Two convergent NPA-like hierarchies for the quantum bilocal scenario

TL;DR

This paper introduces a new hierarchy for characterising quantum correlations in network scenarios, proves its convergence in the bilocal case, and explores its relation to existing hierarchies, advancing the understanding of quantum network correlations.

Contribution

It presents a novel bilocal NPA hierarchy, proves its equivalence to a scalar extension hierarchy, and establishes its convergence for the simplest quantum network scenario.

Findings

The new hierarchy is equivalent to a modified scalar extension hierarchy.

Convergence of the hierarchy is proven in the bilocal scenario.

Relations with other known hierarchies are characterized.

Abstract

Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascu\'es et al., New J. Phys. 10, 073013 (2008)], known as the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.