arXiv:2507.05642 · Journal of Combinatorial Designs

Quantum Latin Squares With all Possible Cardinalities

Ying Zhang, Xin Wang, Lijun Ji

Published May 2, 2026• 0 citations

Abstract

ABSTRACT A quantum Latin square of order (denoted as QLS) is an array whose entries are unit column vectors from the ‐dimensional Hilbert space , such that each row and column forms an orthonormal basis. Two unit vectors are regarded as identical if there exists a real number such that ; otherwise, they are considered distinct. The cardinality of a QLS is the number of distinct vectors in the array. In this paper, we use sub‐QLSs to prove that for any integer , and any integer , there is a QLS with cardinality .

DOI: 10.1002/jcd.70021

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arXiv:2507.05642 · Journal of Combinatorial Designs

Quantum Latin Squares With all Possible Cardinalities

Ying Zhang, Xin Wang, Lijun Ji

Published May 2, 2026• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.1002/jcd.70021

Related papers

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

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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

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

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)

We construct an explicit real quantum Latin square of order six with cardinality

29

, the last unresolved value in the order-six spectrum. We first construct a punctured

6 \times 6

array in

\mathbb{R}^5

whose punctured rows and columns are orthonormal bases, and then adjoin a common diagonal vector in an orthogonal one-dimensional summand. The six diagonal entries lie on one ray. Of the thirty off-diagonal entries, two rays occur twice and the other twenty-six occur once; exact support and coordinate-ratio comparisons establish this count. Together with the known constructions, this closes the cardinality spectrum of quantum Latin squares of order

6

In this paper, we show that for any integer

v \ge 8

with

v \ne 9,11,23

, a quantum Latin square of order

v

exists for every cardinality

c \in [v,\,v^{2}] \setminus \{\,v+1\,\}