Relating Toy Models of Quantum Computation: Comprehension, Complementarity and Dagger Mix Autonomous Categories

Dusko Pavlovic

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

    4 Citations (Scopus)
    70 Downloads (Pure)

    Abstract

    Toy models have been used to separate important features of quantum computation from the rich background of the standard Hilbert space model. Category theory, on the other hand, is a general tool to separate components of mathematical structures, and analyze one layer at a time. It seems natural to combine the two approaches, and several authors have already pursued this idea. We explore categorical comprehension construction as a tool for adding features to toy models. We use it to comprehend quantum propositions and probabilities within the basic model of finite-dimensional Hilbert spaces. We also analyze complementary quantum observables over the category of sets and relations. This leads into the realm of test spaces, a well-studied model. We present one of many possible extensions of this model, enabled by the comprehension construction. Conspicuously, all models obtained in this way carry the same categorical structure, extending the familiar dagger compact framework with the complementation operations. We call the obtained structure dagger mix autonomous, because it extends star autonomous categories, popular in computer science, in a similar way like dagger compact structure extends compact categories. Dagger mix autonomous categories seem to arise quite naturally in quantum computation, as soon as complementarity is viewed as a part of the global structure.
    Original languageEnglish
    Title of host publicationProceedings of the 6th International Workshop on Quantum Physics and Logic (QPL 2009)
    Subtitle of host publication8-9 April 2009 • Oxford, United Kingdom
    EditorsB. Coecke, P. Panangaden, P. Selinger
    Place of PublicationAmsterdam
    PublisherElsevier
    Pages121-139
    Number of pages19
    DOIs
    Publication statusPublished - 14 Feb 2011
    Event6th International Workshop on Quantum Physics and Logic, QPL 2009 - Oxford, United Kingdom
    Duration: 8 Apr 20099 Apr 2009
    Conference number: 6
    http://www.cs.ox.ac.uk/bob.coecke/QPL_09.html

    Publication series

    NameElectronic Notes in Theoretical Computer Science
    PublisherElsevier
    Number2
    Volume270
    ISSN (Print)1571-0661
    ISSN (Electronic)1571-0661

    Workshop

    Workshop6th International Workshop on Quantum Physics and Logic, QPL 2009
    Abbreviated titleQPL
    CountryUnited Kingdom
    CityOxford
    Period8/04/099/04/09
    Internet address

    Keywords

    • Relation
    • Quantum computation
    • Semantics
    • Category
    • Span
    • Complementary observable
    • Comprehension principle
    • Test space
    • Hilbert space

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