Abstract Quantum entanglement is known to be monogamous, i.e. it obeys strong constraints on how the entanglement can be distributed among multipartite systems. Almost all the entanglement monotones so far are shown to be monogamous. We explore here a family of entanglement monotones with the reduced functions are concave but not strictly concave and show that they are not monogamous. They are defined by four kinds of the ‘partial-norm’ of the reduced state, which we call them partial-norm of entanglement, minimal partial-norm of entanglement, reinforced minimal partial-norm of entanglement, and partial negativity, respectively. This indicates that, the previous axiomatic definition of the entanglement monotone needs supplemental agreement that the reduced function should be strictly concave since such a strict concavity can make sure that the corresponding convex-roof extended entanglement monotone is monogamous. Here, the reduced function of an entanglement monotone refers to the corresponding function on the reduced state for the measure on bipartite pure states.
Abstract Quantum entanglement is known to be monogamous, i.e. it obeys strong constraints on how the entanglement can be distributed among multipartite systems. Almost all the entanglement monotones so far are shown to be monogamous. We explore here a family of entanglement monotones with the reduced functions are concave but not strictly concave and show that they are not monogamous. They are defined by four kinds of the ‘partial-norm’ of the reduced state, which we call them partial-norm of entanglement, minimal partial-norm of entanglement, reinforced minimal partial-norm of entanglement, and partial negativity, respectively. This indicates that, the previous axiomatic definition of the entanglement monotone needs supplemental agreement that the reduced function should be strictly concave since such a strict concavity can make sure that the corresponding convex-roof extended entanglement monotone is monogamous. Here, the reduced function of an entanglement monotone refers to the corresponding function on the reduced state for the measure on bipartite pure states.
We show that any measure of entanglement that on pure bipartite states is given by a strictly concave function of the reduced density matrix is monogamous on pure tripartite states. This includes the important class of bipartite measures of entanglement that reduce to the (von Neumann) entropy of entanglement. Moreover, we show that the convex roof extension of such measures (e.g., entanglement of formation) is monogamous also on mixed tripartite states. To prove our results, we use the definition of monogamy without inequalities recently put forward [Gour and Guo, Quantum 2, 81 (2018)]. Our results promote the concept that monogamy of entanglement is a property of quantum entanglement and not an attribute of some particular measures of 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.
Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.
A complete characterization and quantification of entanglement, particularly the multipartite entanglement, remains an unfinished long-term goal in quantum information theory. As long as the multipartite system is concerned, the relation between the entanglement contained in different partitions or different subsystems need to take into account. The complete multipartite entanglement measure and the complete monogamy relation is a framework that just deals with such a issue. In this paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, and tightly complete monogamous according to the feature of the reduced function. Especially, we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced function for the first time. By comparison, it is shown that, for the tripartite case, this class of GMEMs is better than the one defined from the minimal bipartite entanglement in literature under the framework of complete MEM and complete monogamy relation. In addition, the relation between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light of different kinds of MEMs and GMEMs.
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.