We consider linearly independent families of Hermitian matrices {A 1 ,. .. , Am} so that W k (A) is convex. It is shown that m can reach the upper bound 2k(n − k) + 1. A key idea in our study is relating the convexity of W k (A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.

We characterize the relationship between the singular values of a Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its offdiagonal block. We also characterize the eigenvalues of a Hermitian (or real symmetric) matrix C = A+ B in terms of the combined list of eigenvalues of A and B. The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.

In this paper, we discuss the connection between two genuinely quantum phenomena-the discontinuity of quantum maximum entropy inference and quantum phase transitions at zero temperature. It is shown that the discontinuity of the maximum entropy inference of local observable measurements signals the non-local type of transitions, where local density matrices of the ground state change smoothly at the transition point. We then propose to use the quantum conditional mutual information of the ground state as an indicator to detect the discontinuity and the non-local type of quantum phase transitions in the thermodynamic limit.

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