Characterization of fiducial states in prime dimensions via mutually unbiased bases

Abstract

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.