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Optimal nonlocal multipartite entanglement concentration based on projection measurements

We propose an optimal nonlocal entanglement concentration protocol (ECP) for multi-photon 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.