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arXiv:2211.06984

Should monogamy of entanglement be defined via equalities or inequalities?

Alexey Lopukhin

Published Nov 13, 2022• 0 citations
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Abstract

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).].

DOI
10.48550/arxiv.2211.06984

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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).].
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).
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.
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.
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.
Multipartite entanglement includes not only the genuine entanglement but also the kkk-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 kkk-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the kkk-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete kkk-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 kkk-entanglement measures in detail. We find that all these kkk-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.