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.
Highlights
Bell inequalities bound the strength of correlations between the outcomes of measurements performed by distant observers who share a physical system under the assumption of Bell-like locality
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
The observers can perform joint measurements on initially independent systems, which may lead to strong correlations across the whole network
Summary
Bell inequalities bound the strength of correlations between the outcomes of measurements performed by distant observers who share a physical system under the assumption of Bell-like locality. Only very little is known about classical and quantum correlations in networks The latter are generalizations of the Bell scenario to more sophisticated configurations featuring several independent sources. Entropic Bell inequalities has been derived for several networks [13], but are usually not very efficient at capturing classical correlations Another approach to study correlations in networks is from the point of view of Bayesian inference [14–20]. The star inequalities we derive can have any number of settings for all observers Their quantum violations can be directly related to the quantum violation of the initial Bell inequality. We illustrate the relevance of this method by an explicit example in which we start from a Bell inequality with more than two settings and construct the mapping to a particular star inequality and study its violation in correlations in the Bell scenario. Inequalities bounding the strength of N -local correlations arising in a star network are called star inequalities
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