Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer $k$, the rank-$k$ numerical range ${\mathrm{\Lambda }}_k(A)$ of an operator $A$ acting on a Hilbert space $\text{cH}$ of 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 ${\mathrm{\Lambda }}_k(A)$ is established. In particular, for $1\le r<k$ it is shown that $\Lambda_k(A) \subseteq \Lambda_{k-r}(A+F)$foranyoperator$F$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 ${\mathrm{\Lambda }}_k(A)$ can be obtained as the intersection of ${\mathrm{\Lambda }}_{k-r}(A+F)$ 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 ${\mathrm{\Lambda }}_k(A)$ are completely determined. Analogous results are obtained for ${\mathrm{\Lambda }}_{\mathrm{\infty }}(A)$ defined as the set of scalars $\lambda$ such that $PAP = \lambda P$foraninfiniterankorthogonalprojection$P$\mathrm{.}Itisshownthat$\Lambda_\infty(A)$istheintersectionofall$\Lambda_k(A)$for$k = 1, 2,

…$\mathrm{.}If$A - \mu I$isnotcompactforany$\mu \in \IC$, then the closure and the interior of ${\mathrm{\Lambda }}_{\mathrm{\infty }}(A)$ coincide with those of the essential numerical range of $A$. The situation for the special case when $A-\mu I$ is compact for some $\mu \in \text{IC}$ is also studied.