A new method to construct families of complex Hadamard matrices in even dimensions

TL;DR

The paper introduces a novel method for constructing extensive families of complex Hadamard matrices in even dimensions, generalizing previous methods and revealing millions of inequivalent families, with implications for quantum information theory.

Contribution

It presents a new construction technique for complex Hadamard matrices in even dimensions, extending existing methods and discovering millions of new inequivalent families.

Findings

Reproduced known complex Hadamard families

Extended some families to new dimensions

Discovered over 13 million inequivalent families in dimension 32

Abstract

We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with the Di\c{t}\u{a} construction and it generalizes the Sz\"oll\H{o}si's method. We reproduce well-known results, extend some families and present new families of complex Hadamard matrices in even dimensions. In particular, found more than 13 millon inequivalent affine families of complex Hadamard matrices in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six.