The Wasserstein Distance of Order 1 for Quantum Spin Systems on Infinite Lattices
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 transportationcost inequality, which implies the uniqueness of their Gibbs states.
 Publication:

Annales Henri Poincaré
 Pub Date:
 December 2023
 DOI:
 10.1007/s0002302301340y
 arXiv:
 arXiv:2210.11446
 Bibcode:
 2023AnHP...24.4237D
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability;
 Quantum Physics
 EPrint:
 Annales Henri Poincar\'e (2023)