  CO emission has been widely used as a tracer of molecular gas mass. However,
it is a long-standing issue to accurately constrain the CO-to-H$_2$ conversion
factor ($\alpha_{\mathrm{CO}}$) that converts CO luminosity to molecular gas
mass, especially in starburst galaxies. We present the first resolved
$\alpha_{\mathrm{CO}}$ modeling results with multiple ALMA CO and $^{13}$CO
transition observations at both giant molecular cloud (GMC) scale at 150 pc and
kpc scale for one of the closest starburst mergers, the Antennae. By combining
our CO modeling results and measurements of 350 GHz dust continuum, we find
that most GMCs in the Antennae have $\alpha_{\mathrm{CO}}$ values $\sim$4 times
smaller than the commonly adopted Milky Way value (4.3). We find
$\alpha_{\mathrm{CO}}$ at GMC scales shows a strong dependence on CO intensity,
$^{13}$CO/CO ratio and GMC velocity dispersion, which is consistent with
various theoretical and simulation predictions. Specifically, we suggest that
the $^{13}$CO/CO line ratio and the velocity dispersion can be used to infer
$\alpha_{\mathrm{CO}}$ in starburst regions. By applying our modeled
$\alpha_{\mathrm{CO}}$ in GMC analyses, we find that GMCs in the Antennae are
less gravitationally bound than in normal spiral galaxies, which is more
consistent with what is predicted by merger simulations. At kpc scale, we find
that our modeled $\alpha_{\mathrm{CO}}$ values are smaller than the modeled
$\alpha_{\mathrm{CO}}$ at GMC scale by 40%, which can be due to inclusion of a
diffuse gas component with lower $\alpha_{\mathrm{CO}}$ values. We find a
similar correlation of $\alpha_{\mathrm{CO}}$ and CO intensity at kpc scales to
that at GMC scales.
