Spectral presheaves, kochen-specker contextuality, and quantale-valued relations

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Abstract

In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.

Original languageEnglish
Pages (from-to)386-398
Number of pages13
JournalElectronic Proceedings in Theoretical Computer Science, EPTCS
Volume266
DOIs
Publication statusPublished - 27 Feb 2018
Event14th International Conference on Quantum Physics and Logic and IQSA Quantum Structures Workshop - Nijmegen, Netherlands
Duration: 3 Jul 20177 Jul 2017

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Quantum theory
Algebra
Topology

Keywords

  • quantum theory
  • Gelfand spectrum
  • Kochen–Specker Theorem

Cite this

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title = "Spectral presheaves, kochen-specker contextuality, and quantale-valued relations",
abstract = "In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.",
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Spectral presheaves, kochen-specker contextuality, and quantale-valued relations. / Dunne, Kevin.

In: Electronic Proceedings in Theoretical Computer Science, EPTCS, Vol. 266, 27.02.2018, p. 386-398.

Research output: Contribution to journalConference Contribution

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