  We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We
also provide a sufficient condition that guarantees the existence of
equilibrium coefficients so that the model shares the same conservation laws
and $H$-theorem with the quantum Boltzmann equation. Unlike the classical BGK
for gas mixtures, the equilibrium coefficinets of the local equilibiriums for
quantum multi-species gases are defined through highly nonlinear relations that
are not explicitly solvable. We verify in a unified way that such nonlinear
relations uniquely determine the equilibrium coefficients under the condition,
leading to the well-definedness of our model.
