Complete Genuine Multipartite Entanglement Monotone

TL;DR

This paper introduces a new framework for quantifying genuine multipartite entanglement, proposing classes of complete entanglement monotones that improve upon existing measures and clarify monogamy relations in quantum systems.

Contribution

It develops conditions for complete and monogamous multipartite entanglement measures, introducing a novel class based on the maximal reduced function, and compares their effectiveness to existing measures.

Findings

Proposed a class of complete MEMs and GMEMs based on maximal reduced functions.

Showed that the new GMEMs outperform existing bipartite-based measures in tripartite systems.

Clarified the relations among monogamy, complete monogamy, and tight complete monogamy.

Abstract

A complete characterization and quantification of entanglement, particularly the multipartite entanglement, remains an unfinished long-term goal in quantum information theory. As long as the multipartite system is concerned, the relation between the entanglement contained in different partitions or different subsystems need to take into account. The complete multipartite entanglement measure and the complete monogamy relation is a framework that just deals with such a issue. In this paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, and tightly complete monogamous according to the feature of the reduced function. Especially, with the assumption that the maximal reduced function is nonincreasing on average under LOCC, we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced function for the first time. By comparison, it is shown that, for the tripartite case, this class of GMEMs is better than the one defined from the minimal bipartite entanglement in literature under the framework of complete MEM and complete monogamy relation. In addition, the relation between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light of different kinds of MEMs and GMEMs.