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The convex algebraic geometry of higher-rank numerical ranges

Jonathan Nino-Cortes, Cynthia Vinzant

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.

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. 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.

Paper Structure

This paper contains 12 sections, 24 theorems, 65 equations, 8 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries from convex algebraic geometry
  3. Notation
  4. Convex geometry
  5. Duality of plane curves
  6. Coordinate changes
  7. The dual convex cone of rank-$k$ numerical range
  8. A membership test for the rank-$k$ numerical range
  9. The algebraic boundary of $\Lambda_{k}(A)$
  10. Lower dimensional $\Lambda_{k}(A)$
  11. Algorithms
  12. Gallery of Examples

Key Result

Theorem 2.1

For any nonempty convex cone $C$, $(C^{*})^{*} = \overline{C}$.

Figures (8)

  • Figure 1: The curve $f_{A}=0$ and numerical ranges $\Lambda_{1}(A)$ and $\Lambda_{2}(A)$ of the $4\times 4$ matrix $A$ from Example \ref{['ex:quartic1']}. The zero set of the polynomial $g_{A}$ vanishes on the boundaries of $\Lambda _{1}(A)$ and $\Lambda_{2}(A)$.
  • Figure 2: The curves $O_{k}(A)$ from \ref{['ex:pringle']} with the cone $(\widehat{\Lambda_1(A)})^{*}$, the convex hull of $O_{2}(A)$, and higher-rank numerical ranges $\Lambda_{1}(A)$ and $\Lambda_{2}(A)$.
  • Figure 3: The curves $\mathcal{V}(f_{A})$ and $\mathcal{V}(g_{A})$ from \ref{['ex:circleandline']}
  • Figure 4: The curve $\mathcal{V}(f_{A})$ and two $[0:0:1]$-tangent lines from \ref{['ex:quarticPtangent']} in the $\{t=1\}$ and $\{y=1\}$ affine charts and curve $O_{2}(A)$.
  • Figure 5: Kippenhahn curve and dual curve for Example \ref{['ex:tritangent']}. Here we study a family of curves obtained by allowing the circle on the right to be shifted along the line $a = b$. The regions shaded by light and dark blue correspond to $\Lambda_{2}(\tilde{A})$ and $\Lambda_{3}(\tilde{A})$, respectively. As we translate the circle, the rank-3 numerical ranges passes from being 2-dimensional to 0-dimensional to empty.
  • ...and 3 more figures

Theorems & Definitions (63)

  • Example 1.1
  • Theorem 2.1: Biduality
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • ...and 53 more