  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 $2\times2$ 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.
