We present two two-step practical entanglement concentration protocols for concentrating an arbitrary three-particle less-entangled $W$ state into a maximally entangled $W$ state assisted with single photons. The first protocol uses the linear optics and the second protocol adopts the cross–Kerr nonlinearity to perform the protocol. In the first protocol, based on the postselection principle, three parties say Alice, Bob, and Charlie in different distant locations can obtain the maximally entangled $W$ state from the arbitrary less-entangled $W$ state with a certain success probability. In the second protocol, the parties are not required to possess the sophisticated single-photon detectors and the concentrated photon pair can be retained after performing this protocol successfully. Moreover, the second protocol can be repeated to get a higher success probability. Both protocols may be useful in practical quantum-information applications.

• Received 7 December 2011

DOI:https://doi.org/10.1103/PhysRevA.85.042302