Mutually unbiased bases: polynomial optimization and symmetry

Understand

A set of $k$ orthonormal bases of $\mathbb C^d$ is called mutually unbiased if $|\langle e,f\rangle |^2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases.

• A natural question is for which pairs $(d,k)$ there exist~$k$ mutually unbiased bases in dimension $d$.
• The (well-known) upper bound $k \leq d+1$ is attained when~$d$ is a power of a prime.
• For all other dimensions it is an open problem whether the bound can be attained.