On the Cohomology of Contextuality

Abstract

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.

Towards a complete cohomology invariant for non-locality and contextuality

“…The False positive of [Car17] As mentioned before, the false positive of [Car17] (cf. Figure 3) is particularly intereseting because it concerns all the sections of the model.…”

“…5 This means that the obstruction γ(s) vanishes, and cohomology is unable to detect the contextuality of the Hardy model. Another, more extreme, false positive is presented in [Car17], and is depicted in the bundle diagram of Figure 3. The peculiarity of this model is that coho-…”

“…the Hardy model [Har93]). In fact, subsequent work has highlighted many limits of the Čech cohomology approach, including the fact that it does not provide a full invariant for contextuality even in its strongest form [Car17], and even under strong assumptions on the measurement scenarios.…”

On Measures and Measurements: a Fibre Bundle approach to Contextuality

“…Moreover, are those contextual models obtainable from quantum theory? Or is it true that quantum contextuality really depends on the non-triviality of the first homology group [25]?…”

On measures and measurements: a fibre bundle approach to contextuality

“…Moreover, are those contextual models obtainable from quantum theory? Or is it true that quantum contextuality really depends on the non-triviality of the first homology group [27]?…”