We present an efficient multipartite entanglement purification protocol (MEPP) for
N-photon systems in a Greenberger-Horne-Zeilinger state with parity-check detectors. It contains two parts. One is the conventional MEPP with which the parties can obtain a high-fidelity
N-photon ensemble directly, similar to the MEPP with controlled-not gates. The other is our recycling MEPP in which the entanglement link is used to produce some
N-photon entangled systems from entangled ${N}^{\ensuremath{'}}$-photon subsystems ($2\ensuremath{\leqslant}{N}^{\ensuremath{'}}<N$) coming from the instances which are just discarded in all existing conventional MEPPs. The entangled ${N}^{\ensuremath{'}}$-photon subsystems are obtained efficiently by measuring the photons with potential bit-flip errors. With these two parts, the present MEPP has a higher efficiency than all other conventional MEPPs.
We present an efficient scheme for the preparing and purifying of the four-photon Greenberger–Horne–Zeilinger (GHZ) state based on linear optics and postselection. First, we describe how to create a four-photon GHZ state in both polarization and spatial degrees of freedom from two pairs. Moreover, in the presence of depolarization noise our scheme is capable of purifying the desired state. In the regime of weak nonlinearity we design an indirect photon number-resolving detection to distinguish two states of the two pairs. At last, a fourfold coincidence detector click indicates the creation of a polarization-entangled four-photon GHZ state.