Asymptotic equipartition property of subadditive multipartite entanglement measures on pure states

We investigate the asymptotic equipartition property (AEP) in the context of multipartite entanglement measures on pure states. Specifically, we formulate AEP for subadditive entanglement measures that admit certain weak conditions. This is motivated by the uniqueness of the entanglement entropy in the asymptotic limit in the bipartite case. On the other hand, its operational relevance comes from the $\text{LOCC}_q$ scenario (asymptotic local operations and classical communication with a sublinear amount of quantum communication). Analogously to the classical AEP, we prove that the regularization of smooth weakly additive entanglement measures (subject to some weak extra conditions) yields weakly additive and asymptotically continuous entanglement measures. Then evaluate the mentioned regularization and smoothing on known Rényi type multipartite entanglement measures, showing that the resulting regularized entanglement measures reduce to convex combinations of bipartite entanglement entropies.