A quantum Latin square of order
n, denoted by
QLS(n), is an
n×n square whose entries are unit column vectors in the
n-dimensional Hilbert space
Hn, such that each row and each column forms an orthonormal basis of
Hn. The cardinality of a QLS(
n) is the number of distinct vectors up to a global phase in the array. A
QLS(n) whose main diagonal and anti-diagonal each forms an orthonormal basis of
Hn is called a diagonal quantum Latin square (
DQLS(n)). In this paper, we focus on the existence of the
DQLS(n) with maximum cardinality (
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
MCDQLS(n), except for a few exceptional cases. This result is based on the study of the existence of idempotent
QLS(n) with maximum cardinality (
MCQLS(n)), and implies an existence result for pandiagonal quantum Latin squares with maximum cardinality (
MCPQLS(n)).
We give three explicit quantum Latin squares of order
6, with cardinalities
13,
15, and
17. Throughout, vectors differing only by a global phase are counted as identical. The cardinality-
13 construction is based on an orthogonal direct-sum decomposition $\C^6=\C^4\oplus\C^2$. The cardinality-
15 and cardinality-
17 constructions are based on two-dimensional Hadamard pairs supported on coordinate planes.