Geometry and Physics Seminar

In quantum theory, compatible families of observables, known as measurement contexts, can be measured jointly, giving rise to probability distributions on their possible outcomes. The set of all compatible families of such distributions forms a convex polytope. Deterministic distributions, given by a delta distribution on each context, are vertices of this polytope. Their convex hull forms the noncontextual polytope, whose facets are described by the famous Bell inequalities. Distributions outside this subpolytope are called contextual and they display nonclassical features such as not admitting a noncontextual hidden-variable model.

In this talk, I will introduce simplicial distributions, a topological framework for modeling probability distributions on measurement and outcome spaces given by simplicial sets. I will explain how this perspective provides topological methods for studying Bell inequalities and the extremal points of simplicial distribution polytopes. More broadly, our goal is to understand to what extent the geometry of these probability polytopes is controlled by the topology of the simplicial sets modeling the measurement scenario.