This work aims to understand the monogamy of quantum entanglement from a geometrical point of view. By regarding quantum entanglement as a geometrical structure on the state space of quantum systems and attributing all entanglement related properties as emergent from this geometry of entanglement, we assume there exists a genuine general monogamous relation of quantum entanglement w.r.t. a correspondent genuine entanglement measure Q* which possesses an underlying geometrical origin. We speculate that the monogamous relations w.r.t. an entanglement measure Q can be understood by comparing the different dimension dependencies of the measure Q and Q*. We gave evidences of our conjecture by readdressing two observed properties of the monogamy relations from this geometrical standpoint. Besides the phenomenal explanation of the monogamy of entanglement, we also discussed a fibre bundle structure based candidate solution for the geometry of entanglement and explained how this idea is related to the ER=EPR conjecture and other interesting quantum information processing problems including monogamy of entanglement, entanglement distillation, bound entanglement and activation, and entanglement catalyst.
This work aims to understand the monogamy of quantum entanglement from a geometrical point of view. By regarding quantum entanglement as a geometrical structure on the state space of quantum systems and attributing all entanglement related properties as emergent from this geometry of entanglement, we assume there exists a genuine general monogamous relation of quantum entanglement w.r.t. a correspondent genuine entanglement measure Q* which possesses an underlying geometrical origin. We speculate that the monogamous relations w.r.t. an entanglement measure Q can be understood by comparing the different dimension dependencies of the measure Q and Q*. We gave evidences of our conjecture by readdressing two observed properties of the monogamy relations from this geometrical standpoint. Besides the phenomenal explanation of the monogamy of entanglement, we also discussed a fibre bundle structure based candidate solution for the geometry of entanglement and explained how this idea is related to the ER=EPR conjecture and other interesting quantum information processing problems including monogamy of entanglement, entanglement distillation, bound entanglement and activation, and entanglement catalyst.
"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.
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.
This work focuses on the entanglement quantification. Specifically, we will go over the properties of entanglement that should be satisfied by a "good" entanglement measure. We will have a look at some of the propositions of the entanglement measures that have been made over the years. Then we will be ready to discuss the proposals of the mathematical representations of another property of entanglement, called monogamy. We will introduce some definitions of monogamous entanglement measures that were proposed and compare them. As an original observation of mine (see page 15, Proof 1), I will also prove that the inequalities (23) and (24) from [C. Lancien, S. Di Martino, M. Huber, M. Piani, G. Adesso and A. Winter Phys. Rev. Lett., 117:060501 (2016).] automatically show that the entanglement of formation and the regularised entropy of entanglement are monogamous entanglement measures in the sense of the definitions that was given in [G. Gour and G. Yu, Quantum 2, 81 (2018).].
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.
This dissertation will serve as an introduction to entanglement quantification, containing highly detailed proofs ensuring solid understanding of the subject. Specifically, we will review the properties of entanglement that should be satisfied by a "good" entanglement measure. Then we will have a look at some of the propositions of the entanglement measures that have been made over the years. We will review in greater detail the entanglement of formation. We will discuss the proposals of the mathematical representations of another property of entanglement, called monogamy. We will introduce some definitions of monogamous entanglement measures that were proposed and compare them. As an original observation of mine (page 28), I will also show that "Should Entanglement Measures be Monogamous or Faithful?" (arXiv:1604.02189) has already developed most of what is necessary to prove that the entanglement of formation and the regularised entropy of entanglement are monogamous entanglement measures in the sense of the definition that was given in "Monogamy of entanglement without inequalities" (arXiv:1710.03295).
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.