Unlike classical correlations, entanglement cannot be freely shared among multiple parties. This unique feature of quantum systems is known as the monogamy of entanglement. While it holds for all multipartite pure states, its converse -- weak entanglement between two parties enforces strong entanglement with a third party -- occurs only under specific conditions. In particular, Hayashi and Chen [Phys. Rev. A \textbf{84}, 012325 (2011)] demonstrated a qualitative version of the converse monogamy of entanglement (CMoE) for tripartite pure states by employing a hierarchy of bipartite entanglement defined through the relations among various separability criteria, and Singh and Datta [IEEE Trans. Inf. Theory \textbf{69}, 6564 (2023)] later extended this notion of the CMoE from the viewpoint of distillability under one-way or two-way classical communication. In this work, we extend their results to the CMoE with broader conditions, and furthermore show that our extensions are maximal with respect to the hierarchies they considered.
The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are weakly entangled, then either of them is generally strongly entangled with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is an effective way to this longstanding problem compared to the means by stochastic local operations and classical communications. We also systematically indicate the structure of the classified states and generate them.
The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are entangled and nondistillable, then either of them is distillable with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is a novel and effective way to this long-standing problem compared to the means by stochastic local operations and classical communications. Furthermore we systematically indicate the structure of the classified states and generate them. We also extend our results to multipartite states.
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.
The monogamy of entanglement means that entanglement cannot be freely shared. In 2014, Oliveira et al. [T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini, Phys. Rev. A 89, 034303 (2014)] proposed a monogamy relation in the linear version and considered it in terms of entanglement of formation. Here we generalize the above version and consider a multilinear monogamy relation for a multiqubit system in terms of entanglement of formation and concurrence. Based on the above results, we present an entanglement criterion for genuine entangled states; also we consider the absolutely maximally entangled states and present what an absolutely maximally entangled state is for a three-qubit system. Last, we apply our results to a three-qubit pure state in terms of quantum discord.
Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in the literature being not complete. We establish here a strict framework for defining the multipartite entanglement measure (MEM): apart from the postulates of the bipartite measure, i.e., vanishing on separable measures and nonincreasing under local operations and classical communication, a complete MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition) and a tightly complete monogamy relation for the complete MEM (an MEM is called a complete MEM if it satisfies both the unification condition and the hierarchy condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis
q entropy of entanglement, R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, the convex-roof extension of negativity, and negativity. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis
q entropy of entanglement are complete MEMs; (ii) multipartite extensions of R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, negativity, and the convex-roof extension of negativity are unified MEMs but not complete MEMs; and (iii) all these multipartite extensions are completely monogamous, and the ones which are defined by the convex-roof structure (except for the R\'enyi $\ensuremath{\alpha}$ entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, as a byproduct, we find a class of states that satisfy the additivity of EoF. We also find a class of tripartite states one part of which can be maximally entangled with the other two parts simultaneously according to the definition of mixed maximally entangled state (MMES) in Li et al. [Z. Li, M. Zhao, S. Fei, H. Fan, and W. Liu, Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that the only MES is the pure MES.
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.