The Cohomology of Non-Locality and Contextuality

In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual. We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.

💡 Research Summary

The paper “The Cohomology of Non‑Locality and Contextuality” builds on the sheaf‑theoretic framework for quantum non‑locality and contextuality introduced by Abramsky and Brandenburger, and shows how Čech cohomology can be used to define a concrete obstruction that witnesses contextual behaviour.
In the sheaf setting, a measurement scenario is modelled by a finite set X of measurement labels together with a cover U={C₁,…,Cₙ} of compatible contexts. For each context C the presheaf of events E assigns the set O^C of outcome assignments (O is a fixed set of outcomes). An empirical model e consists of a family of probability distributions {e_C} on each E(C) that satisfy the no‑signalling (compatibility) condition. The support of e determines a sub‑presheaf S_e⊂E. By applying the free‑abelian‑group functor F_Z to S_e one obtains an abelian presheaf F=F_Z(S_e) whose sections are formal integer linear combinations of outcome assignments that lie in the support.

Čech cohomology is then defined with respect to the cover U. For each q one forms the cochain group C^q(U,F)=∏_{σ∈N_q}F(|σ|) where N_q is the set of q‑simplices of the nerve of U, and the coboundary maps δ^q are the usual alternating sums of restriction homomorphisms. The zeroth cohomology group H⁰(U,F) is in bijection with compatible families of sections, i.e. with global sections of the original presheaf when they exist.

The central construction proceeds as follows. Choose a particular section s₁∈S_e(C₁). Because the model is compatible, there exist sections s_i∈S_e(C_i) (i=2,…,n) that agree with s₁ on overlaps. Assemble these into a 0‑cochain c=(s₁,…,sₙ)∈C⁰(U,F). Its Čech coboundary z=δ⁰(c) is a 1‑cochain whose components are the differences s_i|{C_i∩C_j}−s_j|{C_i∩C_j}. By the no‑signalling condition these differences vanish when restricted further to C₁, so z actually lives in the relative cochain group C¹(U,F̅_{C₁}), where F̅_{C₁} is the kernel of the restriction to C₁. Consequently z is a 1‑cocycle in the relative cohomology group H¹(U,F̅_{C₁}). The cohomology class γ(s₁)=