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

• URL: http://arxiv.org/abs/2210.11446v2
• Date: Wed, 28 Jun 2023 09:41:41 GMT
• Title: The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• Authors: Giacomo De Palma and Dario Trevisan
• Abstract summary: We show a generalization of the Wasserstein distance of order 1 to quantum spin systems on the lattice $mathbbZd$. We also prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality.
• Score: 13.452510519858995
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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