六阶基数为29的量子拉丁方
A Quantum Latin Square of Order Six with Cardinality 29
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中文总结 AI 辅助
本文构造了六阶基数为29的实量子拉丁方，完善了六阶量子拉丁方的基数谱。
中文摘要 AI 辅助
我们构造了一个明确的六阶实量子拉丁方，其基数为$29$，这是六阶量子拉丁方谱中最后一个未解决的值。我们首先在$\boldsymbol{R}^5$中构造一个带孔的$6\times6$阵列，其带孔行和列均为标准正交基，随后在正交的一维直和项中添加一个公共对角向量。六个对角元位于同一条射线上；在三十个非对角元中，两条射线各出现两次，其余二十六条射线各出现一次，精确的支撑和坐标比比较证实了这一计数。结合已知的构造，这完善了六阶量子拉丁方的基数谱。
英文摘要
We construct an explicit real quantum Latin square of order six with cardinality $29$, the last unresolved value in the order-six spectrum. We first construct a punctured $6 \times 6$ array in $\mathbb{R}^5$ whose punctured rows and columns are orthonormal bases, and then adjoin a common diagonal vector in an orthogonal one-dimensional summand. The six diagonal entries lie on one ray. Of the thirty off-diagonal entries, two rays occur twice and the other twenty-six occur once; exact support and coordinate-ratio comparisons establish this count. Together with the known constructions, this closes the cardinality spectrum of quantum Latin squares of order $6$.
