  We introduce random matrix theory to study the tomographic efficiency of a
wide class of measurements constructed out of weighted 2-designs, including
symmetric informationally complete (SIC) probability operator measurements
(POMs). In particular, we derive analytic formulae for the mean Hilbert-Schmidt
distance and the mean trace distance between the estimator and the true state,
which clearly show the difference between the scaling behaviors of the two
error measures with the dimension of the Hilbert space. We then prove that the
product SIC POMs---the multipartite analogue of the SIC POMs---are optimal
among all product measurements in the same sense as the SIC POMs are optimal
among all joint measurements. We further show that, for bipartite systems,
there is only a marginal efficiency advantage of the joint SIC POMs over the
product SIC POMs. In marked contrast, for multipartite systems, the efficiency
advantage of the joint SIC POMs increases exponentially with the number of
parties.
