We consider the average distance between four bases in dimension six. Thedistance between two orthonormal bases vanishes when the bases are the same,and the distance reaches its maximal value of unity when the bases areunbiased. We perform a numerical search for the maximum average distance andfind it to be strictly smaller than unity. This is strong evidence that no fourmutually unbiased bases exist in dimension six. We also provide a two-parameterfamily of three bases which, together with the canonical basis, reach thenumerically-found maximum of the average distance, and we conduct a detailedstudy of the structure of the extremal set of bases.
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