In this paper, we prove that there exist a symmetric quantum Latin square with maximal cardinality of order
n for all
n≥6, and a symmetric idempotent quantum Latin square with maximal cardinality of order
4k+2 for all
k≥1.
In this paper, we show that for any integer
v≥8 with
v=9,11,23, a quantum Latin square of order
v exists for every cardinality
c∈[v,v2]∖{v+1}.
A quantum Latin square of order
v, QLS(
v), is a
v×v array in which each of entries is a unit column vector from the Hilbert space
Cv, such that every row and column forms an orthonormal basis of
Cv. 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≥4. Moreover, based on Wilson's construction and Direct Product construction, we establish some possible cardinality range of a QLS(
v) for any
v≥4.