  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 determinations 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
founded, 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.
