The Existence of Diagonal Quantum Latin Squares with Maximum Cardinality

A quantum Latin square of order $n$, denoted by $\operatorname{QLS}(n)$, is an $n \times n$ square whose entries are unit column vectors in the $n$-dimensional Hilbert space $\mathcal{H}_n$, such that each row and each column forms an orthonormal basis of $\mathcal{H}_n$. The cardinality of a QLS($n$) is the number of distinct vectors up to a global phase in the array. A $\mathrm{QLS}(n)$ whose main diagonal and anti-diagonal each forms an orthonormal basis of $\mathcal{H}_n$ is called a diagonal quantum Latin square ($\mathrm{DQLS}(n)$). In this paper, we focus on the existence of the $\mathrm{DQLS}(n)$ with maximum cardinality ($\mathrm{MCDQLS}(n)$). By employing direct constructions based on row-quantum Latin rectangle and special complete mapping, together with the recursive techniques such as the singular direct product construction, We have almost completely determined the existence of $\mathrm{MCDQLS}(n)$, except for a few exceptional cases. This result is based on the study of the existence of idempotent $\mathrm{QLS}(n)$ with maximum cardinality ($\mathrm{MCQLS}(n)$), and implies an existence result for pandiagonal quantum Latin squares with maximum cardinality ($\mathrm{MCPQLS}(n)$).