Exclusivity principle, Ramsey theory, and $n$-cycle PR boxes
Raman Choudhary, Rui Soares Barbosa
TL;DR
The paper addresses how the E-principle constrains contextual correlations in KS scenarios, focusing on $n$-cycle PR boxes and activation effects when combining independent copies. It introduces an edge-colored joint exclusivity multigraph Γ^k to preserve the source of exclusivity and connects activation theory with Ramsey theory, enabling noncomputational analysis. The authors show that two copies yield E-principle violations only for $n=4$ (CHSH) and $n=5$ (KCBS), while no activation occurs for $n\ge 6$ with two or even three copies; they derive lower bounds on $n$ as a function of the number of copies and provide bounds showing no violation for larger $k$ when $n$ surpasses certain thresholds. This work reveals a deep link between quantum contextuality, combinatorics, and Ramsey theory, offering a general methodology to rule out E-principle violations in broad KS contexts and guiding future explorations of contextuality via graph-theoretic and combinatorial techniques.
Abstract
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 $n \geq 6$-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.