Higher rank numerical ranges of normal matrices

Abstract

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A\in M_n$ has eigenvalues $a_1,\dot{.}.,a_n$, then its higher rank numerical range ${\Lambda }_k(A)$ is the intersection of convex polygons with vertices $a_{j_1},\dot{.}.,a_{j_{n-k+1}}$, where $1\le j_1<\dot{.}.<j_{n-k+1}\le n$. In this paper, it is shown that the higher rank numerical range of a normal matrix with $m$ distinct eigenvalues can be written as the intersection of no more than $\max \{m,4\}$ closed half planes. In addition, given a convex polygon $\mathcal{P}$ a construction is given for a normal matrix $A\in M_n$ with minimum $n$ such that ${\Lambda }_k(A)=\mathcal{P}$. In particular, if $\mathcal{P}$ has $p$ vertices, with $p\ge 3$, there is a normal matrix $A\in M_n$ with $n\le \max \left\{p+k-1,2k+2\right\}$ such that ${\Lambda }_k(A)=\mathcal{P}$.

12 pages, 9 figures, to appear in SIAM Journal on Matrix Analysis and Applications

References in corpus (4)

• Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations
• Quantum Error Correcting Codes From The Compression Formalism
• Condition for the higher rank numerical range to be non-empty
• Higher rank numerical ranges and low rank perturbations of quantum channels

Cited by in corpus (4)

• Higher-rank Numerical Ranges and Kippenhahn Polynomials
• Higher Rank Numerical Ranges of Normal Operators and unitary dilations
• Private quantum codes: introduction and connection with higher rank numerical ranges
• On the boundary of weighted numerical ranges