Quantum Latin squares with all possible cardinalities

• URL: http://arxiv.org/abs/2507.05642v1
• Date: Tue, 08 Jul 2025 03:40:19 GMT
• Title: Quantum Latin squares with all possible cardinalities
• Authors: Ying Zhang, Xin Wang, Lijun Ji,
• Abstract summary: A quantum Latin square of order $n$ (denoted as QLS$(n)$) is an $ntimes n$ array.<n>Two unit vectors $|urangle, |vranglein mathcalH_n$ are regarded as identical if there exists a real number $theta$.<n>The cardinality $c$ of a QLS$(n)$ is the number of distinct vectors in the array.
• Score: 11.305754681609002
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A quantum Latin square of order $n$ (denoted as QLS$(n)$) is an $n\times n$ array whose entries are unit column vectors from the $n$-dimensional Hilbert space $\mathcal{H}_n$, such that each row and column forms an orthonormal basis. Two unit vectors $|u\rangle, |v\rangle\in \mathcal{H}_n$ are regarded as identical if there exists a real number $\theta$ such that $|u\rangle=e^{i\theta}|v\rangle$; otherwise, they are considered distinct. The cardinality $c$ of a QLS$(n)$ is the number of distinct vectors in the array. In this paper, we use sub-QLS$(4)$s to prove that for any integer $m\geq 2$ and any integer $c\in [4m,16m^2]\setminus \{4m+1\}$, there is a QLS$(4m)$ with cardinality $c$.

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