Eigenvalues and eigenvectors of complex Hadamard matrices
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.10471
- arXiv:
- arXiv:2408.10471
- Bibcode:
- 2024arXiv240810471L
- Keywords:
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- Quantum Physics;
- Mathematical Physics
- E-Print:
- 15 pages,0 figures