Quantum Latin squares of order $6m$ with all possible cardinalities

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 $θ$ such that $|u\rangle=e^{iθ}|v\rangle$; otherwise, they are considered distinct. The cardinality $c$ of a QLS$(n)$ is the number of distinct vectors in the array. In this note,we use sub-QLS$(6)$ to prove that for any integer $m\geq 2$ and any $c\in [6m,36m^2]\setminus \{6m+1\}$, there is a QLS$(6m)$ with cardinality $c$.

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