Exponential Decay of Mutual Information for Gibbs states of local Hamiltonians
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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.
| Original language | English |
|---|---|
| Article number | 650 |
| Journal | Quantum |
| Volume | 6 |
| Pages (from-to) | 1-40 |
| ISSN | 2521-327X |
| DOIs | |
| Publication status | Published - 2022 |
Bibliographical note
Funding Information:
Acknowledgements: The authors would like to thank Álvaro M. Alhambra, in particular for suggesting the use of the geometric Rényi divergences that improved the presentation of Section 3.1; Yoshiko Ogata, whose comments inspired the proof of Theorem 6.2; and also Kohtaro Kato, Angelo Lucia and David Pérez-García for insightful discussions. AB acknowledges support from the VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059) and from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) via the Innovation Fund Denmark. AC is partially supported by a MCQST Distinguished PostDoc fellowship, by the Seed Funding Program of the MCQST and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC-2111 390814868. APH is partially supported by the grants “Juan de la Cierva Formación” (FJC2018-036519-I) and 2021-MAT11 (ETSI Industriales, UNED), as well as by the Spanish Ministerio de Ciencia e Innovación project PID2020-113523GB-I00, by Comunidad de Madrid project QUITEMAD-CM P2018/TCS4342, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648913).
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