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The Logic of Contextuality

Authors Samson Abramsky , Rui Soares Barbosa

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Author Details

Samson Abramsky
  • Department of Computer Science, University of Oxford, UK
Rui Soares Barbosa
  • INL - International Iberian Nanotechnology Laboratory, Braga, Portugal

Funding

  • Abramsky, Samson: This author acknowledges support from EPSRC - Engineering and Physical Sciences Research Council, EP/T00696X/1, Resources and Coresources: a junction between categorical semantics and descriptive complexity.
  • Barbosa, Rui Soares: This author acknowledges support from EPSRC - Engineering and Physical Sciences Research Council, EP/R044759/1, Combining Viewpoints in Quantum Theory (Ext.), and from FCT - Funda{ção} para a Ci{ê}ncia e a Tecnologia, CEECINST/00062/2018.

Acknowledgements

The authors would like to thank Chris Heunen for helpful discussions.

Cite As Get BibTex

Samson Abramsky and Rui Soares Barbosa. The Logic of Contextuality. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CSL.2021.5

Abstract

Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful.
We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including:  
- establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; 
- formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; 
- investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations;
- developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Quantum computation theory
  • Mathematics of computing
Keywords
  • partial Boolean algebras
  • contextuality
  • exclusivity principles
  • Kochen-Specker paradoxes
  • tensor product

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References

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