H₂ -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 the H₂-reducible matrix in the four MUBs has exactly nine 2 × 2 Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric H₂-reducible matrix, the Hermitian matrix, Dita family, Bjorck’s circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs.