Real Entries of Complex Hadamard Matrices and Mutually Unbiased Bases in Dimension Six

TL;DR

This paper analyzes the real entry counts in complex Hadamard matrices of small dimensions, especially six, and explores implications for the existence of four mutually unbiased bases in quantum information.

Contribution

It analytically determines possible real entry counts for complex Hadamard matrices in dimensions 2, 3, 4, and 6, and links these results to the open problem of mutually unbiased bases in dimension six.

Findings

Real entries in 6x6 complex Hadamard matrices can be 0-22, 24, 25, 26, or 30.

If four mutually unbiased bases exist in dimension six, the real entries in related matrices are at most 22.

The results constrain the structure of potential MUBs in quantum information theory.

Abstract

We investigate the number of real entries of an $n\times n$ complex Hadamard matrix (CHM). 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 (MUBs) 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 exists then the real entries in any one of the remaining three matrices does not exceed $22$.