On the ellipticity of the higher rank numerical range

TL;DR

This paper studies the shape of the higher rank numerical range for certain matrices, showing they are often elliptical, which has implications for quantum error correction and uses a unified approach.

Contribution

It provides a unified analysis of the ellipticity of higher rank numerical ranges for 2-by-2 block matrices, extending previous results with a new approach.

Findings

Higher rank numerical ranges can be elliptical for specific matrices.

The approach unifies previous results on the shape of these sets.

Applications in quantum error correction are highlighted.

Abstract

The higher rank numerical range is a concept that generalizes the classical numerical range, and it has application in quantum error correction. We investigate these sets for $2$-by-$2$ block matrices with associated Kippenhahn curves consisting of ellipses (and eventually points). As a consequence, elliptical higher rank numerical range results are derived in a unified way, using an approach developed by Spitkovsky {\it et al}.