On 6× 6 complex Hadamard matrices containing two nonintersecting identical 3× 3 submatrices

It is conjectured that four mutually unbiased bases in dimension 6 do not exist in quantum information. The conjecture is equivalent to the nonexistence of some three (Formula presented.) complex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate the (Formula presented.) CHM U of Schmidt rank 3 containing two nonintersecting identical (Formula presented.) submatrices V, i.e. (Formula presented.). We show that such U exists, V, W, X have rank 2 or 3, and they have rank 2 at the same time. We construct the analytical expressions of U when V is, respectively, of rank 2, unitary and normal. We apply our results to the conjecture by showing that U with some normal V is not one of the three (Formula presented.) CHMs.