New approach to finding the maximum number of mutually unbiased bases in ${\mathbb{C}}^6$

There has been great interest in finding sets of $m$ mutually unbiased bases which are compatible with a given space ${\mathbb{C}}^d$, specially in physics due to their interesting applications in quantum information theory. Several general results have been obtained so far, but surprising results may occur for definite $(m,d)$-values. One such case that has remained an open question (the simplest case) is the one regarding the existence of $m=4$ mutually orthogonal bases for $d=6$. In the present work we introduce a new approach to the problem by translating it into an optimization procedure for a given pair $(m,d)$.