Separability of symmetric states and vandermonde decomposition

Abstract

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.

Constructing unextendible product bases from multiqubit ones

“…So the four vectors form a 4 × 4 singular matrix, which has rank at most three. However, by applying Matlab to matrix (34), we have shown that such a matrix does not exist. Hence, the product vector |u, v, w, y 8 A:D:E:BC does not exist.…”

“…the PPT entangled states are related to the entanglement distillation problem [27,28], the detection of entanglement [29,30], as well as multipartite genuinely entangled states and entanglement-breaking subspaces [31,32]. The study of multipartite PPT states has also been devoted to the separability of completely symmetric states [33], the separability of symmetric states and vandermonde decomposition [34]. Hence, it is an important problem to construct novel PPT entangled states from UPBs, which may evidently motivated the study of fore-mentioned topics.…”

Constructing unextendible product bases from multiqubit ones

“…the PPT entangled states are related to the entanglement distillation problem [27,28], the detection of entanglement [29,30], as well as multipartite genuinely entangled states and entanglement-breaking subspaces [31,32]. The study of multipartite PPT states has also been devoted to the separability of completely symmetric states [33], the separability of symmetric states and vandermonde decomposition [34]. Hence, it is an important problem to construct novel PPT entangled states from UPBs, which may evidently motivate the study of the aforementioned topics.…”

Symmetric inseparability and number entanglement in charge conserving mixed states

“…where ρ i A and ρ i B are density matrices for the two subsystems A and B = Ā, with probabilities ∑ i p i = 1. Determining whether a mixed state is separable is NP hard [6][7][8] . The criterion for a state to have finite negativity, i.e.…”