Quantifying multipartite quantum states by ($k+1$)-partite entanglement measures

In this paper, we investigate how to quantify the quantum states of $n$-particles from the point of $(k+1)$-partite entanglement $(1\leq k\leq n-1)$, which plays an instrumental role in quantum nonlocality and quantum metrology. We put forward two families of entanglement measures termed $q$-$(k+1)$-PE concurrence $(q>1)$ and $\alpha$-$(k+1)$-PE concurrence $(0\leq\alpha<1)$, respectively. As far as the pure state is concerned, they are defined based on the minimum in entanglement. Meanwhile, rigorous proofs showing that both types of quantifications fulfill all the requirements of an entanglement measure are provided. In addition, we also propose two alternative kinds of entanglement measures, named $q$-$(k+1)$-GPE concurrence $(q>1)$ and $\alpha$-$(k+1)$-GPE concurrence $(0\leq\alpha<1)$, respectively, where the quantifications of any pure state are given by taking the geometric mean of entanglement under all partitions satisfying preconditions. Besides, the lower bounds of these measures are presented by means of the entanglement of permutationally invariant (PI) part of quantum states and the connections of these measures are offered. Moreover, we compare these measures and explain the similarities and differences among them. Furthermore, for computational convenience, we consider enhanced versions of the above quantifications that can be utilized to distinguish whether a multipartite state is genuinely strong $k$-producible.