Separability of symmetric states and vandermonde decomposition

Symmetry is one of the central mysteries of quantum mechanics and plays an essential role in multipartite entanglement. In this paper, we consider the separability problem of quantum states in the symmetric space. We establish the relation between the separability of multiqubit symmetric states and the decomposability of Hermitian positive semidefinite matrices. This relation allows us to exchange concepts and ideas between quantum entanglement and Vandermonde decomposition. As an application, we build a suite of tools to investigate the decomposability and show the power of this relation both in theoretical and numerical aspects. For theoretical results, we establish the witness for the decomposability similar to the entanglement witness and characterize the decomposability of some subclasses of matrices. Furthermore, we provide the necessary conditions for the decomposability. Besides, we suggest a numerical algorithm to check whether a given matrix is decomposable. The numerical examples are tested to show the effectiveness.