Abstract It is well known that entanglement is one of the distinctive aspects of the quantum world. The measurement and quantification of entanglement is one of the major problems in understanding and applying this resource. This research aims to investigate entanglement measures and their properties in high-dimensional systems. One such property is entanglement monogamy, viz. quantum correlations are not shareable, different from the classical ones. On the other hand, the study of entanglement depends on the definitions of its measures. Here we examine the applicability of concurrence, an entanglement measure known for its effectiveness in the analysis of qubit entanglement. While the entanglement of two-level systems or qubits is widely understood, its extension to higher dimensions remains challenging. Our research establishes equivalence between different concurrence definitions for bipartite qubit pure states and extends them to qudits. Subsequently, we investigate the monogamous relations of these measurements to explore their characteristics. This approach might enhance our comprehension of quantum traits in qudit systems and advance our understanding of entanglement in broader quantum contexts.
Unlike classical correlation, quantum entanglement cannot be freely shared among many parties. This restricted shareability of entanglement among multi-party systems is known as monogamy of entanglement, which is one of the most fundamental properties of entanglement. Here, we summarize recent theoretical progress in the field of monogamy of entanglement. We firstly review the standard CKW-type monogamy inequalities in terms of various entanglement measures. In particular, the squashed entanglement and one-way distillable entanglement are monogamous for arbitrary dimensional systems. We then introduce some generalized version of monogamy inequalities which extend and sharpen the traditional ones. We also consider the dual polygamy inequalities for multi-party systems. Moreover, we present two new definitions to define monogamy of entanglement. Finally, some challenges and future directions for monogamy of entanglement are highlighted.
Exploring the shareability and distribution of entanglement possesses fundamental significance in quantum information tasks. In this paper, we demonstrate that the square of bipartite entanglement measures
Gq-concurrence, which is the generalization of concurrence, follows a set of hierarchical monogamy relations for any
N-qubit quantum state. On the basis of these monogamy inequalities, we render two kinds of hierarchical indicators that exhibit evident advantages in the capacity of witnessing entanglement. Moreover, we show an analytical relation between
Gq-concurrence and concurrence in
2⊗d systems. Furthermore, we rigorously prove that the monogamy property of squared
Gq-concurrence is superior to that of squared concurrence in
2⊗d2⊗d3⊗⋯⊗dN systems. In addition, several concrete examples are provided to illustrate that for multilevel systems, the squared
Gq-concurrence satisfies the monogamy relation, even if the squared concurrence does not. These results better reveal the intriguing characteristic of multilevel entanglement and provide critical insights into the entanglement distribution within multipartite quantum systems.
It is well known that a particle cannot freely share entanglement with two or more particles. This restriction is generally called monogamy. However the formal quantification of such restriction is only known for some measures of entanglement and for two-level systems. The first and broadly known monogamy relation was established by Coffman, Kundu, and Wootters for the square of the concurrence. Since then, it is usually said that the entanglement of formation is not monogamous, as it does not obey the same relation. We show here that despite that, the entanglement of formation cannot be freely shared and therefore should be said to be monogamous. Furthermore, the square of the entanglement of formation does obey the same relation of the squared concurrence, a fact recently noted for three particles and extended here for N particles. Therefore the entanglement of formation is as monogamous as the concurrence. We also numerically study how the entanglement is distributed in pure states of three qubits and the relation between the sum of the bipartite entanglement and the classical correlation.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
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.