The H₂ -reducible matrix in four six-dimensional mutually unbiased bases

Finding four six-dimensional mutually unbiased bases (MUBs) is a long-standing open problem in quantum information. By assuming that they exist and contain the identity matrix, we investigate whether the remaining three MUBs have an H₂-reducible matrix, namely a 6 × 6 complex Hadamard matrix (CHM) containing a 2 × 2 subunitary matrix. We show that every 6 × 6 CHM containing at least 23 real entries is an H₂-reducible matrix. It relies on the fact that the CHM is complex equivalent to one of the two constant H₂-reducible matrices. They, respectively, have exactly 24 and 30 real entries, and both have more than eighteen 2 × 2 subunitary matrices. It turns out that such H₂- reducible matrices do not belong to the remaining three MUBs. This is the corollary of a stronger claim; namely, any H₂-reducible matrix belonging to the remaining three MUBs has exactly nine or eighteen 2 × 2 subunitary matrices.