On the cardinalities of quantum Latin squares

A quantum Latin square of order $v$, QLS($v$), is a $v\times v$ array in which each of entries is a unit column vector from the Hilbert space $\mathbb{C}^{v}$, such that every row and column forms an orthonormal basis of $\mathbb{C}^{v}$. The cardinality of a QLS($v$) is the number of its vectors distinct up to a global phase, which is the crucial indicator for distinguishing between classical QLSs and non-classical QLSs. In this paper, we investigate the possible cardinalities of a QLS($v$). As a result, we completely resolve the existence of a QLS($v$) with maximal cardinality for any $v\geq 4$. Moreover, based on Wilson's construction and Direct Product construction, we establish some possible cardinality range of a QLS($v$) for any $v\geq 4$.