A combinatorial approach for studying LOCC transformations of multipartite states

We develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among $n$ agents. Further, we discuss a new graph theoretic metric on a class of multi-partite states, and its implications.