arXiv:2605.15540

Three Quantum Latin Squares of Order 6 with Cardinalities 13, 15, and 17

Zhipeng Xu

Published May 15, 2026• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2605.15540

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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:2605.15540

Three Quantum Latin Squares of Order 6 with Cardinalities 13, 15, and 17

Zhipeng Xu

Published May 15, 2026• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2605.15540

Related papers

New Cardinalities for Quantum Latin Squares of Order Six

Zhipeng Xu

We give explicit quantum Latin squares of order $6$ with cardinalities $19$ , $21$ , $23$ , $25$ , $27$ , $32$ , and $35$ , where vectors differing only by a global phase are identified. Cardinalities $19$ and $21$ arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities $23$ and $25$ by a controlled splitting of two pairs of rays. Cardinality $27$ is obtained from mixed Schur products of a $BH(6,6)$ matrix and a row-permuted copy. Two further mixed constructions give cardinalities $32$ and $35$ : the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality $7$ , these constructions realize every order-six cardinality except $7$ and the single currently unresolved value $29$ .We give explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, $23$, $25$, $27$, $32$, and $35$, where vectors differing only by a global phase are identified. Cardinalities $19$ and $21$ arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities $23$ and $25$ by a controlled splitting of two pairs of rays. Cardinality $27$ is obtained from mixed Schur products of a $BH(6,6)$ matrix and a row-permuted copy. Two further mixed constructions give cardinalities $32$ and $35$: the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality $7$, these constructions realize every order-six cardinality except $7$ and the single currently unresolved value $29$.

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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 $6$ .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 $6$.

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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 $v$, 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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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

The Maximal Cardinality of Quantum Latin Squares and Quantum Latin Cubes

Yuyuan Zhang, Yiwen Zhang, Mingzhen Lv +1 more

Journal of Combinatorial DesignsDec 15, 20251 citation

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.

We give explicit quantum Latin squares of order

6

with cardinalities

19

21

23

25

27

32

, and

35

, where vectors differing only by a global phase are identified. Cardinalities

19

and

21

arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities

23

and

25

by a controlled splitting of two pairs of rays. Cardinality

27

is obtained from mixed Schur products of a

BH(6,6)

matrix and a row-permuted copy. Two further mixed constructions give cardinalities

32

and

35

: the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality

7

, these constructions realize every order-six cardinality except

7

and the single currently unresolved value

29

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

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

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)