Convex Geometric Connections to Information Theory

Convex geometry is a field of mathematics that has experienced rapid growth in recent years and has proven to be an extremely useful perspective in areas of research. Problems in many different fields can be interpreted geometrically which often leads to powerful and surprising results. This thesis establishes connections between convex geometry and both classical and quantum information theory. We introduce the mean width bodies and illustrate the geometric interpretation they provide for the relative entropy of cone measures of a convex body and its polar. We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Other connections are made by considering quantum information theory. The fundamental objects in quantum information theory are quantum states. The set of states is convex as are some of its important subsets. Therefore, convex geometry provides a natural approach to explore quantum states. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. Finally, the optimal constants for the spherical isoperimetric inequality are provided and generalizations of a concentration inequality are suggested.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

Case Western Reserve University School of Graduate Studies

Year dc:date

2013

Author dc:creator

• Jenkinson, Justin

Contributors dc:contributor

• Szarek, Stanislaw
• Werner, Elisabeth

• Mathematics

Statement dc:rights

• unrestricted
• This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.

Language dc:language

English

Repository record dc:identifier

http://rave.ohiolink.edu/etdc/view?acc_num=case1365179413

OAI identifier oai:identifier

oai:etd.ohiolink.edu:case1365179413