Numerical evidence for the maximum number of mutually unbiased bases in dimension six

The question of determining the maximal number of mutually unbiased bases in dimension six has received much attention since their introduction to quantum information theory, but a definitive answer has still not been found. In this Letter we move away from the traditional analytic approach and use a numerical approach to attempt to determine this number. We numerically minimise a non-negative function Nd,NNd,N of a set of N+1N+1 orthonormal bases in dimension d which only evaluates to zero if the bases are mutually unbiased. As a result we find strong evidence that (as has been conjectured elsewhere) there are no more than three mutually unbiased bases in dimension six.