Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff--von Neumann Theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries, and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.

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量子幻方：膨胀及其局限性

量子置换矩阵和量子幻方是置换矩阵和幻方的推广，其中条目不再是数字而是来自任意（非交换）代数的元素。著名的 Birkhoff--von Neumann 定理将幻方表征为置换矩阵的凸组合。在非交换情况下，相应的问题是：每个量子幻方是否都属于量子置换矩阵的矩阵凸包？也就是说，每个量子幻方都膨胀成一个量子置换矩阵吗？在这里我们证明即使在最简单的非交换情况下这也是错误的。我们还对膨胀为具有交换项的量子置换矩阵的量子幻方进行了分类，并证明了该集合直径的定量下界。最后，