Characterization of fiducial states in prime dimensions from mutually unbiased bases

D. Goyeneche, R. Salazar and A. Delgado


We derive a parameterization for the probability distributions defining the fiducial operator of a symmetric informationally complete positive-operator-valued measure in its decomposition on mutually unbiased bases for every prime dimension d. This parameterization leads to a (d2 − 1)/2-manifold formed by hermitian operators ρ with tr(ρ) = tr(ρ2) = 1. We show that the existence of quantum states within this mani- fold can be studied through a variational problem. The only possible solutions of this problem are pure quantum states which also are fiducial states. Thus, any solution of this variational problem leads to a symmetric informationally complete positive-operator-valued measure. We have verified that the fiducial states numerically obtained by Scott and Grassl [11] are, up to d = 23, solutions of the variational problem here presented. We derive an upper bound for the probability distributions of any fiducial operator in the manifold and show that they also minimize the quadratic R ́enyi entropy.