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).
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).
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).].
"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.
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.
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.
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.
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.