Unified view of monogamy relations for different entanglement measures

TL;DR

This paper introduces a unified framework for understanding monogamy and polygamy relations in quantum entanglement measures, proposing the concepts of monogamy and polygamy power and establishing their existence across all multipartite systems.

Contribution

It formalizes the conditions under which entanglement measures obey monogamy and polygamy relations, introducing the novel concepts of monogamy and polygamy power and providing rigorous proofs.

Findings

Monogamy power and polygamy power exist in all multipartite systems.

Entanglement measures generally violate the polygamy relation.

The paper formalizes when entanglement measures obey monogamy and polygamy relations.

Abstract

A particularly interesting feature of nonrelativistic quantum mechanics is the monogamy laws of entanglement. Although the monogamy relation has been explored extensively in the last decade, it is still not clear to what extent a given entanglement measure is monogamous. We give here a conjecture on the amount of entanglement contained in the reduced states by observing all the known related results at first. Consequently, we propose the monogamy power of an entanglement measure and the polygamy power for its dual quantity, the assisted entanglement, and show that both the monogamy power and the polygamy power exist in any multipartite systems with any dimension, from which we formalize exactly for the first time when an entanglement measure and an assisted entanglement obey the monogamy relation and the polygamy relation respectively in a unified way. In addition, we show that any entanglement measure violates the polygamy relation, which is misstated in some recent papers. Only the existence of monogamy power is conditioned on the conjecture, all other results are strictly proved.