Tripartite entanglement and quantum relative entropy

We establish relations between tripartite pure stare entanglement and addit ivity properties of the bipartite relative entropy of entanglement. Our res ults pertain to the asymptotic limit of local manipulations on a large numb er of copies of the state. We show that additivity of the relative entropy would imply that there are at least two inequivalent types of asymptotic tr ipartite entanglement. The methods used include the application of some use ful lemmas that enable us to analytically calculate the relative entropy fo r some classes of bipartite states.