Unbiased complex Hadamard matrices and bases

TL;DR

This paper introduces the concept of mutually unbiased complex Hadamard matrices, establishes bounds on their number, and demonstrates their use in constructing unbiased real Hadamard matrices, including a new example of order 36.

Contribution

It defines the class of mutually unbiased complex Hadamard matrices and shows how they can be used to generate unbiased real Hadamard matrices, providing new constructions and bounds.

Findings

Maximum of 2 MUCH matrices for order 2n, n odd

Construction of unbiased real Hadamard matrices of order 36

New bounds and examples for MUCH matrices

Abstract

We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is attained for n = 1,5,9. Furthermore, we prove that certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. As a consequence we generate a new pair of unbiased real Hadamard matrices of order 36.