Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.12607

A Quantum Latin Square of Order Six with Cardinality 29

Aishwarya P. Das, Durgesh Kumar

Published Aug 12, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $6$.

DOI
10.48550/arxiv.2608.12607

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 $29$.

arXiv:2607.11800•Jul 13, 2026•0 citations

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

Ying Zhang, Lijun Ji

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 $c$.

quant-pharXiv:2601.09132•Jan 15, 2026•0 citations

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

Zhipeng Xu

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.

arXiv:2605.15540•May 15, 2026•0 citations

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

arXiv:2606.27758•Jun 26, 2026•0 citations

Fractional and quantum Latin squares: real solutions of quantum 36 officers problem

Mykhailo Hontarenko, Karol Życzkowski

A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order $3^3$ and six of order $6^4$, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.

quant-pharXiv:2610.17170•Oct 13, 2026•0 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 $v\geq 4$.

arXiv:2508.01972•Aug 4, 2025•0 citations
We construct an explicit real quantum Latin square of order six with cardinality 292929, the last unresolved value in the order-six spectrum. We first construct a punctured 6×66 \times 66×6 array in R5\mathbb{R}^5R5 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 666.
We give explicit quantum Latin squares of order 666 with cardinalities 191919, 212121, 232323, 252525, 272727, 323232, and 353535, where vectors differing only by a global phase are identified. Cardinalities 191919 and 212121 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 232323 and 252525 by a controlled splitting of two pairs of rays. Cardinality 272727 is obtained from mixed Schur products of a BH(6,6)BH(6,6)BH(6,6) matrix and a row-permuted copy. Two further mixed constructions give cardinalities 323232 and 353535: 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 777, these constructions realize every order-six cardinality except 777 and the single currently unresolved value 292929.
Quantum Latin squares of order 6m6m6m with all possible cardinalities
A quantum Latin square of order nnn (denoted as QLS(n)(n)(n)) is an n×nn\times nn×n array whose entries are unit column vectors from the nnn-dimensional Hilbert space Hn\mathcal{H}_nHn​, such that each row and column forms an orthonormal basis. Two unit vectors ∣u⟩,∣v⟩∈Hn|u\rangle, |v\rangle\in \mathcal{H}_n∣u⟩,∣v⟩∈Hn​ are regarded as identical if there exists a real number θθθ such that ∣u⟩=eiθ∣v⟩|u\rangle=e^{iθ}|v\rangle∣u⟩=eiθ∣v⟩; otherwise, they are considered distinct. The cardinality ccc of a QLS(n)(n)(n) is the number of distinct vectors in the array. In this note,we use sub-QLS(6)(6)(6) to prove that for any integer m≥2m\geq 2m≥2 and any c∈[6m,36m2]∖{6m+1}c\in [6m,36m^2]\setminus \{6m+1\}c∈[6m,36m2]∖{6m+1}, there is a QLS(6m)(6m)(6m) with cardinality ccc.
We give three explicit quantum Latin squares of order 666, with cardinalities 131313, 151515, and 171717. Throughout, vectors differing only by a global phase are counted as identical. The cardinality-131313 construction is based on an orthogonal direct-sum decomposition $\C^6=\C^4\oplus\C^2$. The cardinality-151515 and cardinality-171717 constructions are based on two-dimensional Hadamard pairs supported on coordinate planes.
A quantum Latin square of order nnn, denoted by QLS⁡(n)\operatorname{QLS}(n)QLS(n), is an n×nn \times nn×n square whose entries are unit column vectors in the nnn-dimensional Hilbert space Hn\mathcal{H}_nHn​, such that each row and each column forms an orthonormal basis of Hn\mathcal{H}_nHn​. The cardinality of a QLS(nnn) is the number of distinct vectors up to a global phase in the array. A QLS(n)\mathrm{QLS}(n)QLS(n) whose main diagonal and anti-diagonal each forms an orthonormal basis of Hn\mathcal{H}_nHn​ is called a diagonal quantum Latin square (DQLS(n)\mathrm{DQLS}(n)DQLS(n)). In this paper, we focus on the existence of the DQLS(n)\mathrm{DQLS}(n)DQLS(n) with maximum cardinality (MCDQLS(n)\mathrm{MCDQLS}(n)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 MCDQLS(n)\mathrm{MCDQLS}(n)MCDQLS(n), except for a few exceptional cases. This result is based on the study of the existence of idempotent QLS(n)\mathrm{QLS}(n)QLS(n) with maximum cardinality (MCQLS(n)\mathrm{MCQLS}(n)MCQLS(n)), and implies an existence result for pandiagonal quantum Latin squares with maximum cardinality (MCPQLS(n)\mathrm{MCPQLS}(n)MCPQLS(n)).
A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order 333^333 and six of order 646^464, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
A quantum Latin square of order vvv, QLS(vvv), is a v×vv\times vv×v array in which each of entries is a unit column vector from the Hilbert space Cv\mathbb{C}^{v}Cv, such that every row and column forms an orthonormal basis of Cv\mathbb{C}^{v}Cv. The cardinality of a QLS(vvv) 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(vvv). As a result, we completely resolve the existence of a QLS(vvv) with maximal cardinality for any v≥4v\geq 4v≥4. Moreover, based on Wilson's construction and Direct Product construction, we establish some possible cardinality range of a QLS(vvv) for any v≥4v\geq 4v≥4.