Eigenvalues and eigenvectors of complex Hadamard matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914917620121600 |
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| author | Liang, Mengfan Chen, Lin |
| author_facet | Liang, Mengfan Chen, Lin |
| contents | Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10471 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvalues and eigenvectors of complex Hadamard matrices Liang, Mengfan Chen, Lin Quantum Physics Mathematical Physics Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs. |
| title | Eigenvalues and eigenvectors of complex Hadamard matrices |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2408.10471 |