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We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F ( a , b ). However, in the main result of this paper we also prove that for any values of the parameters ( a , b ), the standard basis and F ( a , b ) cannot be extended to a MUB-quartet . The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.
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Jaming et al. (2009) studied this question.
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