Complete characterization of LOCC preorder on reference states

Determine a complete, necessary-and-sufficient characterization of the LOCC (Local Operations and Classical Communication) preorder on the family of multipartite reference states defined by weight functions α (i.e., tensor products of GHZ states across subsets of parties), specifying precisely when a transformation ψ_β → ψ_α is possible under LOCC. The goal is to extend the paper’s explicit LO characterization (Theorem LOD) and the partial LOCC protocols (local measurement reductions and quantum teleportation) to a full classification of LOCC convertibility within this class.

Background

The paper introduces a broad class of reference states built from GHZ blocks and gives a complete characterization of the LO preorder (Theorem LOD) in terms of divisibility of weight functions. For LOCC, the authors provide several sufficient conditions and explicit protocols (e.g., local measurement reduction and quantum teleportation) but do not obtain a full classification.

They explicitly state that despite these partial results, a full description of the LOCC preorder among reference states remains unresolved and leave this question open for future work.