The convex algebraic geometry of higher-rank numerical ranges

TL;DR

This paper explores the convex algebraic geometry of higher-rank numerical ranges, generalizing classical concepts, and introduces an algorithm for their explicit computation, with applications in quantum error correction.

Contribution

It provides a new geometric framework for higher-rank numerical ranges and extends Kippenhahn's theorem, along with an explicit algorithm for calculation.

Findings

Generalization of Kippenhahn's theorem

Algorithm for computing higher-rank numerical ranges

Insights into convex algebraic geometry of these sets

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. 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.