  An important measure of bipartite entanglement is the entanglement of
formation, which is defined as the minimum average pure state entanglement of
all decompositions realizing a given state. A decomposition which achieves this
minimum is called an optimal decomposition. However, as for the entanglement of
formation, there is not much known about the structure of such optimal
decompositions, except for some special cases, like states of two qubits or
isotropic states. Here we present a necessary and sufficient condition for a
set of pure states of a finite dimensional bipartite system to form an optimal
decomposition. This condition is well suited to treat the question, whether the
entanglement of formation is additive or not.
