While most fundamental interactions in nature are known to be mediated by
quantized fields, the possibility has been raised that gravity may behave
differently. Making this concept precise enough to test requires consistent
models. Here we construct an explicit example of a theory where a
non-entangling version of an arbitrary two-body potential V(r) arises from
local measurements and feedback forces. While a variety of such theories exist,
our construction causes particularly strong decoherence compared to more subtle
approaches. Regardless, expectation values of observables obey the usual
classical dynamics, while the interaction generates no entanglement. Applied to
the Newtonian potential, this produces a non-relativistic model of gravity with
fundamental loss of unitarity. The model contains a pair of free parameters, a
substantial range of which is not excluded by observations to date. As an
alternative to testing entanglement properties, we show that the entire
remaining parameter space can be tested by looking for loss of quantum
coherence in small systems like atom interferometers coupled to oscillating
source masses.Comment: 15 pages + appendices, 3 figure