Quantum Optimal Transport: Quantum Couplings and Many-Body Problems

This text is a set of lecture notes for a 4.5-hour course given at the Erdős Center (Rényi 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 Golse et al. (Commun. Math. Phys. 343:165–205, 2016), and in Golse and Paul (Arch. Ration. Mech. Anal. 223:57–94, 2017). 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.