Quantum Optimal Transport with Quantum Channels

We propose a new generalization to quantum states of the Wasserstein distance, which is fundamental distance between probability distributions given by minimization transport cost. Our proposal first where plans are in natural correspondence with channels, such that can be interpreted as physical operation on system. main result proof modified triangle inequality for our distance. also prove state and itself intimately connected Wigner-Yanase metric manifold states. then specialize Gaussian systems, provide mathematical model electromagnetic radiation regime. noiseless attenuators amplifiers optimal thermal states, recovers classical semiclassical limit.