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A Greenberger Horne Zeilinger (GHZ) entangled state with a phase is crucialfor realizing desired multipartite quantum states for practical applications.Here, we report violations of the general Bell inequality (GBI) introduced in[1] using the four photon polarization entangled phase GHZ state realized viaintrinsic polarization correlation and collective two-photon coherence in the5S1/2 5P3/2 5D5/2 transition of 87Rb atoms. The phase GHZ state can be achievedby the unitary transformation of only one local phase of the four photons.Theoretically, the GHZ state with the {\pi}/4 phase affords maximal violationof the GBI of 2sqrt(2) at the local measurement settings of the Pauli operators{\sigma}_x and {\sigma}_y. We experimentally demonstrate strong violations ofthe GBI of the phase-GHZ state by 47 standard deviations. In addition to theentanglement witness for the phase GHZ state, the results represent a genuinefour photon entanglement of the phase GHZ state, thereby providing a novelresource for realizing photonic quantum computation, magic state distillationfrom entangled states, and quantum networks based on atom photon interactions.

Violation of Bell inequality by four photon Greenberger Horne Zeilinger state with a phase from a warm atomic ensemble