The Wasserstein distance of order 1 for quantum spin systems on infinite lattices

TL;DR

This paper introduces a quantum version of the Wasserstein distance for spin systems on infinite lattices, establishing its properties, duality with a Lipschitz constant, and applications to entropy continuity and Gibbs state uniqueness.

Contribution

It generalizes the Wasserstein distance to quantum spin systems, links it with quantum Lipschitz constants, and applies it to entropy bounds and phase transition analysis.

Findings

Defined a quantum $W_1$ distance for lattice spin systems.

Proved duality between quantum Lipschitz constant and $W_1$ distance.

Established a transportation-cost inequality for quantum Gibbs states.

Abstract

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.