Conjecture: Nonincrease of E′(n) under LOCC on average

Show that the global multipartite quantity E′(n), defined as the maximal single-party reduced-function value over all subsystems, is nonincreasing on average under LOCC, thereby validating it as an entanglement monotone.

Background

The authors consider E′(n) constructed from the maximal reduced function but acknowledge they cannot prove LOCC-average monotonicity.

They explicitly conjecture this property to hold.

References

We can not prove here ${E'}{(n)}$ is nonincreasing on average under LOCC, but we conjecture that ${E'}{(n)}$ does not increase on average under LOCC.

— Measure of entanglement and the monogamy relation: a topical review  (2512.21992 - Guo et al., 26 Dec 2025) in Section 9.4 MEM from the maximal reduced function