Higher rank numerical ranges and low rank perturbations of quantum channels

TL;DR

This paper explores the properties of higher rank numerical ranges of operators, their behavior under low rank perturbations, and implications for quantum error correction, providing new insights into the structure and stability of these sets.

Contribution

It establishes a connection between low rank perturbations and higher rank numerical ranges, with applications to quantum channels and error correction codes.

Findings

$ ext{Lambda}_k(A) ext{ is contained in } ext{Lambda}_{k-r}(A+F)$ for rank $F ext{ with } ext{rank}(F) ext{ } ext{leq } r$

$ ext{Lambda}_k(A)$ can be expressed as an intersection of $ ext{Lambda}_{k-r}(A+F)$ sets for normal or finite-dimensional $A$

The structure of $ ext{Lambda}_ extinfty(A)$ is characterized as the intersection of all $ ext{Lambda}_k(A)$ sets.

Abstract

For a positive integer $k$, the rank-$k$ numerical range $\Lambda_k(A)$ of an operator $A$ acting on a Hilbert space $\cH$ 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 $\Lambda_k(A)$ is established. In particular, for $1 \le r < k$ it is shown that $\Lambda_k(A) \subseteq \Lambda_{k-r}(A+F)$ 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 $\Lambda_k(A)$ 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 $\Lambda_\infty(A)$ defined as the set of scalars $\lambda$ such that $PAP = \lambda P$ for an infinite rank orthogonal projection $P$. It is shown that $\Lambda_\infty(A)$ is the intersection of all $\Lambda_k(A)$ for $k = 1, 2, >...$. If $A - \mu I$ is not compact for any $\mu \in \IC$, then the closure and the interior of $\Lambda_\infty(A)$ coincide with those of the essential numerical range of $A$. The situation for the special case when $A-\mu I$ is compact for some $\mu \in \IC$ is also studied.