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

TL;DR

This paper uses hypergraph models to analyze LOCC transformations of multipartite CAT states, establishing conditions for LOCC incomparability and characterizing the structure of ensembles with respect to LOCC transformations.

Contribution

It introduces hypergraph-based criteria for LOCC incomparability of multipartite CAT state ensembles, extending previous results and exploring the structure of LOCC transformations.

Findings

LOCC incomparable ensembles correspond to hypergraphs with same number of hyperedges

The largest set of mutually LOCC incomparable ensembles grows exponentially with parties

Partial entropic criteria sometimes fail to determine LOCC incomparability

Abstract

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 $r\geq 2$, 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 $r\geq 2$, 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.