Strictly-complete measurements for bounded-rank quantum-state tomography

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.