We present a violation of the Clauser-Horne-Shimony-Holt and the Clauser-Horne inequalities using heralded single photons entangled in momentum and polarization modes. A Mach-Zehnder interferometer and polarization optics are used to rotate the spatial and polarization bases, respectively. With this setup we were able to test quantum mechanics with the original formulation of the Clauser-Horne inequality. The results rule out a wide class of realistic non-contextual hidden-variable theories.

Born's quantum probability rule is traditionally included among the quantum postulates as being given by the squared amplitude projection of a measured state over a prepared state, or else as a trace formula for density operators. Both Gleason's theorem and Busch's theorem derive the quantum probability rule starting from very general assumptions about probability measures. Remarkably, Gleason's theorem holds only under the physically unsound restriction that the dimension of the underlying Hilbert space H must be larger than two. Busch's theorem lifted this restriction, thereby including qubits in its domain of validity. However, while Gleason assumed that observables are given by complete sets of orthogonal projectors, Busch made the mathematically stronger assumption that observables are given by positive operator-valued measures. The theorem we present here applies, similarly to the quantum postulate, without restricting the dimension of H and for observables given by complete sets of orthogonal projectors. We also show that the Born rule applies beyond the quantum domain, thereby exhibiting the common root shared by some quantum and classical phenomena.

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We report theoretical calculations and experimental observations of Pancharatnam's phase originating from arbitrary SU (2) transformations applied to polarization states of light. We have implemented polarimetric and interferometric methods which allow us to cover the full Poincaré sphere. As a distinctive feature, our interferometric array is robust against mechanical and thermal disturbances, showing that the polarimetric method is not inherently superior to the interferometric one, as previously assumed. Our strategy effectively amounts to feed an interferometer with two copropagating beams that are orthogonally polarized with respect to each other. It can be applied to different types of standard arrays, like a Michelson, a Sagnac, or a Mach-Zehnder interferometer. We exhibit the versatility of our arrangement by performing measurements of Pancharatnam's phases and fringe visibilities that closely fit the theoretical predictions. Our approach can be easily extended to deal with mixed states and to study decoherence effects.