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.

References

The gap between the two is the 'Bell gap' in expressiveness, and quantifying it for specific model classes is an open problem.

— Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework  (2603.28973 - Polson et al., 30 Mar 2026) in Section 7.3, Implications for Machine Learning

Several questions then arise. What consequences does this greater expressive power have for models of learning, and can it provide an advantage in concrete inference tasks? What is the maximal quantum value of $\mathcal B_5$, and which priors and measurements attain it? More generally, how do quantum violations of bounds obeyed by conditionally i.i.d.\ classical models depend on the Hilbert-space dimension, and how robust are they to noise?

— Witnessing Quantum Bayesian Inference beyond Classical Learning  (2610.09293 - Buscemi, 7 Oct 2026) in Discussion