Tripartite genuinely entangled states from entanglement-breaking subspaces

The determination of genuine entanglement is a central problem in quantum information processing. We investigate the tripartite state as the tensor product of two bipartite entangled states by merging two systems. We show that the tripartite state is a genuinely entangled (GE) state when the range of both bipartite states are entanglement-breaking (EB) subspaces. We further investigate the tripartite state when one of the two bipartite states has rank two. Our results provide the latest progress on a conjecture proposed in the paper [Yi Shen et al 2020 J. Phys. A 53 125302]. We apply our results to construct multipartite states whose bipartite reduced density operators have additive entanglement of formation (EOF). Further, such states are distillable across every bipartition under local operations and classical communications.