  Observation of gravitationally induced entanglement between two massive
particles can be viewed as implying the existence of the nonclassical nature of
gravity. However, weak interaction in the gravitational field is extremely
small so that gravitationally induced entanglement is exceptionally challenging
to test in practice. For addressing this key challenge, here we propose a
criterion based on the logical contradictions of weak entanglement, which may
boost the sensitivity of the signal due to the gravitationally induced
entanglement. Specifically, we make use of the weak-value scenario and
Einstein-Podolsky-Rosen steering. We prove that it is impossible for a
classical mediator to act on two local quantum objects to simulate
amplified-weak-value phenomenon in two-setting Einstein-Podolsky-Rosen
steering. Our approach can amplify the signal of gravitationally induced
entanglement that were previously impossible to observe by any desired factor
that depends on the magnitude of the weak value. Our results not only open up
the possibility of exploring nonclassical nature of gravity in the near future,
but they also pave the way for weak entanglement criterion of a more general
nature.
