Exponential decay of mutual information for Gibbs states of local Hamiltonians

Abstract

The thermal equilibrium properties of physical systems can be described using Gibbs states. It is therefore of great interest to know when such states allow for an easy description. In particular, this is the case if correlations between distant regions are small. In this work, we consider 1D quantum spin systems with local, finite-range, translation-invariant interactions at any temperature. In this setting, we show that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespective of the temperature. In order to prove the latter, we show that exponential decay of correlations of the infinite-chain thermal states, exponential uniform clustering and exponential decay of the mutual information are equivalent for 1D quantum spin systems with local, finite-range interactions at any temperature. In particular, Araki's seminal results yields that the three conditions hold in the translation-invariant case. The methods we use are based on the Belavkin-Staszewski relative entropy and on techniques developed by Araki. Moreover, we find that the Gibbs states of the systems we consider are superexponentially close to saturating the data-processing inequality for the Belavkin-Staszewski relative entropy.

Final version: 40 pages; corrected typos, revised notation, added further comments to clarify some points

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Cited by in corpus (22)

• Rapid thermalization of spin chain commuting Hamiltonians
• Quantum many-body systems in thermal equilibrium
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• Continuity of quantum entropic quantities via almost convexity
• Exponential clustering of bipartite quantum entanglement at arbitrary temperatures
• The advantage of quantum control in many-body Hamiltonian learning
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• Universality in the tripartite information after global quenches: (generalised) quantum XY models
• General Continuity Bounds for Quantum Relative Entropies
• A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit
• From decay of correlations to locality and stability of the Gibbs state
• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Subsystem eigenstate thermalization hypothesis for translation invariant systems
• Universality in the tripartite information after global quenches: spin flip and semilocal charges
• Efficient learning of ground & thermal states within phases of matter
• Strong decay of correlations for Gibbs states in any dimension
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium
• A Faster Algorithm for the Free Energy in One-Dimensional Quantum Systems
• Belavkin-Staszewski Quantum Markov Chains
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains