殊的莫比乌斯梯子。所得生成函数方程保留了完整的循环结构，但仍可进行大幅简化，这引出了明确的卡特兰加权求值、细化循环统计量的多项式结果，以及自然对角特化与非交叉树之间的联系。这两个拓扑族表现出截然不同的组合机制，为断点图的局部变换如何控制细化置换枚举提供了互补示例。作为进一步应用，相同的卡特兰加权和出现在线性光学中随机高斯态纠缠的渐近酉-魏因加滕展开中。这些组合结果确定了雷尼熵展开中涉及的主导项和常数阶矩多项式，其中射影平面贡献给出了有限尺寸常数修正。
英文摘要
Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural topological interpretation: their canonical surfaces are, respectively, the sphere and the projective plane. The spherical family is characterized by noncrossing configurations, while the projective-plane family admits a different decomposition involving a distinguished family of Möbius ladders. The resulting generating-function equations retain the full cycle structure but nevertheless admit substantial reductions. This leads to explicit Catalan-weighted evaluations, polynomiality results for refined cycle statistics, and a connection between a natural diagonal specialization and noncrossing trees. The two topological families exhibit markedly different combinatorial mechanisms, providing complementary examples of how local transformations of breakpoint graphs can control refined permutation enumerations. As a further application, the same Catalan-weighted sums arise in asymptotic unitary-Weingarten expansions for entanglement of random Gaussian states in linear optics. The combinatorial results determine the leading and constant-order moment polynomials entering the Rényi entropy expansion, with the projective-plane contribution giving the finite-size constant correction.
