Quantum Optimal Transport: Quantum Couplings and Many-Body Problems

TL;DR

This paper introduces a quantum analogue of the Wasserstein distance, explores its properties and applications, and discusses its implications for many-body quantum problems, based on a series of lecture notes from a summer school.

Contribution

It develops the theory of quantum optimal transport, including defining quantum couplings, and extends classical concepts like Kantorovich duality to the quantum realm.

Findings

Quantum Wasserstein distance of exponent 2 is introduced.

Properties such as triangle inequality and duality are established.

Applications to many-body quantum problems are discussed.

Abstract

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.