We establish a Kantorovich duality for he pseudometric $\cE_\hb$ introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. \textbf{223} (2017), 57--94], obtained from the usual Monge-Kantorovich distance $\MKd$ between classical densities by quantization of one side of the two densities involved. We show several type of inequalities comparing $\MKd$, $\cE_\hb$ and $MK_\hb$, a full quantum analogue of $\MKd$ introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. \textbf{343} (2016), 165--205], including an up to $\hbar$ triangle inequality for $MK_\hb$. Finally, we show that, when nice optimal Kantorovich potentials exist for $\cE_\hb$, optimal couplings induce classical/quantum optimal transports and the potentials are linked by a semiquantum Legendre type transform.