Simplicial quantum contextuality

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.

48 pages, 15 figures

References in corpus (5)

• Logical Bell Inequalities
• Cohomology and the Algebraic Structure of Contextuality in Measurement Based Quantum Computation
• Simplicial distributions, convex categories and contextuality
• Contextuality in the Bundle Approach, n-Contextuality, and the Role of Holonomy
• Mermin polytopes in quantum computation and foundations

Cited by in corpus (2)

• Topological methods for studying contextuality: $N$-cycle scenarios and beyond
• Harvesting Contextuality from the Vacuum