The outcomes of measurements on entangled quantum systems can be nonlocally
correlated. However, while it is easy to write down toy theories allowing
arbitrary nonlocal correlations, those allowed in quantum mechanics are
limited. Quantum correlations cannot, for example, violate a principle known as
macroscopic locality, which implies that they cannot violate Tsirelson's bound.
This work shows that there is a connection between the strength of nonlocal
correlations in a physical theory, and the structure of the state spaces of
individual systems. This is illustrated by a family of models in which local
state spaces are regular polygons, where a natural analogue of a maximally
entangled state of two systems exists. We characterize the nonlocal
correlations obtainable from such states. The family allows us to study the
transition between classical, quantum, and super-quantum correlations, by
varying only the local state space. We show that the strength of nonlocal
correlations - in particular whether the maximally entangled state violates
Tsirelson's bound or not - depends crucially on a simple geometric property of
the local state space, known as strong self-duality. This result is seen to be
a special case of a general theorem, which states that a broad class of
entangled states in probabilistic theories - including, by extension, all
bipartite classical and quantum states - cannot violate macroscopic locality.
Finally, our results show that there exist models which are locally almost
indistinguishable from quantum mechanics, but can nevertheless generate
maximally nonlocal correlations.Comment: 26 pages, 4 figures. v2: Document structure changed. Main theorem has
been extended. It applies to all quantum states now. v3: new abstrac