Existence of incomparable pure bipartite states in infinite dimensional systems

Based on set theoretic ordering properties, a general formulation for constructing a pair of convertibility monotones, which are generalizations of distillable entanglement and entanglement cost, is presented. The new pair of monotones do not always coincide for pure bipartite infinite dimensional states under SLOCC (stochastic local operations and classical communications), demonstrating the existence of SLOCC incomparable pure bipartite states, a new property of entanglement in infinite dimensional systems, with no counterpart in the corresponding finite dimensional systems.