A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
Résumé
In this paper we exhibit the existence of an {\it infinite family of triplets} of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, $F(a,b)$. The emergence of such an infinite family is surprising because only a handful of isolated examples of MUB-triplets have been known in the literature so far. However, we also prove that {\it no triplet of the infinite family can be extended to a MUB-quartet}. We consider this latter result a breakthrough in that the {\it method} of proof may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.