We investigate the quantum nature of gravity in terms of the coherence of
quantum objects. As a basic setting, we consider two gravitating objects each
in a superposition state of two paths. The evolution of objects is described by
the completely positive and trace-preserving (CPTP) map with a
population-preserving property. This property reflects that the probability of
objects being on each path is preserved. We use the $\ell_1$-norm of coherence
to quantify the coherence of objects. In the present paper, the quantum nature
of gravity is characterized by an entangling map, which is a CPTP map with the
capacity to create entanglement. We introduce the entangling-map witness as an
observable to test whether a given map is entangling. We show that, whenever
the gravitating objects initially have a finite amount of the $\ell_1$-norm of
coherence, the witness tests the entangling map due to gravity. Interestingly,
we find that the witness can test such a quantum nature of gravity, even when
the objects do not get entangled. This means that the coherence of gravitating
objects always becomes the source of the entangling map due to gravity. We
further discuss a decoherence effect and an experimental perspective in the
present approach.
