  Newtonian gravity yields specific observable consequences, the most striking
of which is the emergence of a $1/r^2$ force. In so far as communication can
arise via such interactions between distant particles, we can ask what would be
expected for a theory of gravity that only allows classical communication. Many
heuristic suggestions for gravity-induced decoherence have this restriction
implicitly or explicitly in their construction. Here we show that communication
via a $1/r^2$ force has a minimum noise induced in the system when the
communication cannot convey quantum information, in a continuous time analogue
to Bell's inequalities. Our derived noise bounds provide tight constraints from
current experimental results on any theory of gravity that does not allow
quantum communication.
