AI 中文总结
本文解决了悬而未决数十年的六阶复阿达马矩阵分类问题，通过有限关联分类方法，补充了Szöllősi扩张方法的全局步骤，为六阶阿达马矩阵及相关领域研究提供了严谨框架。
AI 中文摘要
复阿达马矩阵编码了完全平衡的酉变换，它们是相互无偏量子测量和多光子干涉测量的基础。其分类在五阶时已完成，但六阶——即首个存在若干连续族与一个孤立解共存的维度——的分类问题已悬而未决数十年。本文中，我们针对标准等价下的六阶复阿达马矩阵给出了完全且精确的有限关联分类。我们首先证明，每一个此类矩阵都可通过有限的分支完备程序由一个初始的去相位3×3角块构造得到，这补充了Szöllősi扩张方法中缺失的全局步骤，并证明了他的猜想：在标准等价下，除Karlsson的三参数族和Tao的孤立矩阵之外的所有类，都可从合适的角块代数地恢复。随后我们描述了由四个初始相位进行重构的几何，表明除Tao的孤立矩阵和一个显式Karlsson矩阵外，每一类都存在一个代表元，该代表元可通过在水平和垂直方向各求解一个二次方程和一个三次方程得到。我们的工作解决了该分类问题，并为进一步研究六阶阿达马矩阵提供了严谨框架，可应用于平衡六模干涉仪和相互无偏基的研究。
英文摘要
Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased $3 \times 3$ corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.
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