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.

Simplicial distributions, convex categories and contextuality

“…Throughout the paper R will denote a commutative semiring. In this section, we introduce simplicial distributions [OKI22] defined over the semiring R. These objects describe distributions on a simplicial set parametrized by another simplicial set. Our main result is that the set of simplicial distributions constitute an R-convex set, in the sense that it is an algebra over the distribution monad.…”

“…Simplicial distributions are first introduced in [OKI22]. In this section we recall the basic definitions.…”

“…We will write x ⊕ y for the z edge, the XOR of x and y. For more details on this interpretation see [OKI22].…”

“…In general, given the four triangles corresponding to the four boxes in the CHSH scenario, any way of assembling them into a simplicial set provides a description as a simplicial distribution. In addition to the realizations given in this example and Example 2.25 there is another realization where X is a punctured torus; see [OKI22] and [OCI22].…”

“…This approach introduces cohomology classes that can detect strong contextuality but fail to capture weaker versions, e.g., the famous Clauser-Horne-Shimony-Holt (CHSH) scenario [CHSH69]. In [OKI22] both approaches are unified under the theory of simplicial distributions. This theory goes beyond the usual assumption that measurements and outcomes are represented by finite sets.…”