The power of being positive: Robust state estimation made possible by quantum mechanics

We study the problem of quantum-state tomography under the assumption that the state of the system is close to pure. In this context, an efficient measurements that one typically formulates uniquely identify a pure state from within the set of other pure states. In general such measurements are not robust in the presence of measurement noise and other imperfections, and therefore are less practical for tomography. We argue here that state tomography experiments should instead be done using measurements that can distinguish a pure state from {\em any} other quantum state, of any rank. We show that such nontrivial measurements follows from the physical constraint that the density matrix is positive semidefinite and prove that these measurements yield a robust estimation of the state. We assert that one can implement such tomography relatively simply by measuring only a few random orthonormal bases; our conjecture is supported by numerical evidence. These results are generalized for estimation of states close to bounded-rank.