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 H-2-reducible matrix in the four MUBs has exactly nine 2 x 2 Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric H-2-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.

Automated summary

The research shows that if there are four MUBs, then the H-2-reducible matrix within them contains exactly nine 2x2 Hadamard submatrices. This result has been utilized to exclude some known CHMs from the four MUBs. These findings represent the latest progress on the existence of six-dimensional MUBs.