We investigate the physically allowed probabilities for transforming one $N$-partite $W$-class state to another by means of local operations assisted with classical communication. Recently,   S. Kintaş and S. Turgut [J. Math. Phys. 51, 092202 (2010)] obtained an upper bound for the maximum probability of transforming two such states. Here, we provide a simple sufficient and necessary condition for when this upper bound can be satisfied and, thus, when optimality of state transformation can be achieved. Our discussion involves obtaining lower bounds for the transformation of arbitrary $W$-class states and showing precisely when this bound saturates the bound of Kintaş and Turgut. Finally, we consider the question of transforming symmetric $W$-class states and find that, in general, the optimal one-shot procedure for converting two symmetric states requires a nonsymmetric filter by all the parties.

• Received 29 September 2010

DOI:https://doi.org/10.1103/PhysRevA.82.062314