  We consider the problem of quantum-state tomography under the assumption that
the state is pure, and more generally that its rank is bounded by a given value
$r$. In this scenario two notions of informationally complete measurements
emerge: rank-$r$ complete measurements and rank-$r$ strictly-complete
measurements. Whereas in the first notion, a rank-$r$ state is uniquely
identified from within the set of rank-$r$ states, in the second notion the
same state is uniquely identified from within the set of all physical states,
of any rank. We argue, therefore, that strictly-complete measurements are
compatible with convex optimization, and we prove that they allow robust
quantum state estimation in the presence of experimental noise. We also show
that rank-$r$ strictly-complete measurements are as efficient as rank-$r$
complete measurements. We construct examples of strictly-complete measurements
and give a complete description of their structure in the context of matrix
completion. Moreover, we numerically show that a few random bases form such
measurements. We demonstrate the efficiency-robustness property for different
strictly-complete measurements with numerical experiments. We thus conclude
that only strictly-complete measurements are useful for practical tomography.
