  We propose two schemes to generate four-photon polarization-entangled states
from the second-order emission of the spontaneous parametric down-conversion
process. By using linear optical elements and the coincidence-detection, the
four indistinguishable photons emitted from parametric down-conversion source
result in the Greenberger-Horne-Zeilinger (GHZ) state or the superposition of
two orthogonal GHZ states. For this superposition state, under particular phase
settings we analyze the quantum correlation function and the local hidden
variable (LHV) correlation. As a result, the Bell inequality derived from the
LHV correlation is violated with the visibility larger than 0.442. It means
that the present four-photon entangled state is therefore suitable for testing
the LHV theory.
