  Optimal generalized measurements for state estimation are well understood.
However, practical quantum state tomography is typically performed using a
fixed set of projective measurements and the question of how to choose these
measurements has been largely unexplored in the literature. In this work we
develop theoretical asymptotic bounds for the average fidelity of pure qubit
tomography using measurement sets whose axes correspond to vertices of Platonic
solids. We also present complete simulations of maximum likelihood tomography
for mixed qubit states using the Platonic solid measurements. We show that
overcomplete measurement sets can be used to improve the accuracy of
tomographic reconstructions.
