  We show that if a set of four mutually unbiased bases (MUBs) in
$\mathbb{C}^6$ exists and contains the identity, then any other basis in the
set contains at most two product states and at the same time has Schmidt rank
at least three. Here both the product states and the Schmidt rank are defined
over the bipartite space $\mathbb{C}^2\otimes\mathbb{C}^3$. We also investigate
the connection of the Sinkhorn normal form of unitary matrices to the fact that
there is at least one vector unbiased to any two orthonormal bases in any
dimension.
