Multipartite Composition of Contextuality Scenarios

Abstract

Contextuality is a particular quantum phenomenon that has no analogue in classical probability theory. Given two independent systems, a natural question is how to represent such a situation as a single test space. In other words, how separate contextuality scenarios combine into a joint scenario. Under the premise that the the allowed probabilistic models satisfy the No Signalling principle, Foulis and Randall defined the unique possible way to compose two contextuality scenarios. When composing strictly-more than two test spaces, however, a variety of possible composition methods have been conceived. Nevertheless, all these formally-distinct composition methods appear to give rise to observationally equivalent scenarios, in the sense that the different compositions all allow precisely the same sets of classical and quantum probabilistic models. This raises the question of whether this property of invarianceunder-composition-method is special to classical and quantum probabilistic models, or if it generalizes to other probabilistic models as well, our particular focus being Q 1 models. Q 1 models are physically important since, when applied to scenarios constructed by a particular composition rule, coincide with the well-defined "Almost Quantum Correlations". In this work we see that some composition rules, however, give rise to scenarios with inequivalent allowed sets of Q 1 models. We find that the non-trivial dependence of Q 1 models on the choice of composition method is apparently an artifact of failure of those composition rules to capture the orthogonality relations given by the Local Orthogonality principle. We prove that Q 1 models satisfy invariance-undercomposition-method for all the constructive compositions protocols which do capture this notion of Local Orthogonality.

Contextuality and Expressivity of Non-locality

“…A natural link to be investigate is the relation between this "conditional contextuality" and almost quantum correlations analysed in [8,10] and also with the notion of "no detection events" used recently by Kunjwal [9] to relate graph contextuality of [6] to hypergraph contextuality.…”

Modelling contextuality by probabilistic programs with hypergraph semantics

“…Perhaps surprisingly it is an issue that has received scant attention to date (Dzhafarov and Kujala, 2014b). When signalling is not present, it would be interesting to investigate how variations of multipartite composition of contextuality being investigated in physics may inspire new experimental designs outside of physics (Sainz and Wolfe, 2017).…”

“…We are now in a position to illustrate the semantics of the P-program of Figure 7 by the corresponding contextuality scenario depicted in Figure 11. Observe how the FR product produces the extra edges that span the events across measurement contexts labeled P1, P2, P3 and P4 when compared with the direct product hypergraph depicted in Figure 10, At first these spanning edges may seem arbitrary, but they happen to guarantee that the allowable probabilistic models over the composite contextuality scenario X A ⊗ FR X B satisfy the "no signalling" condition (Sainz and Wolfe, 2017). By way of illustration, the normalization condition on edges imposes the following constraints (see Figure 11):…”

Almost Quantum Correlations are Inconsistent with Specker's Principle

“…A theory satisfies the principle of consistent exclusivity (CE) if every probabilistic model p admissible by the theory satisfies all the relevant CE inequalities 8 . 7 For a discussion on how to define such Bell hypergraphs we refer the reader to [8,28]. 8 A stronger version of the Consistent Exclusivity principle imposes constraints on the probabilistic models of a contextuality scenario by considering its application in larger scenarios that contain the original one (see [8]).…”

“…For a discussion on how to define such Bell hypergraphs we refer the reader to[8,28] 8. A stronger version of the Consistent Exclusivity principle imposes constraints on the probabilistic models of a contextuality scenario by considering its application in larger scenarios that contain the original one (see[8]).…”