Mixed-state extension of Rènyi-α entanglement for α>1

Develop a definition of the Rènyi-α entanglement measure for mixed bipartite states when α>1 that yields a well-defined entanglement quantifier (e.g., nonincreasing on average under LOCC), since the current pure-state definition does not satisfy the required monotonicity and its extension to mixed states is not known.

Background

The paper reviews Rènyi-α entropy-based entanglement measures and notes that for α>1, Rènyi entropies are Schur concave but not concave, so the corresponding entanglement measure is defined only for pure states and fails the LOCC-average monotonicity criterion for mixed states.

The authors explicitly state that extending this measure to mixed states remains unknown.

References

Interestingly, the R{e}nyi entropies are Schur concave but not concave when $\alpha>1$, thus $E_\alpha$ is an entanglement measure on pure states, but do not satisfy Eq.~\eref{average} in such a case. As pointed in Ref., it still remains unknown how we can extend such measure to mixed states.

— Measure of entanglement and the monogamy relation: a topical review  (2512.21992 - Guo et al., 26 Dec 2025) in Section 2.3 Rènyi-α entanglement