Order-six CHMs containing exactly three distinct elements

Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show that such CHMs can only be complex equivalent to two kind of matrices, one is $H_2$-reducible and the other is the Tao matrix. Using our result one can further narrow the range of MUB trio (a set of four MUBs in $\mathbb{C}^6$ consists of an MUB trio and the identity) since we find that the two CHMs neither belong to MUB trios. Our results may lead to the more complete classification of $6\times 6$ CHMs whose elements in the first row are all 1.