  We investigate how to generate continuous-variable entanglement between
distant optomechanical and spin systems, by transferring input two-mode
squeezed vacuum state to the system. Such a setup has been proposed for
backaction evading gravitational-wave measurement, squeezing the output noise
below the standard quantum limit. We find that the spin cavity entanglement
saturates to a particular value when no mechanics are involved even though the
entanglement of the input beam increases steadily, and drops down when the
mechanical oscillator interacts with the cavity. Our study also reveals that
the spin optical readout rate enables the robustness of the spin-cavity
entanglement with input squeezing whereas the optomechanical coupling strength
disables it. The entanglement reaches its maximum when the effective resonance
frequency and bandwidth of the cavity match the spin system. Determining
collective quadrature fluctuations, our analysis also shows that even though
the entanglement between spin and cavity, and cavity and mechanics is
significantly present; it is still impossible to obtain entanglement between
spin and mechanical oscillator.
