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arXiv:2606.27758

The Existence of Diagonal Quantum Latin Squares with Maximum Cardinality

黄琳, Yang Li

Published Jun 26, 2026• 0 citations
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Abstract

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

DOI
10.48550/arxiv.2606.27758

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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(nnn) 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 (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 (\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 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.
Quantum Latin squares of order 6m6m6m with all possible cardinalities
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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.
In this paper, we prove that there exist a symmetric quantum Latin square with maximal cardinality of order nnn for all n≥6n\geq 6n≥6, and a symmetric idempotent quantum Latin square with maximal cardinality of order 4k+24k+24k+2 for all k≥1k\geq 1k≥1.