In 2022, Golse and Paul defined a pseudometric for quantum optimal transport that extends the classical Wasserstein distance. They proved that the pseudometric satisfies the triangle inequality in certain cases. This thesis reviews their proof in the case where the middle point is a classical density. Motivated by that proof, we formulate the optimal transport problem and propose the quantum Wasserstein distance on the noncommutative 2-torus. This thesis also proves that the proposed quantum Wasserstein distance satisfies the triangle inequality in the case where the middle point is a classical density on the 2-torus.