On the cardinalities of quantum Latin squares
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866918113488928768 |
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| author | Zang, Yajuan Zheng, Meihui Tian, Zihong Shan, Xiuling |
| author_facet | Zang, Yajuan Zheng, Meihui Tian, Zihong Shan, Xiuling |
| contents | 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 $v\geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the cardinalities of quantum Latin squares Zang, Yajuan Zheng, Meihui Tian, Zihong Shan, Xiuling Combinatorics Mathematical Physics 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 $v\geq 4$. |
| title | On the cardinalities of quantum Latin squares |
| topic | Combinatorics Mathematical Physics |
| url | https://arxiv.org/abs/2508.01972 |