Informational completeness in bounded-rank quantum-state tomography
arXiv:1510.02736 · doi:10.1103/PhysRevA.93.052105
Abstract
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. In this scenario, new notions of informationally complete POVMs emerge, which allow for high-fidelity state estimation with fewer measurement outcomes than are required for an arbitrary rank state. We study this in the context of matrix completion, where the POVM outcomes determine only a few of the density matrix elements. We give an analytic solution that fully characterizes informational completeness and elucidates the important role that the positive-semidefinite property of density matrices plays in tomography. We show how positivity can impose a stricter notion of information completeness and allow us to use convex optimization programs to robustly estimate bounded-rank density matrices in the presence of statistical noise.
10 pages, 1 figure. This submission was combined with arXiv:1511.01433 to produce arXiv:1605.02109, which is published in Phys. Rev. A
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