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 = λP 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, forIn 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 = λP 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 − μI 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 − μI is compact for some μ ∈ C is also studied.Published by Elsevier Inc.

Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations

“…Results on Λ k (A) for infinite dimensional operators have been obtained in [10] An open question in [2] concerns the lower bound of dim H which ensures that Λ k (A) is nonempty for every bounded linear operator A acting on H. The following result was proved in [9] and answers the above question. …”

Product of operators and numerical range

“…[9][10][11][12][13][14][15] It has been shown that k (A) possesses many nice properties as W (A). For example, if A ∈ M n and 1 ≤ k < n, then, k (A) is always convex [15] and equals the set of complex number λ ∈ C for which Re (e −iθ λ) ≤ λ k Re e −iθ A for all real θ [11,Theorem 2.2].…”

Spectra, norms and numerical ranges of generalized quadratic operators

“…For an infinite dimensional operator A, one can extend the definition of rank-k numerical range to Λ ∞ (A) defined as the set of scalars λ ∈ C such that P AP = λP for an infinite rank orthogonal projection P on H, see [20,25]. Evidently, Λ ∞ (A) consists of those λ ∈ C for which there exists an infinite orthonormal set {x i ∈ H : i ≥ 1} such that Ax i , x j = δ i j λ for all i, j ≥ 1.…”

“…Evidently, Λ ∞ (A) consists of those λ ∈ C for which there exists an infinite orthonormal set {x i ∈ H : i ≥ 1} such that Ax i , x j = δ i j λ for all i, j ≥ 1. It is shown in [20] that …”