A characterization of complex Hadamard matrices appearing in families of MUB triplets

TL;DR

This paper characterizes certain complex Hadamard matrices of order 6, showing they belong to specific known families, which aids in identifying matrices in mutually unbiased bases triplets.

Contribution

It proves that complex Hadamard matrices with three distinct columns and specific entries must belong to known families, using algebraic and structural techniques.

Findings

Matrices are identified as belonging to the transposed Fourier family or 2-circulant family.

Recognition of these matrices in MUB triplets is simplified.

The approach involves solving polynomial systems with Gröbner bases.

Abstract

It is shown that a normalized complex Hadamard matrix of order $6$ having three distinct columns, each containing at least one $-1$ entry necessarily belongs to the transposed Fourier family, or to the family of $2$-circulant complex Hadamard matrices. The proofs rely on solving polynomial system of equations by Gr\"obner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension $6$.