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arXiv:2408.10471

Eigenvalues and eigenvectors of complex Hadamard matrices

Mengfan Liang, Lin Chen

Published Aug 20, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2408.10471

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Towards a classification of 6×66\times 66×6 complex Hadamard matrices
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Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţă's matrix D6. This answers a recent question of Bengtsson et al. [3].
Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 × 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 × 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.