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.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.

Physical Review AJul 1, 20247 citations

Detection of k-Partite Entanglement.

Yan Hong

International Journal of Software and InformaticsJan 1, 20140 citations

K‐Nonseparable Entanglement Measures Based on the Complete Information of Reduced States

Rui‐Zhou Liang, Zhi‐Xiang Jin, Wei Chen +1 more

Abstract In this work, the framework of ‐nonseparability is utilized to investigate parameterized multipartite entanglement measures. A ‐nonseparable revised informationally complete entanglement measure(‐RICEM) is proposed for ‐partite quantum systems(), capable of identifying all ‐nonseparable states across arbitrary ‐partite configurations. The ‐RICEM offers enhanced resolution in entanglement characterization compared to conventional measures. It is proven that the proposed ‐nonseparable measure is a valid entanglement measure, satisfying entanglement monotonicity, convexity, and nonzero for ‐nonseparable states. Using properties of ‐concurrence under partial transposition and realignment, analytical lower bounds for mixed states in ‐RICEM are derived. Furthermore, operational connections between ‐RICEM and the global negativity measure are established. A comparative analysis between the genuine multipartite entanglement GME‐RICEM and existing genuine multipartite entanglement measures reveals distinct advantages in state discrimination capability.Abstract In this work, the framework of ‐nonseparability is utilized to investigate parameterized multipartite entanglement measures. A ‐nonseparable revised informationally complete entanglement measure(‐RICEM) is proposed for ‐partite quantum systems(), capable of identifying all ‐nonseparable states across arbitrary ‐partite configurations. The ‐RICEM offers enhanced resolution in entanglement characterization compared to conventional measures. It is proven that the proposed ‐nonseparable measure is a valid entanglement measure, satisfying entanglement monotonicity, convexity, and nonzero for ‐nonseparable states. Using properties of ‐concurrence under partial transposition and realignment, analytical lower bounds for mixed states in ‐RICEM are derived. Furthermore, operational connections between ‐RICEM and the global negativity measure are established. A comparative analysis between the genuine multipartite entanglement GME‐RICEM and existing genuine multipartite entanglement measures reveals distinct advantages in state discrimination capability.

Annalen der PhysikSep 16, 20250 citations

(k+1)

-partite entanglement measure of N-partite quantum states

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.