Exponential decay of mutual information for Gibbs states of local Hamiltonians

Andreas Bluhm; Ángela Capel; Antonio Pérez-Hernández

doi:10.22331/q-2022-02-10-650

Exponential decay of mutual information for Gibbs states of local Hamiltonians

Andreas Bluhm¹, Ángela Capel^2,3,4, and Antonio Pérez-Hernández^5,6

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

Published:	2022-02-10, volume 6, page 650
Eprint:	arXiv:2104.04419v2
Doi:	https://doi.org/10.22331/q-2022-02-10-650
Citation:	Quantum 6, 650 (2022).

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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.

Featured image: This image shows a graphical representation of the main result of this paper: For quantum Gibbs states of finite-range, translation-invariant Hamiltonians on quantum spin chains, we show that the mutual information between two spatially separated regions A and C decays exponentially with the size of the middle region B.

Popular summary

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.

► BibTeX data

@article{Bluhm2022exponentialdecayof,
  doi = {10.22331/q-2022-02-10-650},
  url = {https://doi.org/10.22331/q-2022-02-10-650},
  title = {Exponential decay of mutual information for {G}ibbs states of local {H}amiltonians},
  author = {Bluhm, Andreas and Capel, {\'{A}}ngela and P{\'{e}}rez-Hern{\'{a}}ndez, Antonio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {650},
  month = feb,
  year = {2022}
}

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[5] Hamza Fawzi, Omar Fawzi, and Samuel O. Scalet, "A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit", Quantum 7, 1011 (2023).

[6] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé, "Entropy Decay for Davies Semigroups of a One Dimensional Quantum Lattice", Communications in Mathematical Physics 405 2, 42 (2024).

[7] Ivan Bardet, Ángela Capel, Li Gao, Angelo Lucia, David Pérez-García, and Cambyse Rouzé, "Rapid Thermalization of Spin Chain Commuting Hamiltonians", Physical Review Letters 130 6, 060401 (2023).

[8] Álvaro M. Alhambra, "Quantum Many-Body Systems in Thermal Equilibrium", PRX Quantum 4 4, 040201 (2023).

[9] Zhiqiang Huang and Xiao-Kan Guo, "Subsystem eigenstate thermalization hypothesis for translation invariant systems", Physical Review E 109 5, 054120 (2024).

[10] Tomotaka Kuwahara and Keiji Saito, "Exponential Clustering of Bipartite Quantum Entanglement at Arbitrary Temperatures", Physical Review X 12 2, 021022 (2022).

[11] Vanja Marić, "Universality in the tripartite information after global quenches: spin flip and semilocal charges", Journal of Statistical Mechanics: Theory and Experiment 2023 11, 113103 (2023).

[12] Álvaro M. Alhambra and J. Ignacio Cirac, "Locally Accurate Tensor Networks for Thermal States and Time Evolution", PRX Quantum 2 4, 040331 (2021).

[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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