Weak quasi-factorization for the Belavkin-Staszewski relative entropy

Bluhm, Andreas; Capel, Ángela; Pérez-Hernández, Antonio

doi:10.48550/arxiv.2101.10312

Cited by 1 publication

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“…The interest in this quantity has increased in the past few years, with a comprehensive study of the fundamental properties of the maximal f -divergences [Mat18,HM17], of which the BS-entropy is a prominent example. Recently, a recovery condition and a strengthened data-processing inequality for such divergences has been obtained [BC20], as well as some weak quasi-factorization results for the BS-entropy [BCPH21]. Remarkably, a subclass of maximal f -divergences, namely the geometric Rényi divergences, has found applications in estimating channel capacities [FF19].…”

Section: Introductionmentioning

confidence: 99%

Exponential decay of mutual information for Gibbs states of local Hamiltonians

Bluhm,

Capel,

Pérez-Hernández

2021

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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. 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.

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