  We introduce the concept of selective quantum state tomography or SQST, a
tomographic scheme that enables a user to estimate arbitrary elements of an
unknown quantum state using a fixed measurement record. We demonstrate how this
may be done with the following notable advantages (i) a number of state copies
that depends only on the desired precision of the estimation, rather than the
dimension of the unknown state; (ii) a similar reduction in the requisite
classical memory and computational cost; (iii) an approach to state tomography
using $O(\epsilon^{-2}\log d)$ state copies for maximum norm error $\epsilon$,
as well as achieving nearly optimal bounds for full tomography with independent
measurements. As an immediate extension to this technique we proceed to show
that SQST can be used to generate an universal data sample, of fixed and
dimension independent size, from which one can extract the mean values from a
continuous class of operators on demand.
