Contextuality Scenarios Arising from Networks of Stochastic Processes

Rodrigo Iglesias, Fernando Tohmé, Marcelo Auday

An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class 𝒳 of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations are coherent, in the sense that the distribution on a face of a simplex coincides with the marginal of the distribution over the entire simplex. An empirical model is called contextual if its distributions cannot be obtained by marginalizing a joint distribution over 𝒳. Contextual empirical models arise naturally in quantum theory, giving rise to some of its counter -intuitive statistical consequences. In this paper, we present a different and classical source of contextual empirical models: the interaction among many stochastic processes. We attach an empirical model to the ensuing network in which each node represents an open stochastic process with input and output random variables. The statistical behaviour of the network in the long run makes the empirical model generically contextual and even strongly contextual.

Ehtibar N. Dzhafarov

Ehtibar N. Dzhafarov

M. Anoussis, Alexandros Chatzinikolaou, Ivan G. Todorov

We introduce an operator system, universal for the probabilistic models of a contextuality scenario, and identify its maximal C∗-cover via the right C∗-algebra of a canonical ternary ring of operators, arising from a hypergraph version of stochastic operator matrices. We study dilating contextuality scenarios, which have the property that each positive operator representation thereof admits a dilation to a projective representation on a larger Hilbert space, and characterise them via the equality of the aforementioned universal operator system and the operator system arising from the canonical generators of the respective hypergraph C∗-algebra. We characterise the no-signalling probabilistic models over a pair of contextuality scenarios of different types, which arise from either the positive operator representations or from the projective representations of these scenarios, in terms of states on operator system tensor products. Generalising the notion of a synchronous no-signalling correlation to the hypergraph framework, we define coherent probabilistic models associated with a given contextuality scenario and characterise various classes thereof via different types of traces of the hypergraph C∗-algebra, associated with the scenario. We establish several equivalent formulations of the Connes Embedding Problem in terms of no-signalling probabilistic models and hypergraph operator systems.

Robert C. Bishop, Peter beim Graben

Moritz F. Richter, Andrea Smirne, Walter T. Strunz +1 more

Abstract In classical stochastic theory, the joint probability distributions of a stochastic process obey by definition the Kolmogorov consistency conditions. Interpreting such a process as a sequence of physical measurements with probabilistic outcomes, these conditions reflect that the measurements do not alter the state of the underlying physical system. Prominently, this assumption has to be abandoned in the context of quantum mechanics, yet there are also classical processes in which measurements influence the measured system. Here, conditions that characterize uniquely classical processes that are probed by a reasonable class of such invasive measurements are derived. We then analyze under what circumstances such classical processes can simulate the statistics arising from quantum processes associated with informationally‐complete measurements. It is expected that this investigation will help build a bridge between two fundamental traits of non‐classicality, namely, coherence and contextuality.