Quantum nonlocality is possible in the triangle network with no inputs and binary outputs, which is the smallest such scenario by number of variables and outcomes.
Fritz, Beyond Bell’s theorem: correlation scenarios, New J
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abstract
Bell's Theorem witnesses that the predictions of quantum theory cannot be reproduced by theories of local hidden variables in which observers can choose their measurements independently of the source. Working out an idea of Branciard, Rosset, Gisin and Pironio, we consider scenarios which feature several sources, but no choice of measurement for the observers. Every Bell scenario can be mapped into such a \emph{correlation scenario}, and Bell's Theorem then discards those local hidden variable theories in which the sources are independent. However, most correlation scenarios do not arise from Bell scenarios, and we describe examples of (quantum) nonlocality in some of these scenarios, while posing many open problems along the way. Some of our scenarios have been considered before by mathematicians in the context of causal inference.
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Exhaustive search yields conjectured tight Bell inequalities defining the local set for symmetric binary-outcome triangle networks, together with outer approximations used to probe the classical-quantum gap.
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The minimal example of quantum network Bell nonlocality
Quantum nonlocality is possible in the triangle network with no inputs and binary outputs, which is the smallest such scenario by number of variables and outcomes.
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Local models and Bell inequalities for the minimal triangle network
Exhaustive search yields conjectured tight Bell inequalities defining the local set for symmetric binary-outcome triangle networks, together with outer approximations used to probe the classical-quantum gap.