Simplicial quantum contextuality

TL;DR

This paper introduces a novel topological framework for quantum contextuality using simplicial sets, extending measurement models and providing new proofs and witnesses for foundational quantum theorems.

Contribution

It develops a topologically inspired formalism for contextuality based on simplicial sets, generalizing previous models and proofs in quantum foundations.

Findings

Presented a new proof of Fine's theorem using simplicial distributions.

Defined cohomological witnesses for strong contextuality.

Reformulated Gleason's and Kochen-Specker theorems within the simplicial framework.

Abstract

We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason's theorem and Kochen-Specker theorem can be expressed naturally within this new language.