  Understanding the phenomenon of quantum superposition of gravitational fields
induced by massive quantum particles is an important starting point for quantum
gravity. The purpose of this study is to deepen our understanding of the
phenomenon of quantum superposition of gravitational fields. To this end, we
consider a trade-off relation of entanglement (monogamy relation) in a
tripartite system consisting of two massive particles and a gravitational field
that may be entangled with each other. Consequently, if two particles cannot
exchange information mutually, they are in a separable state, and the particle
and gravitational field are always entangled. Furthermore, even when two
particles can send information to each other, there is a trade-off between the
two particles and the gravitational field. We also investigate the behavior of
the quantum superposition of the gravitational field using quantum discord. We
find that quantum discord increases depending on the length scale of the
particle superposition. Our results may help understand the relationship
between the quantization of the gravitational field and the meaning of the
quantum superposition of the gravitational field.
