Generalized Frame Potential and Problems Related to SIC-POVMs

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dc.contributor.advisorBenedetto, John Jen_US
dc.contributor.advisorOkoudjou, Kasso Aen_US
dc.contributor.authorKang, Shujieen_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2020-07-08T05:35:25Z
dc.date.available2020-07-08T05:35:25Z
dc.date.issued2020en_US
dc.description.abstractFrame theory generalizes the idea of bases in Hilbert space, and the frame potential is an important tool when studying frame theory. In this thesis, we first explore the minimization problem of a generalized definition of frame potential, namely the p-frame potential, and show there exists a universal optimizer under certain conditions by applying a method involving ultraspherical polynomials and spherical designs. Next, we further discuss the topic on Grassmannian frames, which are special cases of minimizers of p-frame potentials. We present the construction of equiangular lines in lower dimensions since numerical result showed their connections with Grassmannian frames. We also derive properties of the (6,4)-Grassmannian frame. Then, we obtain lower bounds for the generalized frame potentials in the complex setting. The frame potentials may provide a different approach to determine the existence of Gabor frames that are equiangular. This relates the potential minimization problem to the unsolved Zauner conjecture. In addition, we study the properties of Gramian matrices of Gabor frames in an attempt to search for Gabor frames with a small number of different inner products. We also calculate the number of different inner products in Gabor frames generated by Alltop sequences and Björck sequences. In addition, we also present examples related to a generalized support uncertainty inequality and shift-invariant spaces on LCA groups.en_US
dc.identifierhttps://doi.org/10.13016/ahiu-sdln
dc.identifier.urihttp://hdl.handle.net/1903/26074
dc.language.isoenen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledFrame potentialen_US
dc.subject.pquncontrolledGrassmannian frameen_US
dc.subject.pquncontrolledPotential energy minimizationen_US
dc.subject.pquncontrolledSharp configurationen_US
dc.subject.pquncontrolledSIC-POVMen_US
dc.subject.pquncontrolledSpherical designen_US
dc.titleGeneralized Frame Potential and Problems Related to SIC-POVMsen_US
dc.typeDissertationen_US

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