  In this work we present some new properties of fiducial states in prime
dimensions. We parameterize fiducial operators on eigenvectors bases of
displacement operators, which allows us to find a manifold $\Omega$ of
hermitian operators satisfying $\mathrm{Tr}(\rho)=\mathrm{Tr}(\rho^2)=1$ for
any $\rho$ in $\Omega$. This manifold contains the complete set of fiducial
pure states in every prime dimension. Indeed, any quantum state $\rho\geq0$
belonging to $\Omega$ is a fiducial pure state. Also, we present an upper bound
for every probability associated to mutually unbiased decomposition of fiducial
states. This bound allows us to prove that every fiducial state tends to be
mutually unbiased to the maximal set of mutually unbiased bases in higher prime
dimensions. Finally, we show that any $\rho$ in $\Omega$ minimizes an entropic
uncertainty principle related to the second order R\'enyi entropy.
