A particularly interesting feature of nonrelativistic quantum mechanics is the monogamy laws of entanglement. Although the monogamy relation has been explored extensively in the last decade, it is still not clear to what extent a given entanglement measure is monogamous. We give here a conjecture on the amount of entanglement contained in the reduced states by observing all the known related results at first. Consequently, we propose the monogamy power of an entanglement measure and the polygamy power for its dual quantity, the assisted entanglement, and show that both the monogamy power and the polygamy power exist in any multipartite systems with any dimension, from which we formalize exactly for the first time when an entanglement measure and an assisted entanglement obey the monogamy relation and the polygamy relation respectively in a unified way. In addition, we show that any entanglement measure violates the polygamy relation, which is misstated in some recent papers. Only the existence of monogamy power is conditioned on the conjecture, all other results are strictly proved.
A particularly interesting feature of nonrelativistic quantum mechanics is the monogamy laws of entanglement. Although the monogamy relation has been explored extensively in the last decade, it is still not clear to what extent a given entanglement measure is monogamous. We give here a conjecture on the amount of entanglement contained in the reduced states by observing all the known related results at first. Consequently, we propose the monogamy power of an entanglement measure and the polygamy power for its dual quantity, the assisted entanglement, and show that both the monogamy power and the polygamy power exist in any multipartite systems with any dimension, from which we formalize exactly for the first time when an entanglement measure and an assisted entanglement obey the monogamy relation and the polygamy relation respectively in a unified way. In addition, we show that any entanglement measure violates the polygamy relation, which is misstated in some recent papers. Only the existence of monogamy power is conditioned on the conjecture, all other results are strictly proved.
Monogamy relation is one of the essential properties of quantum entanglement, which characterizes the distribution of entanglement in a multipartite system. By virtual of the unified-(q,s) entropy, we obtain some novel monogamy and polygamy inequalities in general class of entanglement measures. For the multiqubit system, a class of tighter monogamy relations are established in term of the α-th power of unified-(q,s) entanglement for α ≥ 1. We also obtain a class of tighter polygamy relations in the β-th (0 ≤ β ≤ 1) power of unified-(q,s) entanglement of assistance. Applying these results to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance, and Tsallis-q entanglement of assistance, we obtain the corresponding monogamy and polygamy relations. Typical examples are presented for illustration. Furthermore, the complementary monogamy and polygamy relations are investigated for the α-th (0 ≤ α ≤ 1) and β-th (β ≥ 1) powers of unified entropy, respectively, and the corresponding monogamy and polygamy inequalities are obtained.
"Is entanglement monogamous?" asks the title of a popular article [B. Terhal, IBM J. Res. Dev. 48, 71 (2004)], celebrating C. H. Bennett's legacy on quantum information theory. While the answer is affirmative in the qualitative sense, the situation is less clear if monogamy is intended as a quantitative limitation on the distribution of bipartite entanglement in a multipartite system, given some particular measure of entanglement. Here, we formalize what it takes for a bipartite measure of entanglement to obey a general quantitative monogamy relation on all quantum states. We then prove that an important class of entanglement measures fail to be monogamous in this general sense of the term, with monogamy violations becoming generic with increasing dimension. In particular, we show that every additive and suitably normalized entanglement measure cannot satisfy any nontrivial general monogamy relation while at the same time faithfully capturing the geometric entanglement structure of the fully antisymmetric state in arbitrary dimension. Nevertheless, monogamy of such entanglement measures can be recovered if one allows for dimension-dependent relations, as we show explicitly with relevant examples.
Abstract It is well known that any well-defined bipartite entanglement measure E E obeys γ γ th-monogamy relations Eq. (1.1) and assisted measure Ea E a obeys δ δ th-polygamy relations Eq. (1.2). Recently, we presented a class of tighter parameterized monogamy relation for the α α th (α≥γ) ( α ≥ γ ) power based on Eq. 1.1. This study provides a family of tighter lower (resp. upper) bounds of the monogamy (resp. polygamy) relations in a unified manner. In the first part of the paper, the following three basic problems are focused: tighter monogamy relation for the α α th ( 0≤α≤γ 0 ≤ α ≤ γ ) power of any bipartite entanglement measure E E based on Eq. (1.1); tighter polygamy relation for the β β th ( β≥δ β ≥ δ ) power of any bipartite assisted entanglement measure Ea E a based on Eq. (1.2); tighter polygamy relation for the ω ω th ( 0≤ω≤δ 0 ≤ ω ≤ δ ) power of any bipartite assisted entanglement measure Ea E a based on Eq. (1.2). In the second part, using the tighter polygamy relation for the ω ω th ( 0≤ω≤2 0 ≤ ω ≤ 2 ) power of CoA, we obtain good estimates or bounds for the ω ω th ( 0≤ω≤2 0 ≤ ω ≤ 2 ) power of concurrence for any N-qubit pure states ∣ψ⟩AB1⋯BN−1
Monogamy and polygamy relations are essential properties of quantum entanglement, which characterize the distributions of entanglement in multipartite systems. In this paper, we establish the general monogamy relations for γ-th (0 ≤ γ ≤ α, α ≥ 1) power of quantum entanglement based on unified-(q,s) entanglement and polygamy relations for δ-th (δ ≥ β, 0 ≤ β ≤ 1) power of entanglement of assistance based on unified-(q,s) entanglement of assistance, which provides a complement to the previous research in terms of different parameter regions of γ and δ. These results are then applied to specific quantum correlations, e.g., entanglement of formation, Renyi-q entanglement of assistance and Tsallis-q entanglement of assistance to get the corresponding monogamy and polygamy inequalities. Moreover, typical examples are presented for illustration.
For a bipartite entanglement measure E that satisfies the γth-power monogamy inequality (Eq.~\eqref{e:chap1-ineq1}), and for its assisted counterpart Ea that obeys the δth-power polygamy inequality (Eq.~\eqref{e:chap1-ineq2}), we introduce a unified, tunable framework indexed by a parameter m≥1. Within this framework, we derive two hierarchical families of refined inequalities:
a tightened α-power monogamy relation for E, valid for all α≥mγ;
a tightened β-power polygamy relation for Ea, applicable for (m−1)δ<β≤mδ.
As m increases, the bounds become progressively tighter, recovering known results at m=1. Notably, the optimal monogamy bound emerges as a piecewise function of α, with additional correction terms activated as α crosses successive integer thresholds, thereby offering a sharper characterization of entanglement distribution.
We demonstrate that our results generalize and strengthen existing monogamy and polygamy relations through analytical comparisons and numerical evaluations using concurrence and concurrence of assistance. This hierarchical, parameterized approach offers enhanced and flexible tools for applications in quantum communication, quantum networks, and multipartite quantum information processing.
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.