Foundation of quantum optimal transport and applications

Quantum optimal transportation seeks an operator which minimizes the total cost of transporting a quantum state to another state, under some constraints that should be satisfied during transportation. We formulate this issue by extending the Monge–Kantorovich problem, which is a classical optimal transportation theory, and present some applications. As an example, we address infinitely repeated quantum games and establish the folk theorem of the quantum prisoners’ dilemma, which claims mutual cooperation can be an equilibrium of the infinitely repeated quantum game. We also exhibit a series of examples which show generic and practical advantages of the abstract quantum optimal transportation theory.