Strong decay of correlations for Gibbs states in any dimension

TL;DR

This work investigates how correlations in quantum Gibbs states decay with distance, introducing a strong mixing condition that bounds deviations from independence between distant regions. The authors develop two notions of local effective Hamiltonians (strong and weak) and prove that strong locality implies exponential mixing under a commuting hypothesis via cluster expansions and Araki expansionals; they extend the analysis to the weak setting using local indistinguishability and quantum belief propagation. They provide explicit, quantitative conditions under which strong local effective Hamiltonians exist (finite-degree and exponentially decaying interactions) and derive corresponding mixing bounds, along with a local indistinguishability framework applicable to high-temperature regimes. The results unify several correlation measures (covariance, mutual information, mixing, and local indistinguishability) in quantum Gibbs states under high-temperature/short-range assumptions, with implications for rapid mixing and stability, while highlighting open questions about the necessity and generality of local effective Hamiltonians in non-commuting settings.

Abstract

Quantum systems in thermal equilibrium are described using Gibbs states. The correlations in such states determine how difficult it is to describe or simulate them. In this article, we show that if the Gibbs state of a quantum system satisfies that each of its marginals admits a local effective Hamiltonian with short-range interactions, then it satisfies a mixing condition, that is, for any regions $A$, $C$ the distance of the reduced state $ρ_{AC}$ on these regions to the product of its marginals, $$\| ρ_{AC} ρ_A^{-1} \otimes ρ_C^{-1} - 1_{AC}\|\, ,$$ decays exponentially with the distance between regions $A$ and $C$. This mixing condition is stronger than other commonly studied measures of correlation. In particular, it implies the exponential decay of the mutual information between distant regions. The mixing condition has been used, for example, to prove positive log-Sobolev constants. On the way, we prove that the the condition regarding local effective Hamiltonian is satisfied if the Hamiltonian of the system is commuting and also commutes with every marginal of the Gibbs state. The proof of these results employs a variety of tools such as Araki's expansionals, quantum belief propagation and cluster expansions.

Figures (8)

• Figure 1: Example of configuration of regions $Z \subset Y \subset Y'$ in Proposition \ref{['theo:localityEstimates']}. If the local interaction of the system is exponentially decaying, then the evolutions of an observable supported in $Z$ under $H_{Y}$ and $H_{Y'}$, respectively, are exponentially close to each other in the distance from $Z$ to the complement of $Y$.
• Figure 2: Example of configuration of the three disjoint regions $A,B,C$ in Proposition \ref{['prop:estimates_expansionals_normal']}
• Figure 3: On the left picture, we represent three distinct dispositions of a subset $X$ with respect to the tracing region $L^{c}$. In particular, the local interactions of the effective Hamiltonian must satisfy $\widetilde{\Phi}^{L, \beta}_{X_{2}} = \Phi_{X_{2}}$, while $\widetilde{\Phi}^{L, \beta}_{X_{3}}$ is a multiple of the identity. On the right picture, the coincidence $X \cap L = X \cap L'$ yields that $\widetilde{\Phi}^{L, \beta}_{X} = \widetilde{\Phi}^{L', \beta}_{X}$.
• Figure 4: On the left-hand side, a multiset $\gamma =[X_{1}, X_{2}, X_{3}, X_{4}, X_{5}]$ that is connected (polymer). On the right-hand side, an homonymous multiset that is disconnected, as it can be decomposed as $\gamma = \gamma_{1} \vee \gamma_{2}$ where $\gamma_{1} = [X_{1}, X_{2}, X_{4}]$ and $\gamma_{2} = [X_{3}, X_{5}]$ satisfy $\gamma_{1} \wedge \gamma_{2} = \emptyset$.
• Figure 5: Display of two sublattices $A$, $B$ of $\Lambda$ such that $\mathrm{dist}(A,B) \geq \ell$.

Theorems & Definitions (41)

• Proposition 2.1
• proof
• Proposition 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• proof : Proof of \ref{['prop:estimates_expansionals_normal']}
• Corollary 2.6
• proof
• Definition 3.1: Strong form