[Paper Review] From decay of correlations to locality and stability of the Gibbs state

Ángela Capel, Massimo Moscolari+2 more|arXiv (Cornell University)|Oct 13, 2023

Quantum many-body systemsPhysics and Astronomy3 citations

TL;DR

This paper establishes that decay of correlations in quantum Gibbs states implies both locality and stability under local perturbations, using Lieb-Robinson bounds and quantum belief propagation. The key contribution is proving the equivalence of decay of correlations, local indistinguishability, and local perturbation response in general quantum spin systems, valid in any dimension and at high temperatures.

ABSTRACT

We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb-Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.

Motivation & Objective

• To establish a rigorous connection between decay of correlations and the locality/stability of Gibbs states in quantum many-body systems.
• To demonstrate that local perturbations affect only local observables (LPPL) when correlations decay exponentially.
• To show that local indistinguishability—where local observables cannot distinguish between full and truncated Gibbs states—follows from decay of correlations.
• To extend these results to finite one-dimensional spin chains with short-range, exponentially decaying interactions, proving decay of correlations above a temperature threshold.
• To unify and generalize existing concepts of locality and stability in thermal quantum states using a common framework based on quantum belief propagation.

Proposed method

• Analyzes the locality properties of quantum belief propagation (QBP) as a tool to study the response of Gibbs states to perturbations.
• Employs Lieb-Robinson bounds to control the propagation speed of information in quantum spin systems, ensuring effective locality.
• Uses differential equations for QBP to track the time evolution of local observables under perturbations.
• Derives uniform decay of correlations from local indistinguishability via reverse implications in the framework of quasi-local maps.
• Applies the theory to quantum spin systems with short-range interactions at high temperature, where decay of correlations is known to hold.
• Establishes uniform decay of correlations in finite 1D spin chains with translation-invariant, exponentially decaying interactions above a temperature threshold.

Experimental results

Research questions

• RQ1 Does decay of correlations in a Gibbs state imply that local perturbations affect only local observables?
• RQ2 Can local indistinguishability of a Gibbs state be derived from decay of correlations?
• RQ3 Is the stability of Gibbs states under local perturbations equivalent to decay of correlations in general quantum systems?
• RQ4 What is the temperature threshold above which decay of correlations holds in finite one-dimensional spin chains with short-range interactions?
• RQ5 How do Lieb-Robinson bounds and quantum belief propagation jointly enable the derivation of locality and stability from decay of correlations?

Key findings

• Decay of correlations in a Gibbs state implies local perturbations affect only local observables (LPPL), with the response bounded by exponential decay in distance.
• Local indistinguishability—where local observables cannot distinguish the full Gibbs state from its truncated version—follows from uniform LPPL and is equivalent to decay of correlations under the given conditions.
• For quantum spin systems with short-range interactions at high enough temperature, decay of correlations holds, implying both locality and stability of the Gibbs state.
• In finite one-dimensional spin chains with translation-invariant, exponentially decaying interactions, decay of correlations is proven to hold above a temperature threshold that vanishes in the limit of finite-range interactions.
• The quantum belief propagation formalism provides a dynamical framework to derive and analyze locality and stability, with its differential equations capturing the time evolution of local observables.
• The equivalence of decay of correlations, LPPL, and local indistinguishability is established in any dimension, relying only on locality of the Hamiltonian and Lieb-Robinson bounds.