Higher rank numerical ranges and low rank perturbations of quantum channels

[摘要] For a positive integer k, the rank-k numerical range Lambda(k)(A) or an operator A acting on a Hilbert space H or dimension at least k is the set of scalars lambda such that PAP = lambda P for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Lambda(k)(A) is established. In particular, for 1 <= r < k it is shown that Lambda(k)(A) subset of Lambda(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 Lambda(k)(A) can be obtained as the intersection of Lambda(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 Lambda(k)(A) are completely determined. Analogous results are obtained for Lambda(infinity)(A) defined as the set of scalars X such that PAP = lambda P for an infinite rank orthogonal projection P. It is shown that Lambda(infinity)(A) is the intersection of all Lambda(k)(A) for k = 1, 2..... If A - mu I is not compact for all mu is an element of C. then the closure and the interior of A,.(A) coincide with those of the essential numerical range of A. The situation for the special case when A - mu I is compact for some mu is an element of C is also studied. Published by Elsevier Inc.

[发布日期] 2008-12-15

[关键词] Hilbert space;bounded linear operators;higher rank numerical range;quantum error correcting codes;quantum channels