  The construction of physically relevant low dimensional state models, and the
design of appropriate measurements are key issues in tackling quantum state
tomography for large dimensional systems. We consider the statistical problem
of estimating low rank states in the set-up of multiple ions tomography, and
investigate how the estimation error behaves with a reduction in the number of
measurement settings, compared with the standard ion tomography setup. We
present extensive simulation results showing that the error is robust with
respect to the choice of states of a given rank, the random selection of
settings, and that the number of settings can be significantly reduced with
only a negligible increase in error. We present an argument to explain these
findings based on a concentration inequality for the Fisher information matrix.
In the more general setup of random basis measurements we use this argument to
show that for certain rank $r$ states it suffices to measure in $O(r\log d)$
bases to achieve the average Fisher information over all bases. We present
numerical evidence for states upto 8 atoms, supporting a conjecture on a lower
bound for the Fisher information which, if true, would imply a similar
behaviour in the case of Pauli bases. The relation to similar problems in
compressed sensing is also discussed.
