Yiu-Tung Poon

Another proof of a result of Westwick

Let be the set of all n × n unitary matrices and A, C two n × n complex matrices with C hermitian. In this note, the author gives another proof of the following result: the set is convex.

Condition for the higher rank numerical range to be non-empty

It is shown that the rank-k numerical range of every n-by-n complex matrix is non-empty if k < n/3 + 1. The proof is based on a recent characterization of the rank-k numerical range by Li and Sze, the Hellys theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that rank-2 numerical range is non-empty if n ≥ 4. This confirms a conjecture of Choi et al. If k ≥ n/3 + 1, an n-by-n complex matrix is given for which the rank-k numerical range is empty. An extension of the result of bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.

Convexity of the Joint Numerical Range

Let A=(A1, . . ., Am) be an m-tuple of n × n Hermitian matrices. For

The automorphism group of separable states in quantum information theory

1 \le k \le n

HIGHER RANK NUMERICAL RANGES AND LOW RANK PERTURBATIONS OF QUANTUM CHANNELS

, the

A K-theoretic invariant for dynamical systems

k

Recovery in quantum error correction for general noise without measurement

{\rm th} joint numerical range of A is defined by

Principal Submatrices of a Hermitian Matrix

W_k(A) = \{ ({\rm \tr}(X^*A_1X), \dots, {\rm \tr}(X^*A_mX) ): X \in {\bf C}^{n\times k}, X^*X = I_k \}.