Higher rank numerical ranges and low rank perturbations of quantum channels

Abstract

For a positive integer k, the rank- k numerical range Λ k ( A ) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that P A P = λ P for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λ k ( A ) is established. In particular, for 1 ⩽ r < k it is shown that Λ k ( A ) ⊆ Λ k − r ( A + F ) for any operator F with rank ( F ) ⩽ r . In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a ( k − r ) -dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λ k ( A ) can be obtained as the intersection of Λ k − r ( A + F ) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λ k ( A ) are completely determined. Analogous results are obtained for Λ ∞ ( A ) defined as the set of scalars λ such that P A P = λ P for an infinite rank orthogonal projection P. It is shown that Λ ∞ ( A ) is the intersection of all Λ k ( A ) for k = 1 , 2 , … . If A − μ I is not compact for all μ ∈ C , then the closure and the interior of Λ ∞ ( A ) coincide with those of the essential numerical range of A. The situation for the special case when A − μ I is compact for some μ ∈ C is also studied.