  We propose a generalization of the Wasserstein distance of order 1 to the
quantum states of $n$ qudits. The proposal recovers the Hamming distance for
the vectors of the canonical basis, and more generally the classical
Wasserstein distance for quantum states diagonal in the canonical basis. The
proposed distance is invariant with respect to permutations of the qudits and
unitary operations acting on one qudit and is additive with respect to the
tensor product. Our main result is a continuity bound for the von Neumann
entropy with respect to the proposed distance, which significantly strengthens
the best continuity bound with respect to the trace distance. We also propose a
generalization of the Lipschitz constant to quantum observables. The notion of
quantum Lipschitz constant allows us to compute the proposed distance with a
semidefinite program. We prove a quantum version of Marton's transportation
inequality and a quantum Gaussian concentration inequality for the spectrum of
quantum Lipschitz observables. Moreover, we derive bounds on the contraction
coefficients of shallow quantum circuits and of the tensor product of one-qudit
quantum channels with respect to the proposed distance. We discuss other
possible applications in quantum machine learning, quantum Shannon theory, and
quantum many-body systems.
