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