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Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer $k$, the rank-$k$ numerical range $Λ_k(A)$ of an operator $A$ acting on a Hilbert space $\cH$ 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 $1 \le r < k$ it is shown that $Λ_k(A) \subseteq Λ_{k-r}(A+F)$ for any operator $F$ with $\rank (F) \le r$. In quantum computing, this result implies that a quantum channel with a $k$-dimensional error correcting code under a perturbation of rank $\le r$ 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 $Λ_\infty(A)$ defined as the set of scalars $λ$ such that $PAP = λP$ for an infinite rank orthogonal projection $P$. It is shown that $Λ_\infty(A)$ is the intersection of all $Λ_k(A)$ for $k = 1, 2, >...$. If $A - μI$ is not compact for any $μ\in \IC$, then the closure and the interior of $Λ_\infty(A)$ coincide with those of the essential numerical range of $A$. The situation for the special case when $A-μI$ is compact for some $μ\in \IC$ is also studied.