Violation of a Bell-like Inequality in Neutron Optical Experiments: Quantum contextuality

We report a single-neutron optical experiment to demonstrate the violation of a Bell-like inequality. Entanglement is achieved not between particles, but between the degrees of freedom, in this case, for a single-particle. The spin-{\small 1/2} property of neutrons are utilized. The total wave function of the neutron is described in a tensor product Hilbert space. A Bell-like inequality is derived not by a non-locality but by a contextuality. Joint measurements of the spinor and the path properties lead to the violation of a Bell-like inequality. Manipulation of the wavefunction in one Hilbert space influences the result of the measurement in the other Hilbert space. A discussion is given on the quantum contextuality and an entanglement-induced correlation in our experiment.