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 ∈ M [sub n] has eigenvalues $a₁ ,...,a [sub n], then its higher rank numerical range A [sub k](A) is the intersection of convex polygons with vertices a [sub j1],...,a [sub j] [sub n-k+1], where $1 ≤ j [sub 1] <...< j [sub n-k+1]. 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 P, a construction is given for a normal matrix A ∈ M [sub n] with minimum n such that A [sub k](A)=P. In particular, if P has p vertices, with p⋝3, there is a normal matrix A ∈ M [sub n] with n ≤ max{p+k-1,2k+2} such that A [sub k](A)=P.