A characterization of complex Hadamard matrices appearing in families of MUB triplets
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910631770193920 |
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| author | Matszangosz, Ákos K. Szöllősi, Ferenc |
| author_facet | Matszangosz, Ákos K. Szöllősi, Ferenc |
| contents | It is shown that a normalized complex Hadamard matrix of order $6$ having three distinct columns, each containing at least one $-1$ entry necessarily belongs to the transposed Fourier family, or to the family of $2$-circulant complex Hadamard matrices. The proofs rely on solving polynomial system of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of complex Hadamard matrices appearing in families of MUB triplets Matszangosz, Ákos K. Szöllősi, Ferenc Combinatorics 05B20, 81P45 It is shown that a normalized complex Hadamard matrix of order $6$ having three distinct columns, each containing at least one $-1$ entry necessarily belongs to the transposed Fourier family, or to the family of $2$-circulant complex Hadamard matrices. The proofs rely on solving polynomial system of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension $6$. |
| title | A characterization of complex Hadamard matrices appearing in families of MUB triplets |
| topic | Combinatorics 05B20, 81P45 |
| url | https://arxiv.org/abs/2405.09991 |