Mutually unbiased bases in six dimensions: The four most distant bases

We consider the average distance between four bases in six dimensions. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in six dimensions. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically found maximum of the average distance, and we conduct a detailed study of the structure of the extremal set of bases.

Figure 1

Histogram of the maximum values of the ASD found during a numerical search for 10 000 randomly chosen initial four bases. The search converges to one of the local maxima in about 30% of all runs, and to the global maximum of ${\stackrel{̲}{D^2}}_{\max }=0.9983$ for the other 70% of initial bases.

Figure 2

Contour plot of the ASD for the two-parameter family. Along the dashed curves, relation (7) holds. The four single-parameter families—one for each dashed curve—are equivalent to the two-parameter family in the sense that the maximal and minimal value of the ASD can be found by searching along one of the dashed lines only. The arrows point to the location of one of the eight maxima at $({\theta }_x,{\theta }_t)=(0.9852,1.0094)$, marked by a cross.