  We investigate quantum coherences in the presence of noise by entangling the
spin and path degrees of freedom of the output neutron beam from a noisy
three-blade perfect crystal neutron interferometer. We find that in the
presence of dephasing noise on the path degree of freedom the entanglement of
the output state reduces to zero, however the quantum discord remains non-zero
for all noise values. Hence even in the presence of strong phase noise
non-classical correlations persist between the spin and path of the neutron
beam. This indicates that measurements performed on the spin of the neutron
beam will induce a disturbance on the path state. We calculate the effect of
the spin measurement by observing the changes in the observed contrast of the
interferometer for an output beam post-selected on a given spin state. In doing
so we demonstrate that these measurements allow us to implement a quantum
eraser, and a which-way measurement of the path taken by the neutron through
the interferometer. While strong phase noise removes the quantum eraser, the
spin-filtered which-way measurement is robust to phase noise. We experimentally
demonstrate this disturbance by comparing the contrasts of the output beam with
and without spin measurements of three neutron interferometers with varying
noise strengths. This demonstrates that even in the presence of noise that
suppresses path coherence and spin-path entanglement, a neutron interferometer
still exhibits uniquely quantum behaviour.
