标题： 群不变量子拉丁方 显示英文标题

标题： Group Invariant Quantum Latin Squares

摘要： 一个量子拉丁方是一个由单位向量组成的 $n \times n$ 数组，其中每一行和每一列都构成固定复向量空间的规范正交基。 我们引入了有限群 $G$ 和 $G'$ 的 $(G,G')$-不变量子拉丁方的概念。 这些是量子拉丁方，其行和列分别由 $G$ 和 $G'$ 索引，使得 $a,b$-元数与 $c,d$-元数的内积仅依赖于 $a^{-1}c \in G$ 和 $b^{-1}d \in G'$。 这一定义受到 [Roberson & Schmidt (2020)] 中引入的群不变双射相关性的启发，并且每个群不变量子拉丁方都会产生一个群不变双射相关性，但反之则不然。 在这项工作中，我们研究了这些群不变量子拉丁方及其对应的关联性。 我们的主要结果是：对于一个$(G,G')$-不变的量子拉丁方中的每个向量施加一个全局等距变换后，这些对象与$G$和$G'$的群代数之间的保迹和共轭转置保持同构之间存在自然的双射关系。 这尤其证明了，存在一个$(G,G')$-不变的量子拉丁方当且仅当$G$和$G'$的不可约表示的度数多重集相等。 这一研究方向的另一个动机是，每当群 $G$ 和 $G'$ 的Cayley图量子同构时，总存在一个 $(G,G')$-不变的量子关联来证明这一点，因此在寻找量子同构的Cayley图时，只需考虑这些关联即可。 给定一个群不变的量子关联，我们展示如何构造所有使该关联成为量子同构的图对。 显示英文摘要

摘要： A quantum Latin square is an $n \times n$ array of unit vectors where each row and column forms an orthonormal basis of a fixed complex vector space. We introduce the notion of $(G,G')$-invariant quantum Latin squares for finite groups $G$ and $G'$. These are quantum Latin squares with rows and columns indexed by $G$ and $G'$ respectively such that the inner product of the $a,b$-entry with the $c,d$-entry depends only on $a^{-1}c \in G$ and $b^{-1}d \in G'$. This definition is motivated by the notion of group invariant bijective correlations introduced in [Roberson \& Schmidt (2020)], and every group invariant quantum Latin square produces a group invariant bijective correlation, though the converse does not hold. In this work we investigate these group invariant quantum Latin squares and their corresponding correlations. Our main result is that, up to applying a global isometry to every vector in a $(G,G')$-invariant quantum Latin square, there is a natural bijection between these objects and trace and conjugate transpose preserving isomorphisms between the group algebras of $G$ and $G'$. This in particular proves that a $(G,G')$-invariant quantum Latin square exists if and only if the multisets of degrees of irreducible representations are equal for $G$ and $G'$. Another motivation for this line of work is that whenever Cayley graphs for groups $G$ and $G'$ are quantum isomorphic, then there is a $(G,G')$-invariant quantum correlation witnessing this, and thus it suffices to consider such correlations when searching for quantum isomorphic Cayley graphs. Given a group invariant quantum correlation, we show how to construct all pairs of graphs for which it gives a quantum isomorphism.