The Cohomology of Non-locality and Contextuality

Non-locality and contextuality are key features of the empirical predictions of quantum mechanics that do not arise classically. We will explain how the language of sheaf theory can provide a natural and appealing framework in which to analyse the structure of non-locality and contextuality. Empirical correlations form compatible families or sections on a certain presheaf, and classical correlations are characterised by the existence of global sections.

Cohomology theories can, very roughly, be thought of as descriptions of obstructions to solving some kind of equation. It was first suggested by Isham & Butterfield in 1997 that cohomology may be useful to the study of contextuality. This is achieved within our framework: we will show that the Cech cohomology for the presheaf in question may be used to identify obstructions to global sections, thereby providing witnesses of non-locality and contextuality. It will be seen that the approach provides cohomological witnesses corresponding to many of the familiar proofs of non-locality and contextuality.