We show that the maximum quantum violation of the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality is exactly the maximum value satisfying the following principle: The sum of probabilities of pairwise exclusive events cannot exceed 1. We call this principle "global exclusivity," since its power shows up when it is applied to global events resulting from enlarged scenarios in which the events in the inequality are considered jointly with other events. We identify scenarios in which this principle singles out quantum contextuality, and show that a recent proof excluding nonlocal boxes follows from the maximum violation imposed by this principle to the KCBS inequality.
Why do correlations between the results of measurements performed on physical systems violate Bell and non-contextuality inequalities up to some specific limits? The answer may follow from the observation that in quantum theory, unlike in other theories, whenever there is an experiment to measure
A simultaneously with
B, another to measure
B with
C, and another to measure
A with
C, there is always an experiment to measure all of them simultaneously. This property implies that quantum theory satisfies a seemingly irrelevant restriction called the exclusivity (E) principle which, surprisingly, explains the set of quantum correlations in some fundamental scenarios. An open problem is whether the E principle explains the maximum quantum violation of the Bell-CHSH inequality. Here we show experimentally that the E principle imposes an upper bound to the violation of the Bell-CHSH inequality that matches the maximum predicted by quantum theory. For that, we use the result of an independent experiment testing a specific non-contextuality inequality. We perform both experiments: the Bell-CHSH inequality experiment on polarization entangled states of pairs of photons in Stockholm and, to demonstrate independence, the non-contextuality inequality experiment on single photons' orbital angular momentum states in Rome. The observed results provide the first experimental evidence that the E principle determines the limits of quantum correlations and prove that hypothetical super-quantum violations for either experiment would violate the E principle. This supports the conclusion that the E principle captures a fundamental limitation of nature. If this is true, much of quantum theory trivially follow from merely taking the E principle to be a fundamental truth, and various information-theoretic postulates are also simplified and/or strengthened.
In a recent article entitled "A simple explanation of the quantum violation of a fundamental inequality," Cabello proposes a condition on a class of probabilistic models that, he claims, gives the same bound on contextuality for the KCBS inequality as quantum mechanics, and also rules out PR-box nonlocality. He conjectures that the condition will also reproduce quantum limits on contextuality in other scenarios. Here we show that the proposed principle is actually too weak to derive these results. Cabello has implicitly assumed in the proofs that if all pairs in a set of events are pairwise exclusive (so that their probabilities must sum to less than 1), the set can itself be considered exclusive. Perhaps surprisingly, this is not in general true in the general probabilistic framework under discussion. We call this hidden assumption "Consistent Exclusivity" or CE. With this extra assumption Cabello's proofs are sound. Furthermore, it is established that CE holds in quantum mechanics, providing a reasonable and simple new principle that may characterise quantum non-contextuality in many scenarios.
The exclusivity principle (E-principle) states that the sum of probabilities of pairwise exclusive events cannot exceed 1. Unlike other principles proposed to characterize quantum correlations, its intrinsically non-bipartite formulation enables its application in more general contextuality scenarios. Although equivalent to the no-signalling condition for any bipartite Bell scenario, this equivalence breaks down for two independent copies of the same scenario. Such violation of the E-principle due to multiple copies, known as its activation effect, was studied in [Nat Commun 4, 2263 (2013)] for the nonlocal extremal boxes of
(2,m,2),
(2,2,d), and
(3,2,2) Bell scenarios. The authors mapped the problem of exhibiting activation effects to finding certain cliques inside joint exclusivity graphs. In this work, we refine the joint exclusivity structure to be an edge-colored exclusivity multigraph. This allows us to draw a novel connection to Ramsey theory, which guarantees the existence of certain monochromatic subgraphs in sufficiently large edge-colored cliques, providing a powerful tool for ruling out E-principle violations. We then exploit this connection, drawing on Ramsey-theoretic results to study violations of the E-principle by multiple copies of the contextual extremal boxes of
n-cycle scenarios, called
n-cycle PR boxes. For the usual (
n=4) PR box we show that the known E-principle violation of
5/4 is the maximal achievable using two copies, and that this same upper bound applies to two copies of the KCBS (
n=5) PR box. We then prove that
n≥6-cycle PR boxes, unlike the extremal boxes of the aforementioned Bell scenarios, do not exhibit activation effects with two or three copies. Finally, for any number of independent copies
k, we establish a lower bound on
n above which
n-cycle PR boxes do not exhibit activation effects with
k copies.
A fundamental problem is to understand why quantum theory only violates some noncontextuality (NC) inequalities and identify the physical principles that prevent higher-than-quantum violations. We prove that quantum theory only violates those NC inequalities whose exclusivity graphs contain, as induced subgraphs, odd cycles of length five or more, and/or their complements. In addition, we show that odd cycles are the exclusivity graphs of a well-known family of NC inequalities and that there is also a family of NC inequalities whose exclusivity graphs are the complements of odd cycles. We characterize the maximum noncontextual and quantum values of these inequalities, and provide evidence supporting the conjecture that the maximum quantum violation of these inequalities is exactly singled out by the exclusivity principle.
Recently, many simple principles have been proposed that can explain quantum limitations on possible sets of experimental probabilities in nonlocality and contextuality experiments. However, few implications between these principles are known. Here it is shown that the lack of irreducible third-order interference (a generalization of the idea that no probabilistic interference remains unaccounted for once we have taken into account interference between pairs of slits in a n-sit experiment) implies the principle known as the E principle or consistent exclusivity (that, if each pair of a set of experimental outcomes are exclusive alternatives in some measurement, then their probabilities are consistent with the existence of a further measurement in which they are all exclusive). This is a step towards a more unified understanding of quantum nonlocality and contextuality, which promises to allow derivations of important results from minimal, easily grasped assumptions. As one example, this result implies that lack of third-order interference bounds violation of the Clauser-Horne-Shimony-Holt-Bell inequality to 2.883.