📝 SummarXiv

Abstract

Multipartite entanglement has much more complex structures than bipartite entanglement, such as the semiseparable state. The multipartite state absent of multipartite entanglement is called a 2-producible state, which is a tensor product of at most 2-partite states. Recently, it is proved that a tripartite pure state is 2-producible if and only if the gap between entanglement of purification and its lower bound vanishes. Here, we show that the entanglement of purification gap is not sufficient to detect more than tripartite entanglement with 4-partite random stabilizer states. We then generalize entanglement of purification to the multipartite case, where the gap between generalized entanglement of purification and its lower bound quantifies the quantum communication cost for distributing one part of the multipartite system to the other parts. We also demonstrate that a multipartite state is 2-producible if and only if the generalized entanglement of purification gaps vanish. In addition, we show that the generalized entanglement of purification gaps are related to the local recoverability of the multipartite state from its marginal state on some parts of the system and the distance between the state and the 2-producible states with the relative entropy. Moreover, we calculate the generalized entanglement of purification gaps for the states fulfilling the generalized Schmidt decomposition, which implies that the 4-partite stabilizer states do not always have the generalized Schmidt decomposition. Our results provide a quantitive characterization of multipartite entanglement in multipartite system, which will promote further investigations and understanding of multipartite entanglement.