Multipartite Entanglement Measure and Complete Monogamy Relation

TL;DR

This paper develops a comprehensive framework for multipartite entanglement measures, establishing complete monogamy relations and classifying various extensions of entanglement measures as genuine or unified, with implications for the structure of maximally entangled states.

Contribution

It introduces a strict framework for multipartite entanglement measures and derives complete monogamy relations, classifying extensions as genuine or unified, and refines the concept of maximally entangled states.

Findings

Multipartite entanglement measures satisfy new hierarchy and unification conditions.

Complete monogamy relations are established for these measures.

Only pure states can be maximally entangled; no mixed maximally entangled states exist.

Abstract

Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in literatures are not complete. We establish here a strict framework for defining multipartite entanglement measure (MEM): apart from the postulates of bipartite measure, a genuine MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM and a tightly complete monogamy relation for the genuine MEM. Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis $q$-entropy of entanglement, R\'{e}nyi $\alpha$-entropy of entanglement, the convex-roof extension of negativity and negativity, respectively. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis $q$-entropy of entanglement are genuine MEMs, (ii) multipartite extensions of R\'{e}nyi $\alpha$-entropy of entanglement, negativity and the convex-roof extension of negativity are unified MEMs but not genuine MEMs, and (iii) all these multipartite extensions are completely monogamous and the ones which are defined by the convex-roof structure (except for the R\'{e}nyi $\alpha$-entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, we find a class of tripartite states that one part can maximally entangled with other two parts simultaneously according to the definition of maximally entangled mixed state (MEMS) in [Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that there is no MEMS and that the only MES is the pure MES.