  This text is a set of lecture notes for a 4.5-hour course given at the
Erd\"os Center (R\'enyi Institute, Budapest) during the Summer School "Optimal
Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I
introduces the quantum analogue of the Wasserstein distance of exponent $2$
defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016),
165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017)
57-94]. Lecture II discusses various applications of this quantum analogue of
the Wasserstein distance of exponent $2$, while Lecture III discusses several
of its most important properties, such as the triangle inequality, and the
Kantorovich duality in the quantum setting, together with some of their
implications.
