  We report the feasibility of detecting the gravity-induced entanglement (GIE)
with optomechanical systems, which is the first investigation that clarifies
the feasible experimental parameters to achieve a signal-to-noise ratio of
S/N=1. Our proposal focuses on GIE generation between optomechanical mirrors,
coupled via gravitational interactions, under continuous measurement, feedback
control, and Kalman filtering process, which matured in connection with the
field of gravitational wave observations. We solved the Riccati equation to
evaluate the time evolution of the conditional covariance matrix for
optomechanical mirrors that estimated the minimum variance of the motions. The
results demonstrate that GIE is generated faster than a well-known time scale
without optomechanical coupling. The fast generation of entanglement is
associated with quantum-state squeezing by the Kalman filtering process, which
is an advantage of using optomechanical systems to experimentally detect GIE.
