Quantum optimal transport: an invitation

Abstract

The optimal mass transport problem was formulated centuries ago, but only recently there has been a surge in its applications, particularly in functional inequalities, geometry, stochastic analysis, and numerical solutions for partial differential equations. Quantum optimal transport aims to extend this success story to non-commutative systems, where density operators replace probability measures. This brief review paper aims to describe the latest approaches, highlighting their advantages, disadvantages, and open mathematical problems relevant to applications.