The non-\(H_2\)-reducible matrices and some special complex Hadamard matrices

Characterizing the \(6\times 6\) $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the \(6\times 6\) $6\times 6$ CHMs except the \(H_2\) $H_2$ -reducible matrices and the Tao matrix as the non- \(H_2\) $H_2$ -reducible matrices. As far as we know, no non- \(H_2\) $H_2$ -reducible matrices with analytic form have been found. In this paper, we find some non- \(H_2\) $H_2$ -reducible matrices with analytic form. We also characterize some special \(6\times 6\) $6\times 6$ CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in \(\mathbb {C}^6\) ${\mathbb{C}}^6$ consists of an MUB trio and the identity). Our results may lead to the complete classification of \(6\times 6\) $6\times 6$ CHMs and the solution of MUB problem in \(\mathbb {C}^6\) ${\mathbb{C}}^6$ .