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

A neutron interferometer experiment which clearly demonstrates the violation of a Bell-like inequality is reported. Entanglement is achieved not between particles, but between the degrees of freedom, in this case, for a single particle. The total wave function of spin-1/2 neutron is described in a tensor product Hilbert space. Appropriate combinations of the direction of spin analysis and the position of the phase shifter allow experimental verification of the violation of a Bell-like inequality. Manipulation of the wave function in one Hilbert space influences the result of the measurement in the other Hilbert space: manipulation without touch! The violation of the Bell-like inequality is discussed in terms of quantum contextuality and an entanglement-induced correlation.