Order-six CHMs containing exactly three distinct elements

TL;DR

This paper classifies order-six complex Hadamard matrices with exactly three distinct elements, showing they are equivalent to two known types and implications for mutually unbiased bases in quantum information.

Contribution

It proves that such matrices are only equivalent to two specific types, refining the classification of 6x6 complex Hadamard matrices with certain element constraints.

Findings

CHMs with three elements are equivalent to $H_2$-reducible or Tao matrix

These matrices do not belong to MUB trios in $\

Results narrow the classification of 6x6 CHMs with first row all ones.

Abstract

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.