The Convex Algebraic Geometry of Higher-Rank numerical Ranges

Abstract

The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. In this thesis, we will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn’s theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix. We will also discuss the inverse field of values problem, an inverse problem on the numerical range. We focus on the geometric properties of the set of solutions. Finally, we consider an analogous problem for higher-rank numerical ranges and show how to solve it using the ideas behind the proof of convexity for these sets.