[math] \newcommand{\mathds}{\mathbb}[/math]

Prove that the only complex Hadamard matrices at [math]N=4[/math] are, up to the standard equivalence relation, the matrices

[[math]] F_4^q=\begin{pmatrix} 1&1&1&1\\ 1&-1&1&-1\\ 1&q&-1&-q\\ 1&-q&-1&q \end{pmatrix} [[/math]]

with [math]q\in\mathbb T[/math], which appear as Di\c t\u a deformations of [math]W_4=F_2\otimes F_2[/math].