Preface -- Chapter 1. An Introduction to Optimal Transport and Wasserstein Gradient Flows by Alessio Figalli -- Chapter 2. Dynamics and Quantum Optimal Transport:Three Lectures on Quantum Entropy and Quantum Markov Semigroups by Eric A. Carlen -- Chapter 3. Quantum Couplings and Many-body Problems by Francois Golse -- Chapter 4. Quantum Channels and Qubits by Giacomo De Palma and Dario Trevisan -- Chapter 5. Entropic Regularised Optimal Transport in a Noncommutative Setting by Lorenzo Portinale -- Chapter 6. Logarithmic Sobolev Inequalities for Finite Dimensional Quantum Markov Chains by Cambyse Rouzé.

The flourishing theory of classical optimal transport concerns mass transportation at minimal cost. This book introduces the reader to optimal transport on quantum structures, i.e., optimal transportation between quantum states and related non-commutative concepts of mass transportation. It contains lecture notes on classical optimal transport and Wasserstein gradient flows dynamics and quantum optimal transport quantum couplings and many-body problems quantum channels and qubits These notes are based on lectures given by the authors at the "Optimal Transport on Quantum Structures" School held at the Erdös Center in Budapest in the fall of 2022. The lecture notes are complemented by two survey chapters presenting the state of the art in different research areas of non-commutative optimal transport.