Abstract. The results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in C. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix and to verify the solvability of certain matrix equations.

We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to preservers of the product numerical range.

It is shown that the rank-$k$ numerical range of every $n$-by-$n$ complex matrix is non-empty if $n \ge 3k - 2$. The proof is based on a recent characterization of the rank-$k$ numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that $\Lambda_2(A)$ is non-empty if $n \ge 4$. This confirms a conjecture of Choi et al. If $3k-2>n>0$, an $n$-by-$n$ complex matrix is given for which the rank-$k$ numerical range is empty. Extension of the result to bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.Comment: 4 pages; to appear in LAM

For a positive integer k, the rank-k numerical range Λ k (A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that P A P = 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, forIn 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 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 ∞ (A) defined as the set of scalars λ such that P A P = λP for an infinite rank orthogonal projection P . It is shown that Λ ∞ (A) is the intersection of all Λ k (A) for k = 1, 2, . . . . If A − μI is not compact for all μ ∈ C, then the closure and the interior of Λ ∞ (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 ∈ C is also studied.Published by Elsevier Inc.

Let M n be the set of n × n complex matrices, and for every A ∈ M n , let Sp(A) denote the spectrum of A. For various types of products A 1 * • • • * A k on M n , it is shown that a mapping φ : M n → M n satisfying Sp(A 1 * • • • * A k) = Sp(φ(A 1) * • • • * φ(A k)) for all A 1 ,. .. , A k ∈ M n has the form X → ξS −1 XS or A → ξS −1 X t S for some invertible S ∈ M n and scalar ξ. The result covers the special cases of the usual product A