Multipartite entanglement measures based on geometric mean

In this paper, we investigate $k$-nonseparable $(2\leq k\leq n)$ entanglementmeasures based on geometric mean of all entanglement values of $k$-partitionsin $n$-partite quantum systems. We define a class of entanglement measurescalled $k$-GM concurrence which explicitly detect all $k$-nonseparable statesin multipartite systems. It is rigorously shown that the $k$-GM concurrencecomplies with all the conditions of an entanglement measure. Compared to $k$-MEconcurrence[\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 severaldifferent aspects, embodying that (i) $k$-GM concurrence can reflect thedifferences in entanglement but $k$-ME concurrence fails at times, (ii) $k$-GMconcurrence does not arise sharp peaks when the pure state being measuredvaries continuously, while $k$-ME concurrence appears discontinuity points,(iii) the entanglement order is sometimes distinct. In addition, we establishthe relation between $k$-ME concurrence and $k$-GM concurrence, and furtherderive a strong lower bound on the $k$-GM concurrence by exploiting thepermutationally invariant part of a quantum state. Furthermore, we parameterize$k$-GM concurrence to obtain two categories of more generalized entanglementmeasures, $q$-$k$-GM concurrence $(q>1, 2\leq k\leq n)$ and $\alpha$-$k$-GMconcurrence $(0\leq\alpha<1, 2\leq k\leq n)$, which fulfill the propertiespossessed by $k$-GM concurrence as well. Moreover, $\alpha$-$2$-GM concurrence$(0<\alpha<1)$, as a type of genuine multipartite entanglement measures, isproven in detail satisfying the requirement that the GHZ state is moreentangled than the $W$ state in multiqubit systems.