Exponential decay of mutual information for Gibbs states of local Hamiltonians

Andreas Bluhm1, Ángela Capel2,3,4, and Antonio Pérez-Hernández5,6

1QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
2Fachbereich Mathematik, Universität Tübingen, 72076 Tübingen, Germany
3Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany
4Munich Center for Quantum Science and Technology (MCQST), München, Germany
5Departamento de Matemática Aplicada I, Escuela Técnica Superior de Ingenieros Industriales, Universidad Nacional de Educación a Distancia, calle Juan del Rosal 12, 28040 Madrid (Ciudad Universitaria), Spain
6Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

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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.

Quantum spin chains are used to model atoms sitting on a line. If such systems have translation-invariant short-ranged interactions and are in thermal equilibrium, it was shown by Araki decades ago that correlations between separated regions decrease rapidly (exponentially fast) with the distance between these regions. This result is independent of the temperature and implies that such systems are easy to describe.

Our work studies a stronger measure of correlation than the one studied by Araki, namely the mutual information, which has a clear operational interpretation. We show that also the mutual information between distant regions decays exponentially fast, also independently of the temperature. To derive these results, we use a quantity that has not been used in quantum information theory so much, the Belavkin-Staszewski relative entropy, which gives one way to quantify the difference between quantum states. Moreover, we show that states which model the thermal equilibrium on quantum spin chains with translation-invariant short-ranged interactions almost factorize into smaller states, which should be of independent interest.

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[13] Ángela Capel, Massimo Moscolari, Stefan Teufel, and Tom Wessel, "From decay of correlations to locality and stability of the Gibbs state", arXiv:2310.09182, (2023).

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