Complete parameterization of complex Hadamard matrices in every dimension from pairs of columns

What is it about?

In this work we completely solve the problem of constructing families of complex Hadamard matrices in every dimension from considering pairs of columns. This is is equivalent to start from an orthonormal base (Hadamard) and consider rotations of pairs of vectors (columns) in complex 2D planes such that both vectors of every pair have unimodular entries and we still having an orthonormal base.

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Why is it important?

We demonstrate that such a kind of families exist for every even dimension and do not exist in any odd dimension. Additionally, we impose analytical restrictions for the existence of 4 MUBs in dimension 6 and we found more than 13 millon inequivalent families of complex Hadamard matrices in dimension 32. This is the first complete characterization (of a specific kind of complex Hadamard matrices) in every dimension. This work defines an ordered way for completing the full classification.