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 transportation-cost inequality, which implies the uniqueness of their Gibbs states.

arXiv commentAnnales Henri Poincaré (2023)

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Cited by in corpus (6)

• Quantum Optimal Transport: Quantum Channels and Qubits
• Order $p$ quantum Wasserstein distances from couplings
• Classical shadows meet quantum optimal mass transport
• Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach
• Perturbative criteria for the ergodicity of interacting dissipative quantum lattice systems
• Critical Scaling of the Quantum Wasserstein Distance