Quantum Monge-Kantorovich Problem and the Time Dependent Two Qubit Marginal Problem

The quantum Monge-Kantorovich problem concerns the transportation of one density matrix to another, minimizing an associated transportation cost. We investigate the optimal transport problem between a pair of two level systems from the perspective of the time dependent marginal problem afforded by structure of the transport plans. Applications to approximating semidefinite programs and quantum thermodynamics will be discussed.