The logic of contextuality
- URL: http://arxiv.org/abs/2011.03064v1
- Date: Thu, 5 Nov 2020 19:04:04 GMT
- Title: The logic of contextuality
- Authors: Samson Abramsky, Rui Soares Barbosa
- Abstract summary: Contextuality is a key signature of quantum non-classicality.
We study the logic of contextuality in the setting of partial Boolean algebras.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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 \`a 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.
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