On the Cohomology of Contextuality

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.Comment: In Proceedings QPL 2016, arXiv:1701.0024

On Measures and Measurements: a Fibre Bundle approach to Contextuality

Contextuality is the failure of "local" probabilistic models to become global ones. In this paper we introduce the notions of \emph{measurable fibre bundles}, \emph{probability fibre bundles}, and \emph{sample fibre bundle} which capture and make precise the former statement. The central notions of contextuality are discussed under this formalism, examples worked out, and some new aspects pointed out.Comment: 14 pages; no figures; Purdue-Winer Memorial Lectures 2018; submitted to Phil. Trans. Roy. Soc. A. Comments are very welcom

Contextuality and bundle diagrams

Contextuality describes the local consistency but global inconsistency of data. In this paper, we review and generalize the bundle diagram representation introduced in [S. Abramsky, R. S. Barbosa, K. Kishida, R. Lal, and S. Mansfield, Contextuality, Cohomology and Paradox, 24th EACSL Annual Conference on Computer Science Logic, Vol. 41 (CSL, 2015), p. 211-228 ] to graphically demonstrate the contextuality of diverse empirical models (a collection of probability distributions for jointly measurable observables)