$$H_2$$-reducible matrices in six-dimensional mutually unbiased bases

Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then $$H_2$$ -reducible MUBs has exactly nine $$2\times 2$$ Hadamard submatrices. apply our result to exclude from some known CHMs, such as symmetric matrix, Hermitian Dita family, Bjorck’s circulant and Szollosi family. Our results represent latest progress on existence of MUBs.