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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 threecomplex Hadamard matrices (CHMs) with Schmidt rank at least 3. We investigate theCHM  U  of Schmidt rank 3 containing two nonintersecting identicalsubmatrices  V , i.e.. 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 threeCHMs.

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