Should monogamy of entanglement be defined via equalities or inequalities?

TL;DR

This paper examines the properties and mathematical representations of monogamy in entanglement measures, comparing equality and inequality-based definitions, and proves certain measures are monogamous under specific conditions.

Contribution

It introduces new comparisons of monogamy definitions for entanglement measures and proves that certain measures satisfy monogamy properties under established inequalities.

Findings

Inequalities (23) and (24) imply entanglement of formation is monogamous.

Regularised entropy of entanglement is shown to be monogamous.

Comparison of different definitions of monogamous entanglement measures.

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).].