发表机构
University of Waterloo; Czech Technical University in Prague; The Ohio State University（滑铁卢大学; 布拉格捷克理工大学; 俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推广了单项式矩阵在复阿达马矩阵上的量子群作用，提出量子对称性与$s$-量子等价概念，证明特定作用下同规模阿达马矩阵量子等价，还通过图形演算证明有限量子群量子对称性必为经典等结论。
AI 中文摘要
我们研究单项式矩阵在复阿达马矩阵上作用的量子群推广，这引出了阿达马矩阵的多种量子对称性与量子等价概念。我们证明，若以某一最大单项式量子群作用，给定规模的所有阿达马矩阵都将成为量子等价的。采用更具限制性的量子化则引出了$s$-量子等价的概念。我们给出了相同规模和阶数的Butson矩阵的例子，其对任意$s$都不是$s$-量子等价的。我们还证明，$s$-量子等价可由同步非局域“阿达马等价”游戏进行操作建模。我们的方法主要基于图形演算，使用由互补蜘蛛生成的范畴。我们还利用这些相同工具研究量子群的量子仿射等价性，并证明所有相同规模的有限量子群都是量子仿射等价的。我们还给出了Kasprzak--Sołtan--Woronowicz结果的简短图形证明，该结果断言有限量子群的量子对称性必须是经典的。
英文摘要
We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local ``Hadamard equivalence'' game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak--Sołtan--Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.
Comments32 pages. Some typos corrected in v2. This version submitted for publication
