On the ellipticity of the higher rank numerical range

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}.

16 pages, 1 figure; revised text, corollary and remark moved from Section 2 to Section 3, previous Theorem 3.7 replaced by new Remark 3.2, restatement and condensed proofs for the last theorem and corollary