Detecting Contextuality: Sheaf Cohomology and All vs Nothing Arguments

The main topic of this thesis is the study of contextuality, a key characteristic feature of quantum mechanics that represents one of the most valuable resources for quantum computation. We will adopt the sheaf-theoretic approach to non-locality and contextuality introduced in [AB11]. This elegant mathematical theory will enable us to consider contextuality on a higher scale, as a general, abstract mathematical property. We will specifically focus on mathematical methods to detect the contextuality of empirical models. In particular, the first part of the thesis is dedicated to All vs Nothing (AvN) arguments [ABK+15], a generalisation of a proof originally formulated by Mermin to demonstrate the strong contextuality of the GHZ model [Mer90]. We develop an algorithm capable of identifying a large number of quantumrealisable states that admit such proofs of contextuality, thus providing a significant amount of concrete examples of strongly contextual models. The second part of the project inspects an algebraic-topological method based on sheaf cohomology [AMSB12, ABK+15]. Contributions in this field include counterexamples to some recently advanced hypothesis (e.g. Conjecture 8.1. in [AMSB12]) as well as new implications and characterisations concerning SC and CSC models based on the injectivity of the cohomological obstructions homomorphisms, and an alternative description of the first cohomology group based on torsors of presheaves. This discussion naturally leads us to a new compact and natural approach to obstructions in higher cohomology groups. We define the higher counterpart of cohomological obstructions and investigate its properties. Of particular interest is the fact that these obstructions are organised in a hierarchy of logical implications, which unfortunately turns out to be not suitable for the study of no-signaling models. In the final chapter, we bring together cohomology and AvN arguments in a unique discussion following the ideas already expressed in [ABK+15].