Journal of Combinatorial Designs

The Maximal Cardinality of Quantum Latin Squares and Quantum Latin Cubes

Yuyuan Zhang, Yiwen Zhang, Mingzhen Lv, Haitao Cao

Published Dec 15, 2025• 1 citation

Abstract

ABSTRACT In this paper, by extending the classical singular direct product construction to the quantum setting we resolve the existence problem for quantum Latin squares with maximal cardinality except for 11 possible exceptions. We also prove that there exists a quantum Latin cube of order with maximal cardinality for all positive integers , except when is a prime or , where are prime numbers and .

DOI: 10.1002/jcd.70002

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Journal of Combinatorial Designs

The Maximal Cardinality of Quantum Latin Squares and Quantum Latin Cubes

Yuyuan Zhang, Yiwen Zhang, Mingzhen Lv, Haitao Cao

Published Dec 15, 2025• 1 citation

Publisher

Abstract

DOI: 10.1002/jcd.70002

Related papers

On the cardinalities of quantum Latin squares

Yajuan Zang, Mei Chai Zheng, Zihong Tian +1 more

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

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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)

In this paper, we prove that there exist a symmetric quantum Latin square with maximal cardinality of order

n

for all

n\geq 6

, and a symmetric idempotent quantum Latin square with maximal cardinality of order

4k+2

for all

k\geq 1

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

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\,\}