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International Journal of Theoretical Physics

Some characterizations of fully separability for multipartite quantum states

Yinzhu Wang, Huimin Wu, Fangyu Zhou, Lili Yang, Donghua Yan

Published Mar 1, 2022• 2 citations
Publisher
DOI
10.1007/s10773-022-05083-0

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Let H 1 , H 2 , …, H n , K ( n ⩾ 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state ρ in H 1 ⊗ H 2 ⊗⋅⋅⋅⊗ H n ⊗ K is fully separable if and only if (Λ⊗Id)ρ ⩾ 0 for all (finite-rank) LPP elementary operators Λ from into . Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.
The maximal overlap with the fully separable state for the multipartite entangled pure state with translational invariance is studied explicitly by some exact and numerical evaluations, focusing on the one-dimensional qubit system and some representative types of translational invariance. The results show that the translational invariance of the multipartite state could have an intrinsic effect on the determination of the maximal overlap and the nearest fully separable state for multipartite entangled states. Furthermore, a hierarchy of the basic entangled states with translational invariance is found, from which one could readily find the maximal overlap and a related fully separable state for the multipartite state composed of different translational invariance structures.