Complete $k$-partite entanglement measure

The $k$-partite entanglement, which focus on at most how many particles in the global system are entangled but separable from other particles, is complementary to the $k$-entanglement that reflects how many splitted subsystems are entangled under partitions of the systems in characterizing multipartite entanglement. Very recently, the theory of the complete $k$-entanglement measure has been established in [Phys. Rev. A 110, 012405 (2024)]. Here we investigate whether we can define the complete measure of the $k$-partite entanglement. Consequently, with the same spirit as that of the complete $k$-entanglement measure, we present the axiomatic postulates that a complete $k$-partite entanglement measure should require. Furthermore, we present two classes of $k$-partite entanglement measures and show that one is complete while the other one is unified but not complete except for the case of $k=2$.