Real entries of complex Hadamard matrices and mutually unbiased bases in dimension six

We investigate the number of real entries of an $n\times n$ complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.