Bounds on polygons of higher rank numerical ranges

Abstract

Research in higher rank numerical ranges has originally been motivated by problems in quantum information theory, particularly in quantum error correction. The higher rank numerical range generalizes the classical numerical range of an operator. The higher rank numerical range is typically not a polygon, however when we consider normal operators the higher rank numerical range is a polygon in the complex plane C. In this article, we give a new proof of an upper bound on the number of sides of the higher-rank numerical range of a normal operator and we also find a lower bound for the number of sides of the higher-rank numerical range of a unitary operator. We show that these bounds are the best possible.