Optimal values of bipartite entanglement in a tripartite system

For a general tripartite system in some pure state, an observer possessing any two parts will see them in a mixed state. By the consequence of Hughston-Jozsa-Wootters theorem, each basis set of local measurement on the third part will correspond to a particular decomposition of the bipartite mixed state into a weighted sum of pure states. It is possible to associate an average bipartite entanglement ($\bar{\mathcal{S}}$) with each of these decompositions. The maximum value of $\bar{\mathcal{S}}$ is called the entanglement of assistance ($E_A$) while the minimum value is called the entanglement of formation ($E_F$). An appropriate choice of the basis set of local measurement will correspond to an optimal value of $\bar{\mathcal{S}}$; we find here a generic optimality condition for the choice of the basis set. In the present context, we analyze the tripartite states $W$ and $GHZ$ and show how they are fundamentally different.