There are practical motivations to construct genuine tripartite entangled states based on the collective use of two bipartite entangled states. Here, the case that the states are two‐qubit Werner states is considered. The intervals of parameters of two‐qubit Werner states are revealed such that the tripartite state is genuinely entangled. Furthermore, we also investigate the lower bound of genuine multipartite entanglement concurrence for tripartite qudit states. Several examples are given to show the effectiveness of the lower bound.

No abstract

The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen (2011 Phys. Rev. A 84 012325). We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.

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.

The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size 6 and six UPBs of size 9 in H = C 2 ⊗ C 2 ⊗ C 4 , three UPBs of size 9 in K = C 4 ⊗ C 4 , and no UPB of size 7 in H and K. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state ρ of rank seven, and show that it is also a PPT entangled state in H and K, respectively. We analytically derive the geometric measure of entanglement of a special ρ.