Towards a classification of 6× 6 complex Hadamard matrices
Abstract
Complex Hadamard matrices have received considerable attention in the past few years due to their appearance in quantum information theory. While a complete characterization is currently available only up to order 5 (in haagerup), several new constructions of higher order matrices have appeared recently dita, karol, BN, MM, sz. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in BN, 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 Dita's matrix D6. This answers a recent question of Bengtsson & al. in BB.
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