Quantifying the Bell gap in expressiveness of machine learning models

Quantify, for specific machine learning architectures, the gap in representable distributions between models restricted to commutative operations (which lie inside the local-realist polytope) and models with non-commutative operations (which can represent distributions outside this polytope).

Background

The authors propose viewing model expressiveness through the lens of correlation polytopes: commutative classical models correspond to the local-realist polytope, while non-commutative mechanisms (e.g., quantum circuits) can access distributions beyond it.

They define the "Bell gap" as the expressiveness difference and state that measuring it for concrete architectures remains unresolved.