An invertible map between Bell non-local and contextuality scenarios

Seminar author:Máté Farkas

Event date and time:2023-01-12 03:00

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Bell non-locality and generalised contextuality are both fundamental quantum phenomena that can tell classical and quantum theories apart in an a priori theory-independent manner. At first sight the two notions might appear unrelated, as Bell non-locality addresses bipartite systems, while generalised contextuality addresses single systems. It has, however, been noticed before that Bell non-locality can be connected to generalised contextuality in some simple cases. In this work, we completely formalise this connection. In particular, we map every correlation in a Bell scenario to a behaviour in an indexed family of contextuality scenarios in an invertible way. Furthermore, our construction maps every quantum realisation of a correlation to a quantum realisation of a behaviour in the corresponding contextuality scenario. As such, the set of quantum correlations in a given Bell scenario is isomorphic to an indexed family of quantum contextual behaviours. Furthermore, the set of local correlations is isomorphic to the set of non-contextual behaviours and the set of no-signalling correlations is isomorphic to the set of contextual behaviours. Our map allows us to derive some fundamental properties of the set of quantum contextual behaviours: In general, the membership problem of this set is undecidable, it is not closed, and finite-dimensional quantum systems are not sufficient for realising every behaviour. Finally, due to the result MIP*=RE, there exist no computable sequence of outer approximations of the quantum contextual set that converges to the set itself.