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.

Quantum many-body systems in thermal equilibrium

“…• For 1D systems at all temperatures [67]. This has been proven using the locality estimates from III B, and in particular the properties of the operator E A in Eq.…”

“…The more general one, however, is essentially the same and can be found in [43]. It uses previously mentioned results, and shares some steps and ideas that appear in other perhaps more fundamental questions including, for instance, the proof of the absence of phase transitions in 1D [33,44] or of decay of correlations [33,67]. It will also be a key ingredient in the algorithm of Sec.…”

Efficient learning of ground & thermal states within phases of matter

“…Condition A.1 is satisfied by many classes of Gibbs states, including high-temperature Gibbs states [HMS20, KKBa20] and 1D Gibbs states at any constant temperature [HMS20,BCPH22]. It is also known to hold for ground states of gapped Hamiltonians [HK06].…”

“…Remark D.8. For 1D, translationally invariant Hamiltonians, the Gibbs state has exponential decay of correlations for all temperatures [BCPH22] and hence the phase can be learned efficiently everywhere.…”

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