Monogamy of negativity and relative entropy of entanglement

Prove or disprove the monogamy of the negativity N and of the relative entropy of entanglement E_r under the improved (disentangling-condition-based) monogamy definition.

Background

The improved monogamy definition avoids inequality checks and has been established for many convex-roof measures.

The authors explicitly state that monogamy for non–convex-roof measures like negativity and relative entropy still remains open.

References

the monogamy of the measures that are not defined via the convex-roof extension such as the negativity and the relative entropy of entanglement still remains open, which is also an interesting mathematical problem.

— Measure of entanglement and the monogamy relation: a topical review  (2512.21992 - Guo et al., 26 Dec 2025) in Section Outlook