It is proved that all states of three spin-21 particles exhibiting an ``all versus nothing'' contradiction between quantum mechanics and the local realism of Einstein, Podolsky, and Rosen are exactly the Greenberger-Horne-Zeilinger (GHZ) states and the states obtained from them by local unitary transformations. The proof is obtained by showing that there are at most four elements (except for a different sign) in a set of mutually commuting nonlocal spin observables in the three-qubit system and using the certain algebraic properties that Pauli's matrices satisfy. We show that only does such a set of four nonlocal spin observables present a Greenberger-Horne-Zeilinger-Mermin-like argument. This also reveals the equivalence between the GHZ theorem and maximal violation of the Bell inequality.
Nonlocality is the defining feature of quantum entanglement. Entangled states with multiple particles are of crucial importance in fundamental tests of quantum physics as well as in many quantum information tasks. One of the archetypal multipartite quantum states, Greenberger-Horne-Zeilinger (GHZ) state, allows one to observe the striking conflict of quantum physics to local realism in the so-called all-versus-nothing way. This is profoundly different from Bell's theorem for two particles, which relies on statistical predictions. Here, we demonstrate an integrated photonic chip capable of generating and manipulating the four-photon GHZ state. We perform a complete characterization of the four-photon GHZ state using quantum state tomography and obtain a state fidelity of 0.729±0.006. We further use the all-versus-nothing test and the Mermin inequalities to witness the quantum nonlocality of GHZ entanglement. Our work paves the way to perform fundamental tests of quantum physics with complex integrated quantum devices.