Limits on non-local correlations from the structure of the local state space

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Nonlocality is arguably one of the most remarkable features of
quantum mechanics. On the other hand nature seems to forbid other
no-signaling correlations that cannot be generated by quantum systems.
Usual approaches to explain this limitation is based on information
theoretic properties of the correlations without any reference to
physical theories they might emerge from. However, as shown in [PRL 104,
140401 (2010)], it is the structure of local quantum systems that
determines the bipartite correlations possible in quantum mechanics. We
investigate this connection further by introducing toy systems with
regular polygons as local state spaces. This allows us to study the
transition between bipartite classical, no-signaling and quantum
correlations by modifying only the local state space. It turns out that
the strength of nonlocality of the maximally entangled state depends
crucially on a simple geometric property of the local state space, known
as strong self-duality. We prove that the limitation of nonlocal
correlations is a general result valid for the maximally entangled state
in any model with strongly self-dual local state spaces, since such
correlations must satisfy the principle of macroscopic locality. This
implies notably that Tsirelson’s bound for correlations of the maximally
entangled state in quantum mechanics can be regarded as a consequence of
strong self-duality of local quantum systems. Finally, our results also
show that there exist models which are locally almost identical to
quantum mechanics, but can nevertheless generate maximally nonlocal