Preparation and purification of four-photon Greenberger-Horne-Zeilinger state

We present an efficient scheme for preparing and purifying of 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 creating of a polarization-entangled four-photon GHZ state.