  This paper points out the importance of the quantum nature of the
gravitational interaction with matter in a linearized theory of quantum gravity
induced entanglement of masses (QGEM). We will show how the quantum interaction
entangles the steady states of a closed system (eigenstates) of two test masses
placed in the harmonic traps, and how such a quantum matter-matter interaction
emerges from an underlying quantum gravitational field. We will rely upon
quantum perturbation theory highlighting the critical assumptions for
generating a quantum matter-matter interaction and showing that a classical
gravitational field does not render such an entanglement. We will consider two
distinct examples; one where the two harmonic oscillators are static and the
other where the harmonic oscillators are non-static. In both the cases it is
the quantum nature of the gravitons interacting with the harmonic oscillators
that are responsible for creating an entangled state with the ground and the
excited states of harmonic oscillators as the Schmidt basis. We will compute
the concurrence as a criterion for the above entanglement and highlight the
role of the spin-2 nature of the graviton for entangling the two harmonic
oscillators.
