arXiv:2609.24653

Symmetric Quantum Latin Squares with Maximal Cardinality

Yiwen Zhang, Haitao Cao

Published Sep 21, 2026• 0 citations

Abstract

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 $k\geq 1$.

DOI: 10.48550/arxiv.2609.24653

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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 ($\mathrm{MCPQLS}(n)$).

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arXiv:2609.24653

Symmetric Quantum Latin Squares with Maximal Cardinality

Yiwen Zhang, Haitao Cao

Published Sep 21, 2026• 0 citations

arXiv HTML PDF

Abstract

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

DOI: 10.48550/arxiv.2609.24653

Related papers

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Journal of Combinatorial DesignsDec 15, 20251 citation

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Yuyuan Zhang, Mingzhen Lv, H Cao

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

arXiv:2607.19969Jul 22, 20260 citations

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quant-pharXiv:2601.09132Jan 15, 20260 citations

On the cardinalities of quantum Latin squares

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

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 $v$, 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 $v\geq 4$.

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The Existence of Diagonal Quantum Latin Squares with Maximum Cardinality

黄琳, Yang Li

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 ( $\mathrm{MCPQLS}(n)$ ).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 ($\mathrm{MCPQLS}(n)$).

arXiv:2606.27758Jun 26, 20260 citations

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

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\,\}

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)