The N × N complex Hadamard matrices form a real algebraic manifold CN. We have CN=MN(T)∩NUN, and following Tadej and Życzkowski we investigate here the computation of the enveloping tangent space T̃HCN=THMN(T)∩THNUN, and notably of its dimension d(H)=dim(T̃HCN), called undephased defect of H. Our main result is an explicit formula for the defect of the Fourier matrix FG associated to an arbitrary finite abelian group G=Z N1×⋯×ZNr. We also comment on the general question "does the associated quantum permutation group see the defect", with a probabilistic speculation involving Diaconis-Shahshahani type variables.
Banica, T. (2013). The defect of generalized Fourier matrices. Linear Algebra and Its Applications, 438(9), 3667–3688. https://doi.org/10.1016/j.laa.2013.01.011