  We develop an entanglement criterion with third- and fourth-order cumulants
to detect the entanglement of non-Gaussian states. The efficiency of the
entanglement criterion is investigated for gravitating mirrors in
optomechanical systems. We show that the entangled regime of the mirrors is
enlarged by the third- and fourth-order cumulants. We also discuss the
limitations of the entanglement criterion for mirrors in a highly non-Gaussian
state.
