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.

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations

“…To prove the theorem, we need the following lemma, which can be found in [1]. We give a short proof using the QR decomposition.…”

“…In addition, from our results one can derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix. As shown in [1], the convexity of the higher rank numerical range is closely related to the study of solvability of matrix equations. Following the idea in [1], we study the solvability of certain matrix equations including those of the form (1.1), (1.2) and…”

“…When k = 1, this concept reduces to the classical numerical range, which is well known to be convex by the Toeplitz-Hausdorff theorem; for example, see [8] for a simple proof. In [1] the authors conjectured that Λ k (A) is convex and reduced the convexity problem to the problem of showing that 0 ∈ Λ k (T ) for…”

Generalized interlacing inequalities

“…Here, we briefly describe the background and refer the readers to [4] for details. Quantum states are represented as density matrices in M n , i.e., positive semidefinite matrices with trace one.…”

On the rank-k numerical range of matrices polynomials

“…The concept of higher rank numerical range of matrices has been studied extensively by Choi et al in [4,5,8,16] and later by the authors in [2,3]. We should note that for k = 1, Λ k (L A ) yields the classical numerical range…”