A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations

TL;DR

This paper introduces a complete hierarchy of semidefinite programs to characterize quantum correlations, enabling finite tests to certify quantum behavior and reconstruct quantum states and measurements.

Contribution

It proves the hierarchy's completeness, allowing finite verification of quantum correlations and providing methods to reconstruct underlying quantum states and measurements.

Findings

Hierarchy is complete for quantum correlation characterization

Finite tests can certify quantum correlations in some cases

Method to reconstruct quantum states and measurements from correlations

Abstract

We are interested in the problem of characterizing the correlations that arise when performing local measurements on separate quantum systems. In a previous work [Phys. Rev. Lett. 98, 010401 (2007)], we introduced an infinite hierarchy of conditions necessarily satisfied by any set of quantum correlations. Each of these conditions could be tested using semidefinite programming. We present here new results concerning this hierarchy. We prove in particular that it is complete, in the sense that any set of correlations satisfying every condition in the hierarchy has a quantum representation in terms of commuting measurements. Although our tests are conceived to rule out non-quantum correlations, and can in principle certify that a set of correlations is quantum only in the asymptotic limit where all tests are satisfied, we show that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests. We provide a criterion to detect when such a situation arises, and we explain how to reconstruct the quantum states and measurement operators reproducing the given correlations. Finally, we present several applications of our approach. We use it in particular to bound the quantum violation of various Bell inequalities.