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arXiv:2104.11411

Characterization of Contextuality with Semi-Module Čech Cohomology and its Relation with Cohomology of Effect Algebras

Sidiney Bruno Montanhano

Published Apr 23, 2021• 2 citations
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Abstract

I present a generalized notion of obstruction in Čech cohomology on semi-modules, which allows one to characterize non-disturbing contextual behaviors with any semi-field. This framework generalizes the usual Čech cohomology used in the sheaf approach to contextuality, avoiding false positives. I revise a similar work done in the framework of effect algebras with cyclic and order cohomologies and explore the relationship with the one presented here.

DOI
10.48550/arxiv.2104.11411

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I present a generalized notion of obstruction in Čech cohomology on semi-modules, which allows one to characterize non-disturbing contextual behaviors with any semi-field. This framework generalizes the usual Čech cohomology used in the sheaf approach to contextuality, avoiding false positives. I revise a similar work done in the framework of effect algebras with cyclic and order cohomologies and explore the relationship with the one presented here.
Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computation. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to the Cech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof that Cech cohomology is at least as powerful.
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.
We present two results on the subject of quantum contextuality and cohomology, and non-locality and quantum advantage with shallow circuits. Abramsky et al. showed that a range of examples of quantum contextuality is detected by a cohomological invariant based on Čech cohomology. However, the approach does not give a complete cohomological characterisation of contextuality. A different cohomological approach to contextuality was introduced by Okay et al. Their approach exploits the algebraic structure of the Pauli operators and their qudit generalisations known as Weyl operators. We give an abstract account of this structure, then generalise their approach to any example of contextuality with this structure. We prove at this general level that the approach does not give a more complete characterisation of contextuality than the Čech cohomology approach. Bravyi, Gosset, and König (BGK) gave the first unconditional proof that a restricted class of quantum circuits is more powerful than its classical analogue. The result, for the class of circuits of bounded depth and fan-in (shallow circuits), exploits a particular family of examples of contextuality. BGK's quantum circuit and computational problem are derived from a family of non-local games related to the well-known GHZ non-local game. We present a generalised version of their construction. A systematic way of taking examples of contextuality and producing unconditional quantum advantage results with shallow circuits.
We present two results on the subject of quantum contextuality and cohomology, and non-locality and quantum advantage with shallow circuits. Abramsky et al. showed that a range of examples of quantum contextuality is detected by a cohomological invariant based on Čech cohomology. However, the approach does not give a complete cohomological characterisation of contextuality. A different cohomological approach to contextuality was introduced by Okay et al. Their approach exploits the algebraic structure of the Pauli operators and their qudit generalisations known as Weyl operators. We give an abstract account of this structure, then generalise their approach to any example of contextuality with this structure. We prove at this general level that the approach does not give a more complete characterisation of contextuality than the Čech cohomology approach. Bravyi, Gosset, and König (BGK) gave the first unconditional proof that a restricted class of quantum circuits is more powerful than its classical analogue. The result, for the class of circuits of bounded depth and fan-in (shallow circuits), exploits a particular family of examples of contextuality. BGK's quantum circuit and computational problem are derived from a family of non-local games related to the well-known GHZ non-local game. We present a generalised version of their construction. A systematic way of taking examples of contextuality and producing unconditional quantum advantage results with shallow circuits.
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.
Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.