  We establish relations between tripartite pure state entanglement and
additivity properties of the bipartite relative entropy of entanglement. Our
results pertain to the asymptotic limit of local manipulations on a large
number 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
tripartite entanglement. The methods used include the application of some
useful lemmas that enable us to analytically calculate the relative entropy for
some classes of bipartite states.
