Higher rank numerical ranges and low rank perturbations of quantum channels

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• 作者：Chi-Kwong Li ; Yiu-Tung Poon ; Nung-Sing Sze
• 关键词：Hilbert space ; Bounded linear operators ; Higher rank numerical range ; Quantum error correcting codes ; Quantum channels
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2008
• 出版时间：15 December 2008
• 年：2008
• 卷：348
• 期：2
• 页码：843-855
• 全文大小：244 K

文摘

For a positive integer k, the rank-k numerical range Λ_k(A) of an operator A acting on a Hilbert space 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 Λ_k(A) is established. In particular, for 1r<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 PAP=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 , 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 is also studied.