thermal fluctuations are common to both outputs (Fig. \[correlations\]c). The difference between the observed correlations and their maximum classical value, $\langle X_\mathrm{e}X_\mathrm{o}\rangle - \sqrt{\langle (X^{\mathrm{(th)}}_\mathrm{e})^2\rangle}\sqrt{\langle (X^{\mathrm{(th)}}_\mathrm{o})^2\rangle} = -0.2\pm 0.3~\mathrm{photons}$, is consistent with zero, indicating that thermal noise is perfectly correlated between the two outputs, as expected from optomechanical theory (see supplement). Away from mechanical resonance the cross-correlation vanishes, as expected for uncorrelated noise from the independent measurement chains. Similar behavior is observed for $\langle Y_\mathrm{e}(\omega) Y_\mathrm{o}(\omega)\rangle$ (Fig. \[correlations\]d).

Harnessing the observed correlations, we use classical feedforward to recover a weak upconverted signal. With the weak signal incident on the microwave port and detuned from the pump by $\delta/2\pi = f_\mathrm{m}+5~\mathrm{Hz}$, microwave reflection and converter transmission to the optical port are simultaneously measured. The microwave reflection is fed forward to remove noise from the upconverted optical signal. In a quadrature picture (see supplement) where demodulated microwave and optical fields are described in the time domain by $( q(t),p(t) )$ position-momentum pairs, the fed-forward optical quadrature $\check{q}_\mathrm{o}$ is given by $$\label{eq:ffquad}
\check{q}_\mathrm{o} = q_\mathrm{o} - w \sqrt{ \frac{ n_\mathrm{o} }{ n_\mathrm{e} } } q_\mathrm{e},$$ where $q_\mathrm{o}$ is the measured optical quadrature, $q_\mathrm{e}$ is the microwave quadrature, and $w$ is the feedforward weight; 