Southwest Quantum Information and Technology

Informationally complete measurements from compressed sensing methodology

Amir Kalev, Center for Quantum Information and Control, University of New Mexico

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Determining quantum states and processes from a set of measurements is a fundamental problem in quantum information science. A set of such measurements is said to be informationally complete (IC) if, given sufficient statistics, they uniquely distinguish the desired density or process matrix from the set of all physical matrices. Because the standard protocols for quantum tomography (QT) scale poorly---growing exponentially with the number of subsystems---it is important to develop techniques that minimize the resources necessary for tomography. To this end, the methodology of compressed sensing (CS) has been ported from classical signal recovery and has been applied to the problem of QT. The CS paradigm applies under the assumption that there is an a priori knowledge that the signal has a concise representation, e.g., that it is a sparse vector or a low-rank matrix. Then, according to the CS methodology, one can reconstruct the signal, with very high accuracy, with a substantially reduced number of measurements, as long as the latter satisfy a restricted isometry property (RIP). However, for QT, the detailed nature of the relation between the CS measurements and IC measurements has not been made explicit. In this work, we rigorously establish the connection between RIP and IC through a key feature that arises in the quantum context: the positive semidefinite property of the density matrix (or of the process matrix). We show that due to the positivity of the density matrix, the CS measurements satisfy a special type of IC measurements. This relation has far reaching consequences for QT. First, it enables us to construct special type of IC measurements with tools provided by the CS methodology. By construction, the measurements are robust to noise. And second, with this result in hand, we are able to use simpler and more widespread numerical methods, as opposed to the specialized CS solvers, to achieve similar results. Hence, it becomes easier to employ algorithms that efficiently analyze QT data on large dimensional systems.