Matrix Analysis and Operator Theory with Applications to Quantum Information Theory

dc.contributor.advisorKribs, David
dc.contributor.authorPlosker, Sarah
dc.date.accessioned2013-07-12T15:02:25Z
dc.date.available2013-07-12T15:02:25Z
dc.date.copyright2013-07
dc.date.created2013-07-09
dc.date.issued2013-07-12
dc.degree.departmentDepartment of Mathematics and Statisticsen_US
dc.degree.grantorUniversity of Guelphen_US
dc.degree.nameDoctor of Philosophyen_US
dc.degree.programmeMathematics and Statisticsen_US
dc.description.abstractWe explore the connection between quantum error correction and quantum cryptography through the notion of conjugate (or complementary) channels. This connection is at the level of subspaces and operator subsystems; if we use a more general form of subsystem, the link between the two topics breaks down. We explore both the subspace and subsystem settings. Error correction arises as a means of addressing the issue of the introduction of noise to a message being sent from one party to another. Noise also plays a role in quantum measurement theory: If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Such a setup amounts to the introduction of noise to the measurement process, yet has the advantage of preserving the measurement statistics. Preprocessing by a quantum channel leads to the partial order "cleaner than" on quantum probability measures. Other meaningful partial orders on quantum probability measures exist, and we shall investigate that of cleanness as well as that of absolute continuity. Lastly, we investigate partial orders on vectors corresponding to quantum states; such partial orders, namely majorization and trumping, have been linked to entanglement theory. We characterize trumping first by means of yet another partial order, power majorization, which gives rise to a family of examples. We then characterize trumping through the complete monotonicity of certain Dirichlet polynomials corresponding to the states in question. This not only generalizes a recent characterization of trumping, but the use of such mathematical objects simpli es the derivation of the result.en_US
dc.description.sponsorshipNatural Sciences and Engineering Research Council of Canada
dc.identifier.urihttp://hdl.handle.net/10214/7281
dc.language.isoenen_US
dc.publisherUniversity of Guelphen_US
dc.rights.licenseAll items in the Atrium are protected by copyright with all rights reserved unless otherwise indicated.
dc.subjectquantum error correctionen_US
dc.subjectquantum cryptographyen_US
dc.subjectprivate quantum channelsen_US
dc.subjectmajorizationen_US
dc.subjecttrumpingen_US
dc.subjectpositive operator valued measuresen_US
dc.subjectquantum probability measuresen_US
dc.titleMatrix Analysis and Operator Theory with Applications to Quantum Information Theoryen_US
dc.typeThesisen_US

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