The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices

We propose a generalization of the Wasserstein distance of order 1 to quantum spin systems on the lattice Z(d), which we call specific quantum W-1 distance. The proposal is based on the W-1 distance for qudits of De Palma et al. (IEEE Trans Inf Theory 67(10):6627-6643, 2021) and recovers Ornstein's (d) over bar -distance for the quantum states whose marginal states on any finite number of spins are diagonal in the canonical basis. We also propose a generalization of the Lipschitz constant to quantum interactions on Z(d) and prove that such quantum Lipschitz constant and the specific quantum W-1 distance are mutually dual. We prove a new continuity bound for the von Neumann entropy for a finite set of quantum spins in terms of the quantum W-1 distance, and we apply it to prove a continuity bound for the specific von Neumann entropy in terms of the specific quantum W-1 distance for quantum spin systems on Z(d). Finally, we prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality, which implies the uniqueness of their Gibbs states.