  We develop an operator-based description of two types of multimode-entangled
single-neutron quantum optical devices: Wollaston prisms and radio-frequency
spin flippers in inclined magnetic field gradients. This treatment is similar
to the approach used in quantum optics, and is convenient for the analysis of
quantum contextuality measurements in certain types of neutron interferometers.
We describe operationally the way multimode-entangled single-neutron states
evolve in these devices, and provide expressions for the associated operators
describing the dynamics, in the limit in which the neutron state space is
approximated by a finite tensor product of distinguishable subsystems. We
design entangled-neutron interferometers to measure entanglement witnesses for
the Clauser, Horne, Shimony and Holt, and Mermin inequalities, and compare the
theoretical predictions with recent experimental results. We present the
generalization of these expressions to $n$ entangled distinguishable
subsystems, which could become relevant in the future if it becomes possible to
add neutron orbital angular momentum to the experimentally-accessible list of
entangled modes. We view this work as a necessary first step towards a
theoretical description of entangled neutron scattering from strongly entangled
matter, and we explain why it should be possible to formulate a useful
generalization of the usual Van Hove linear response theory for this case. We
also briefly describe some other scientific extensions and applications which
can benefit from interferometric measurements using the types of single-neutron
multimode entanglement described by this analysis.
