Monogamy of k -entanglement

Multipartite entanglement includes not only the genuine entanglement but also the $k$-entanglement $(k\ensuremath{\geqslant}2)$. It is known that the most bipartite entanglement measures have been shown to be monogamous, but the monogamy relation is involved in the bipartite entanglement measures rather than the multipartite ones. So how we can explore the monogamy relation for $k$-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the $k$-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete $k$-entanglement measure and the associated complete monogamy relation according to the framework of the complete multipartite entanglement measure we established in [Phys. Rev. A 101, 032301 (2020)], which is shown to be an efficient tool for characterizing the multipartite quantum correlation as complementary to the monogamy relation associated with the bipartite measure. Consequently, the relation and the difference between monogamy, complete monogamy, and tightly complete monogamy are clearly depicted in light of the three types of coarser relation of the system partition. We then illustrate our approach with two classes of $k$-entanglement measures in detail. We find that all these $k$-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.