Convex Analysis in Quantum Information

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atmire.migration.oldid5841
dc.contributor.advisorGour, Gilad
dc.contributor.authorGirard, Mark
dc.contributor.committeememberSanders, Barry
dc.contributor.committeememberCunningham, Clifton
dc.contributor.committeememberCockett, Robin
dc.contributor.committeememberGühne, Otfried
dc.contributor.committeememberZinchenko, Yuriy
dc.date.accessioned2017-08-04T15:47:58Z
dc.date.available2017-08-04T15:47:58Z
dc.date.issued2017
dc.date.submitted2017en
dc.description.abstractConvexity arises naturally in the study of quantum information. As a result, many useful tools from convex analysis can be used to give important results regarding aspects of quantum information. This thesis builds up methods using core concepts from convex analysis, including convex optimization problems, convex roof constructions, and conic programming, to study mathematical problems related to quantum entanglement. In this thesis, I develop a method for solving convex optimization problems that arise in quantum information theory by analyzing the corresponding converse problem. That is, given an element in a convex set, I determine a family of convex functions that are minimized at this point. This method is used find explicit formulas for the relative entropy of entanglement, as well as other important quantities used to quantify entanglement, and allows one to show important relationships between them. I also construct a practical algorithm that can be used to compute these quantities. This thesis also presents a method to compute convex roofs of arbitrary entanglement measures evaluated on highly symmetric bipartite states. I also establish a framework for completely characterizing quantum resource theories that are convex. For resource theories with a simple mathematical structure, this gives rise to a complete set of resource monotones that can be computed in practice using semidefinite programs. This has applications to the study of entanglement transformations.en_US
dc.identifier.citationGirard, M. (2017). Convex Analysis in Quantum Information (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/25461en_US
dc.identifier.doihttp://dx.doi.org/10.11575/PRISM/25461
dc.identifier.urihttp://hdl.handle.net/11023/4001
dc.language.isoeng
dc.publisher.facultyGraduate Studies
dc.publisher.institutionUniversity of Calgaryen
dc.publisher.placeCalgaryen
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.
dc.subjectMathematics
dc.subject.otherQuantum Information
dc.titleConvex Analysis in Quantum Information
dc.typedoctoral thesis
thesis.degree.disciplineMathematics and Statistics
thesis.degree.grantorUniversity of Calgary
thesis.degree.nameDoctor of Philosophy (PhD)
ucalgary.item.requestcopytrue
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