Multigraph approach to quantum non-locality

Abstract

Non-contextuality (NC) and Bell inequalities can be expressed as bounds $\Omega$ for positive linear combinations $S$ of probabilities of events, $S\le \Omega$. Exclusive events in $S$ can be represented as adjacent vertices of a graph called the exclusivity graph of $S$. In the case that events correspond to the outcomes of quantum projective measurements, quantum probabilities are intimately related to the Grötschel-Lovász-Schrijver theta body of the exclusivity graph. Then, one can easily compute an upper bound to the maximum quantum violation of any NC or Bell inequality by optimizing $S$ over the theta body and calculating the Lovász number of the corresponding exclusivity graph. In some cases, this upper bound is tight and gives the exact maximum quantum violation. However, in general, this is not the case. The reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in Bell-inequality (and similar) scenarios. An interesting question is whether there is a graph-theoretical concept which accounts for this problem. Here we show that, for any given $N$-partite Bell inequality, an edge-coloured multigraph composed of $N$ single-colour graphs can be used to encode the relationships of exclusivity between each party's parts of the events. Then, the maximum quantum violation of the Bell inequality is exactly given by a refinement of the Lovász number that applies to these edge-coloured multigraphs. We show how to calculate upper bounds for this number using a hierarchy of semi-definite programs and calculate upper bounds for $I_3$, $I_{3322}$ and the three bipartite Bell inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles.

21 pages, 5 figures

References in corpus (27)

• Bell inequalities for arbitrarily high dimensional systems
• Information Causality as a Physical Principle
• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• A simple test for hidden variables in spin-1 system
• Bounding the set of quantum correlations
• Experimentally testable state-independent quantum contextuality
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• Graph-Theoretic Approach to Quantum Correlations
• Quantum non-locality in two three-level systems
• A glance beyond the quantum model
• Local orthogonality as a multipartite principle for quantum correlations
• Preparation contextuality powers parity-oblivious multiplexing
• Simple explanation of the quantum violation of a fundamental inequality
• Maximal violation of the I3322 inequality using infinite dimensional quantum systems
• "All versus nothing" inseparability for two observers
• Maximal violation of the CGLMP inequality for infinite dimensional states
• Partial list of bipartite Bell inequalities with four binary settings
• Fully nonlocal quantum correlations
• Quantum Correlations are Tightly Bound by the Exclusivity Principle
• Experimental fully contextual correlations
• The exclusivity principle forbids sets of correlations larger than the quantum set
• Basic exclusivity graphs in quantum correlations
• Hardy's Paradox for High-Dimensional Systems: Beyond Hardy's Limit
• Violating Bell Inequalities Maximally for Two $d$-Dimensional Systems
• Bell inequalities for the simplest exclusivity graph
• Twin inequality for fully contextual quantum correlations
• Quantum theory realises all joint measurability graphs

Cited by in corpus (14)

• Kochen-Specker Contextuality
• Synchronous observation of Bell nonlocality and state-dependent Kochen-Specker contextuality
• Simple explanation of the quantum limits of genuine $n$-body nonlocality
• Exclusivity principle and the quantum bound of the Bell inequality
• Quantum theory allows for absolute maximal contextuality
• Noncontextuality Inequalities from Antidistinguishability
• Testing noncontextuality inequalities that are building blocks of quantum correlations
• Necessary Conditions for Extended Noncontextuality in General Sets of Random Variables
• Graph Approach to Extended Contextuality
• Experimental test of high-dimensional quantum contextuality based on contextuality concentration
• Quantum sets of the multicolored-graph approach to contextuality
• Graphical Framework for Non-Gaussian Quantum States
• The quantum maxima for the basic graphs of exclusivity are not reachable in Bell scenarios
• Trade-off relations between Bell nonlocality and local Kochen-Specker contextuality in generalized Bell scenarios