Multipartite entanglement measures based on geometric mean

In this paper, we investigate $k$-nonseparable $(2\leq k\leq n)$ entanglement measures based on geometric mean of all entanglement values of $k$-partitions in $n$-partite quantum systems. We define a class of entanglement measures called $k$-GM concurrence which explicitly detect all $k$-nonseparable states in multipartite systems. It is rigorously shown that the $k$-GM concurrence complies with all the 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 measures proposed by us emerge several different aspects, embodying that (i) $k$-GM concurrence can reflect the differences in entanglement but $k$-ME concurrence fails at times, (ii) $k$-GM concurrence does not arise sharp peaks when the pure state being measured varies continuously, while $k$-ME concurrence appears discontinuity points, (iii) the entanglement order is sometimes distinct. 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 parameterize $k$-GM concurrence to obtain two categories of more generalized entanglement measures, $q$-$k$-GM concurrence $(q>1, 2\leq k\leq n)$ and $\alpha$-$k$-GM concurrence $(0\leq\alpha<1, 2\leq k\leq n)$, which fulfill the properties possessed by $k$-GM concurrence as well. Moreover, $\alpha$-$2$-GM concurrence $(0<\alpha<1)$, as a type of genuine multipartite entanglement measures, is proven in detail satisfying the requirement that the GHZ state is more entangled than the $W$ state in multiqubit systems.