Maximal extension on converse monogamy of entanglement for tripartite pure states
Junhyeong An, Soojoon Lee
TL;DR
The problem addressed is the extent of the converse monogamy of entanglement ($CMoE$) within a hierarchy of bipartite entanglement for tripartite pure states. The authors extend the prior qualitative ($CMoE$) and distillability-based formulations to two maximal extensions, proving maximality through spectral relations between reduced states and providing explicit counterexamples to delineate boundaries. Key contributions include a unified maximal-extension framework that encompasses both Hayashi–Chen and Singh–Datta results, plus corollaries linking bound entanglement and quantum capacities. Overall, the work clarifies how strong and weak entanglement distribute under hierarchical criteria and identifies open questions about the precise inclusion relations between hierarchy branches such as $RED$ and $UND_{ ightarrow}$.
Abstract
Unlike classical correlations, entanglement cannot be freely shared among multiple parties. This unique feature of quantum systems is known as the monogamy of entanglement. While it holds for all multipartite pure states, its converse -- weak entanglement between two parties enforces strong entanglement with a third party -- occurs only under specific conditions. In particular, Hayashi and Chen [Phys. Rev. A \textbf{84}, 012325 (2011)] demonstrated a qualitative version of the converse monogamy of entanglement (CMoE) for tripartite pure states by employing a hierarchy of bipartite entanglement defined through the relations among various separability criteria, and Singh and Datta [IEEE Trans. Inf. Theory \textbf{69}, 6564 (2023)] later extended this notion of the CMoE from the viewpoint of distillability under one-way or two-way classical communication. In this work, we extend their results to the CMoE with broader conditions, and furthermore show that our extensions are maximal with respect to the hierarchies they considered.