Measure of entanglement and the monogamy relation: a topical review

TL;DR

This paper reviews the development of entanglement measures and monogamy relations, highlighting their mathematical properties, interrelations, and the challenges in quantifying entanglement in complex quantum systems.

Contribution

It provides a comprehensive overview of entanglement measures and monogamy relations, emphasizing their characteristics, interrelations, and areas needing improvement in finite-dimensional systems.

Findings

Most measures of entanglement have been historically developed and compared.

Monogamy relations help clarify the distribution of entanglement.

Challenges remain in quantifying entanglement in mixed, high-dimensional states.

Abstract

Characterizing entanglement, including quantifying and distribution of entanglement, which lies at heart of the quantum resource theory, have been investigated extensively ever since Bennett \etal proposed three seminal measures of entanglement in 1996. Up to now, there are numerous measures of entanglement that have been proposed from different point of view and plenty of monogamy relations have been explored which make the distribution of entanglement became more and more clear. While this is relatively easy in the case of pure states, it is much more intricate for the case of mixed quantum states especially with higher dimension and more particles in the system. We present here an overview of the theory along this line. We outline most of the results in this field historically and focus on the finite-dimensional systems. In particular we emphasize the point of view that (i) which yardsticks haven been applied in quantifying entanglement and its distribution, (ii) what are the substantive characteristics and interrelations of these measures and their monogamy relations mathematically by comparing, and (iii) which concepts should be improved or revised and how they were developed accordingly.