Entanglement hierarchies in multipartite scenarios

In this paper, we investigate the hierarchical structure of the $n$-partite quantum states. We present a whole set of hierarchical quantifications as a method of characterizing quantum states, which go beyond genuine multipartite entanglement measures and allow for fine identification among distinct entanglement contributions. This kind of quantifications, termed $k$-GM concurrence, can unambiguously classify entangled states into $(n-1)$ distinct classes from the perspective of $k$-nonseparability with $k$ running from $n$ down to 2, and comply with the axiomatic conditions of an entanglement measure. Compared to $k$-ME concurrence [\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.062323} {Phys. Rev. A \textbf{86}, 062323 (2012)}], the hierarchical measures proposed by us embody advantages in distinguishing same class entangled state and measuring continuity. In addition, we establish the relation between $k$-ME concurrence and $k$-GM concurrence, and further derive a strong lower bound on the $k$-GM concurrence by exploiting the permutationally invariant part of a quantum state. Furthermore, we parametrize $k$-GM concurrence to obtain two more general and complete categories of quantifications, $q$-$k$-GM concurrence $(q>1)$ and $\alpha$-$k$-GM concurrence $(0\leq\alpha<1)$, which obey the properties enjoyed by $k$-GM concurrence as well. In particular, $\alpha$-$2$-GM concurrence $(0<\alpha<1)$ determines that the GHZ state and the $W$ state belong to the same hierarchy, and it is proven in detail satisfying the requirement that the GHZ state is more entangled than the $W$ state in multiqubit systems.