Dagger Linear Logic and Categorical Quantum Mechanics

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dc.contributor.advisorCockett, Robin
dc.contributor.advisorGour, Gilad
dc.contributor.authorSrinivasan, Priyaa Varshinee
dc.contributor.committeememberWoelfel, Philipp
dc.contributor.committeememberBauer, Kristine
dc.date2021-11
dc.date.accessioned2021-10-05T16:53:18Z
dc.date.available2021-10-05T16:53:18Z
dc.date.issued2021-09
dc.description.abstractThis thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distributive categories, and *-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. The proof theory of non-compact dagger linear logic is obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences to give a dagger LDC. From every (isomix) dagger LDC one can extract a canonical "unitary core" which up to equivalence is the traditional CQM framework of dagger monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs): every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix dagger LDC. Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis. This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.en_US
dc.identifier.citationSrinivasan, P. V. (2021). Dagger linear logic and categorical quantum mechanics (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.en_US
dc.identifier.doihttp://dx.doi.org/10.11575/PRISM/39337
dc.identifier.urihttp://hdl.handle.net/1880/114030
dc.language.isoengen_US
dc.publisher.facultyScienceen_US
dc.publisher.institutionUniversity of Calgaryen
dc.rightsUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.en_US
dc.subjectCategorical Quantum Mechanicsen_US
dc.subjectCategory Theoryen_US
dc.subjectDagger linear logicen_US
dc.subjectQuantum physicsen_US
dc.subjectFrobenius algebrasen_US
dc.subjectMonoidal categoriesen_US
dc.subjectLinearly distributive categoriesen_US
dc.subject.classificationEducation--Mathematicsen_US
dc.subject.classificationPhysicsen_US
dc.subject.classificationComputer Scienceen_US
dc.titleDagger Linear Logic and Categorical Quantum Mechanicsen_US
dc.typedoctoral thesisen_US
thesis.degree.disciplineComputer Scienceen_US
thesis.degree.grantorUniversity of Calgaryen_US
thesis.degree.nameDoctor of Philosophy (PhD)en_US
ucalgary.item.requestcopytrueen_US
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