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arXiv:1702.03866 · New Journal of Physics

Correlations in star networks: from Bell inequalities to network inequalities

Armin Tavakoli, Marc-Olivier Renou, Nicolas Gisin, Nicolas Brunner

Published Jul 5, 2017• 52 citations
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Abstract

The problem of characterizing classical and quantum correlations in networks is considered. Contrary to the usual Bell scenario, where distant observers share a physical system emitted by one common source, a network features several independent sources, each distributing a physical system to a subset of observers. In the quantum setting, the observers can perform joint measurements on initially independent systems, which may lead to strong correlations across the whole network. In this work, we introduce a technique to systematically map a Bell inequality to a family of Bell-type inequalities bounding classical correlations on networks in a star-configuration. Also, we show that whenever a given Bell inequality can be violated by some entangled state ρ , then all the corresponding network inequalities can be violated by considering many copies of ρ distributed in the star network. The relevance of these ideas is illustrated by applying our method to a specific multi-setting Bell inequality. We derive the corresponding network inequalities, and study their quantum violations.

DOI
10.1088/1367-2630/aa7673

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Bell inequalities bound the strength of classical correlations between observers measuring on a shared physical system. However, studies of physical correlations can be considered beyond the standard Bell scenario by networks of observers sharing some configuration of many independent physical systems. Here, we show how to construct Bell-type inequalities for correlations arising in any tree-structured network, i.e., networks without cycles. This is achieved by an iteration procedure that in each step allows one to add a branch to the tree-structured network and construct a corresponding Bell-type inequality. We explore our inequalities in several examples, in all of which we demonstrate strong violations from quantum theory.
This work investigates network nonlocality sharing for a broad class of bipartite Bell inequalities in a generalized star network with an (n,m,k)(n,m,k)(n,m,k) configuration, comprising nnn independent branches, mmm sequential Alices per branch, and kkk measurement settings per party. On each branch, the intermediate Alices implement optimal weak measurements, whereas the final Alice and the central Bob perform sharp projective measurements. Network nonlocality sharing is witnessed when the quantum values of the network correlations associated with relevant parties simultaneously violate a star-network Bell inequality generated from the given class of bipartite Bell inequalities. We streamline the calculation of the quantum values of the network correlations and derive an analytical expression for the bipartite quantum correlator, valid for arbitrary measurement settings and weak-measurement strengths. The network nonlocality sharing for Vértesi inequalities has been studied within the framework, and simultaneous violations are found in (2,2,6)(2,2,6)(2,2,6) and (2,2,465)(2,2,465)(2,2,465) cases, with the latter exhibiting greater robustness. Our approach suggests a practical route to studying network nonlocality sharing by utilizing diverse bipartite Bell inequalities beyond the commonly used CHSH-type constructions.
Abstract We initiate the study of the nonlocal correlations in generic asymmetric quantum networks in a star configuration. Therein, the diverse unrelated sources can emit either partially or maximally entangled states, while the observers employ varying numbers of measurement settings. We propose nonlinear Bell inequalities tailored to the distributed entangled states. Specifically, we demonstrate that the algebraic maximal violations of the proposed nonlinear Bell inequalities are physically achievable within the quantum region. To achieve this, we construct the segmented Bell operators through the cut-graft-mix method applied to the Bell operators in the standard Bell tests. Furthermore, we devise the fitting Bell operators using the sum-of-square approach.