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Nonlocal photon correlations and violation of Bell inequalities for spatially separated classical light fields

It is theoretically and experimentally shown that photons emitted by statistically independent incoherent classical light sources and measured in the far field in spatially separated modes may display spatial correlations akin to path-entanglement of photons produced by quantum sources. By measuring higher order photon-correlations at different locations, i.e., $m$ photons in one mode and one photon in another mode, we experimentally demonstrate for $m \geq 6$ a violation of Bell-type inequalities for spatial degrees of freedom. The spatial correlations among the photons can be understood from state projection where the detection of the first $m$ photons projects the sources onto a state which emits the subsequent photon in a strongly correlated manner. From this perspective the entanglement and violation of Bell's inequalities appears as a consequence of nonvanishing cross correlations between noncommuting quadrature phase components of the two spatially separated fields after $m$ photons have been recorded. In this way we show that classical systems may produce spatial field correlations violating local realistic theories.