We propose a generalization of the Wasserstein distance of order $1$ to
quantum spin systems on the lattice $\mathbb{Z}^d$, 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, 6627 (2021)] and recovers
Ornstein’s $\bar{d}$-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
$\mathbb{Z}^d$ 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 $\mathbb{Z}^d$. 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.
