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

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 Gröbner 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 $6$.

arXiv comment17 pages, preprint

References in corpus (5)

• On mutually unbiased bases
• Constructing Mutually Unbiased Bases in Dimension Six
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Complex Hadamard matrices of order 6: a four-parameter family
• Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six