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 etal 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.

真正纠缠的确定是量子信息处理中的核心问题。我们通过合并两个系统来研究作为两个二分纠缠态的张量积的三重态。我们表明，当两个二分态的范围都是纠缠破裂（EB）子空间时，三重态是一个真正的纠缠（GE）状态。当两个二分之一状态排名第二时，我们将进一步研究三重状态。我们的结果提供了论文中提出的猜想的最新进展[Yi Shen et al 2020 J. Phys。A 53125302]。我们将我们的结果应用于构造多部分态，该多部分态的二部分降密度算符具有加性形成纠缠（EOF）。此外，在本地操作和经典通信的情况下，此类状态在每个分区中都是可提取的。