Violation of a Bell-like inequality in neutron optical experiments: quantum contextuality

We report on 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-1/2 property of neutrons is utilized. The total wavefunction of the neutron is described in a tensor product Hilbert space. A Bell-like inequality is derived not via a non-locality but via 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