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 PAP = λP 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 ≤ 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 PAP = λP 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 − μI is not compact for any 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−μI is compact for some μ ∈ C is also studied.