We propose an optimal nonlocal entanglement concentration protocol (ECP) for multiphoton systems in a partially entangled pure state, resorting to the projection measurement on an additional photon. One party in quantum communication first performs a parity-check measurement on her photon in an
N-photon system and an additional photon, and then she projects the additional photon into an orthogonal Hilbert space for dividing the original
N-photon systems into two groups. In the first group, the
N parties will obtain a subset of
N-photon systems in a maximally entangled state. In the second group, they will obtain some less-entangled
N-photon systems which are the resource for the entanglement concentration in the next round. By iterating the entanglement concentration process several times, the present ECP has the maximal success probability, which is just equivalent to the entanglement of the partially entangled state. That is, this ECP is an optimal one.
We propose two optimal entanglement concentration protocols (ECPs) for arbitrary single-photon multi-mode W state and multi-photon polarization W state, respectively. In both ECPs, we only require one pair of partially entangled W state, and do not consume any auxiliary photon. Both ECPs are based on the linear optics which can be easily realized. On the other hand, the concentrated maximally entangled states can be remained, which are quite different from the previous ECPs. Moreover, for the concentration of the arbitrary single-photon N-mode W state or N-mode polarization W state, the total success probability is equal to Nth the modulus square of the Schmidt coefficient of the smallest magnitude. It makes both ECPs optimal than all the previous ECPs. Our ECPs may be useful in current quantum communication fields.