A combinatorial approach for studying LOCC transformations of multipartite states

We present a general framework for studying multi-particle entanglement using graph-theoretic methods. We introduce a technique called bicolored merging for determining combinatorial conditions that must be satisfied for two multiparticle states to be comparable under LOCC. We present several results concerning the possibility and impossibility of various state transformations of multi-partite states represented by EPR graphs and entangled hypergraphs under LOCC. We show that there are exponentially many LOCC-incomparable pure multipartite states. We also demostrate a partial ordering of multi-partite states. All the above results have been derived using only the monotonicity postulate for entanglement measures. Further, we discuss a new graph theoretic metric on a class of multi-partite states and its implications.