On the ellipticity of the higher rank numerical range

TL;DR

The paper develops a unified geometric description of the higher rank numerical ranges $\Lambda_k(A)$ for block matrices with off-diagonal blocks, by exploiting Kippenhahn curves $C(A)$ that decompose into ellipses. When $A$ has a structure that yields a reducible $\Re(e^{-i\theta}A)$, the boundary of $W(A)$ and the associated $\Lambda_k(A)$ are described as unions and intersections of elliptical discs centered near $\frac{\alpha+\beta}{2}$ with foci at $\alpha$ and $\beta$, determined by singular values of $C\!+\!D^*$ or related quantities. The authors provide explicit formulas for the ellipses, establish when $\Lambda_k(A)$ reduces to line segments or circular discs, and derive several corollaries for structured matrices (tridiagonal, arrowhead, shift), including cases with θ-independent spectra. These results yield a clear, scalable framework for understanding the ellipticity of higher rank numerical ranges with potential applications in quantum information theory and matrix analysis.

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}.

Figures (1)

• Figure 3.1: Boundaries and foci of $\Lambda_3(A)\subset {\color{blue} \Lambda_2(A)} \subset {\color{red} \Lambda_1(A)}$ for $A$ in \ref{['ex']}

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Remark 2.1
• Corollary 3.1
• proof