Separability criteria based on Heisenberg–Weyl representation of density matrices

Abstract

Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for bipartite quantum systems. It is shown that this criterion can be better than the previous ones in detecting entanglement. The results are generalized to multipartite quantum states.

Detection of multipartite entanglement based on Heisenberg-Weyl representation of density matrices

“…Our main structural matrices in the algorithm differ from the Ref. [17] both in structure and in form. We only use a part of correlation tensors in the HW representation to study the entanglement and genuine multipartite entanglement.…”

“…Remark The authors [17] studied the separability for bipartite quantum systems by all correlation tensors and only gave a necessary condition of fully separable for multipartite quantum states. While in our current approach, we study the entanglement under different partition and present fully separable, bi-separable and tri-separable necessary conditions in tripartite and four-partite quantum systems.…”

“…In [16], generalized Pauli matrices based on the HW operators were introduced and a criterion to detect entanglement by the bounds of the sum of expectation values of any set of anti-commuting observables was given. Moreover, separability criteria in terms of the HW observable basis for multipartite quantum systems were presented in [17].…”

Separability Criteria Based on the Weyl Operators

“…Our method further demonstrates that Weyl operators can be widely applied in quantum information realm [ 13 , 14 , 15 ]. Particularly, Weyl operators also play important roles in entanglement detection [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ]. In fact, Ref.…”

“…In fact, Ref. [ 16 ] provided the generalized Pauli matrices based on the Weyl operators, and proposed a criterion to detect entanglement by the bounds of the sum of expectation values of any set of anti-commuting observables. Moreover, the separability criteria in terms of the Weyl operators for bipartite and multipartite quantum systems were presented in Ref.…”