In this work, the concept of mutually unbiased frames is introduced as the
most general notion of unbiasedness for sets composed by linearly independent
and normalized vectors. It encompasses the already existing notions of
unbiasedness for orthonormal bases, regular simplices, equiangular tight
frames, positive operator valued measure, and also includes symmetric
informationally complete quantum measurements. After introducing the tool, its
power is shown by finding the following results about the last mentioned class
of constellations: (i) real fiducial states do not exist in any even dimension,
and (ii) unknown $d$-dimensional fiducial states are parameterized, a priori,
with roughly $3d/2$ real variables only, without loss of generality.
Furthermore, multi-parametric families of pure quantum states having minimum
uncertainty with regard to several choices of $d+1$ orthonormal bases are
shown, in every dimension $d$. These last families contain all existing
fiducial states in every finite dimension, and the bases include maximal sets
of $d+1$ mutually unbiased bases, when $d$ is a prime number.
