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

TL;DR

This paper explores how the structure of local state spaces in probabilistic theories influences the strength of nonlocal correlations, revealing geometric properties that determine whether such correlations can violate quantum bounds like Tsirelson's.

Contribution

It establishes a connection between local state space geometry and nonlocal correlation strength, including a general theorem limiting violations of macroscopic locality.

Findings

Nonlocal correlations depend on the geometric property of strong self-duality.

Models with polygonal local state spaces can exhibit varying degrees of nonlocality.

Certain models can produce maximally nonlocal correlations while remaining locally similar to quantum mechanics.

Abstract

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.