Hypergraph-theoretic charaterizations for LOCC incomparable ensembles of multipartite CAT states

Using graphs and hypergraphs to systematically model collections of arbitrary subsets of parties representing {\it ensembles (or collections)} of shared multipartite CAT states, we study transformations between such {\it ensembles} under {\it local operations and classical communication (LOCC)}. We show using partial entropic criteria, that any two such distinct ensembles represented by {\it r-uniform hypergraphs} with the same number of hyperedges (CAT states), are LOCC incomparable for even integers rgeq 2, generalizing results in \cite{mscthesis,sin:pal:kum:sri}. We show that the cardinality of the largest set of mutually LOCC incomparable ensembles represented by r-uniform hypergraphs for even rgeq 2, is exponential in the number of parties. We also demonstrate LOCC incomparability between two ensembles represented by 3-uniform hypergraphs where partial entropic criteria do not help in establishing incomparability. Further we characterize LOCC comparability of EPR graphs in a model where LOCC is restricted to teleportation and edge destruction. We show that this model is equivalent to one in which LOCC transformations are carried out through a sequence of operations where each operation adds at most one new EPR pair.