Multigraph approach to quantum non-locality

Abstract

Abstract. Non-contextuality (NC) and Bell inequalities can be expressed as bounds Ω for positive linear combinations S of probabilities of events, S ≤ Ω. 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.

Simple Explanation of the Quantum Limits of Genuinen-Body Nonlocality

“…In contrast, the "sharp measurements," "graph-theoretic," or "contextuality" program [16][17][18] deals with correlations produced by "sharp" measurements, as defined in the framework of general probabilistic theories (GPTs) in Ref. [19], namely, a measurement is sharp when it is repeatable (i.e., it does not perturb any subsequent implementation of the same measurement) and minimally disturbing (i.e., it does not disturb the other measurements with which it is compatible); see also Ref.…”

Exclusivity principle and the quantum bound of the Bell inequality

“…Moreover, by its very definition, the black box approach cannot explain quantum contextual correlations [2]. In addition, there is increasing evidence that the black box approach, with independence of the principle invoked, cannot explain even quantum nonlocal correlations [10].On the other hand, the "sharp measurements," "graphtheoretic," or "contextuality" approach [11][12][13] focus on correlations among the outcomes of sharp measurements. For general probabilistic theories (GPTs), sharp measurements are nondemolition measurements that are repeatable and cause minimal disturbance [14,15].…”

“…On the other hand, the "sharp measurements," "graphtheoretic," or "contextuality" approach [11][12][13] focus on correlations among the outcomes of sharp measurements. For general probabilistic theories (GPTs), sharp measurements are nondemolition measurements that are repeatable and cause minimal disturbance [14,15].…”

On Quantum Possibilistic Paradoxes and Logical Contextuality

“…Further developments on such CSW graph approach, including a recent proposed multigraph approach [82] (but not the multicolored-graph approach [83]), can be found in the textbook of Ref. [19] and in the (in preparation) review [84].…”