Operator systems, contextuality and non-locality
Abstract
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.
