Higher rank numerical ranges and low rank perturbations of quantum channels

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 $μ\in \IC$, 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 $μ\in \IC$ is also studied.

21 pages

References in corpus (2)

• Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations
• Quantum Error Correcting Codes From The Compression Formalism

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