Informational completeness in bounded-rank quantum-state tomography

TL;DR

This paper explores efficient quantum-state tomography for low-rank states, introducing new informationally complete measurements and convex optimization techniques that improve state estimation with fewer measurements.

Contribution

It introduces novel notions of informational completeness for bounded-rank states and provides an analytic characterization and convex methods for robust state estimation.

Findings

New informationally complete POVMs for low-rank states

Analytic characterization of informational completeness

Convex optimization methods for noisy data

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.