Persona: Pérez Hernández, Antonio
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antperez@ind.uned.es
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0000-0001-8600-7083
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Pérez Hernández
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Antonio
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Publicación Strong Decay of Correlations for Gibbs States in Any Dimension(Springer, 2025-09-29) Bluhm, Andreas; Capel, Ángela; Pérez Hernández, Antonio; Agencia Estatal de Investigación; Comunidad de MadridQuantum systems in thermal equilibrium are described using Gibbs states. The correlations in such states determine how difficult it is to describe or simulate them. In this article, we show that if the Gibbs state of a quantum system satisfies that each of its marginals admits a local effective Hamiltonian with short-range interactions, then it satisfies a mixing condition, that is, for any regions A, C the distance of the reduced state on these regions to the product of its marginals, ρACρ−1 A ⊗ρ−1 C −1AC , decays exponentially with the distance between regions A and C. This mixing condition is stronger than other commonly studied measures of correlation. In particular, it implies the exponential decay of the mutual information between distant regions. The mixing condition has been used, for example, to prove positive log-Sobolev constants. On the way, we prove that the the condition regarding local effective Hamiltonian is satisfied if the Hamiltonian only has commuting interactions which also commute with every marginal of their products. The proof of these results employs a variety of tools such as Araki’s expansionals, quantum belief propagation and cluster expansions.Publicación Variants of a multiplier theorem of Kislyakov(Cambridge University Press, 2022-09-01) Defant, Andreas; Mastyło, Mieczysław; Pérez Hernández, Antonio; Agencia Estatal de InvestigaciónWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislaykov and the Kahane–Salem–Zygmund inequality. As a by-product we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus Tn or Boolean cubes {−1, 1}N. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application we show that various recent ℓ1-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.Publicación Exponential Decay of Mutual Information for Gibbs states of local Hamiltonians(Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2022-02-10) Bluhm, Andreas; Capel, Ángela; Pérez Hernández, Antonio; Agencia Estatal de Investigación; European Union’s Horizon 2020; Comunidad de MadridThe 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.Publicación Thermalization in Kitaev’s quantum double models via tensor network techniques(Cambridge Universirty Press, 2023-11-28) Lucia, Angelo; Pérez García, David; Pérez Hernández, Antonio; Agencia Estatal de Investigación; Comunidad de MadridWe show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.Publicación Locality Estimates for Complex Time Evolution in 1D(Springer, 2023-01-03) Pérez García, David; Pérez Hernández, Antonio; Agencia Estatal de Investigación; Comunidad de MadridIt is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translationally invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analyticity on the whole complex plane, whenappliedtoalocalobservable. However, as for nowthere is no mathematical proof of the absence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Araki’s result to include exponential (or faster) tails. Our main result is the analyticity of the timeevolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Araki’s result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.Publicación Continuity of Quantum Entropic Quantities via Almost Convexity(IEEE, 2023-09-01) Bluhm, Andreas; Capel, Angela; Gondolf, Paul; Pérez Hernández, Antonio; Agencia Estatal de Investigación; Comunidad de MadridBased on the proofs of the continuity of the conditional entropy by Alicki, Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF) method. This method allows us to prove a great variety of continuity bounds for the derived entropic quantities. First, we apply the ALAFF method to the Umegaki relative entropy. This way, we recover known almost tight bounds, but also some new continuity bounds for the relative entropy. Subsequently, we apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This yields novel explicit bounds in particular for the BS-conditional entropy, the BS-mutual and BS-conditional mutual information. On the way, we prove almost concavity for the Umegaki relative entropy and the BS-entropy, which might be of independent interest. We conclude by showing some applications of these continuity bounds in various contexts within quantum information theory.