Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(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 Λk(A) is established. In particular, for 1 ≤ r < k it is shown that Λk(A) ⊆ Λk - 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 Λk(A) can be obtained as the intersection of Λk - 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 Λk(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 Λk(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.