arXiv subjects

Antonio Pérez-Hernández

Publications and source records attributed to Antonio Pérez-Hernández.

11 recordsLinked to original sources

arXiv2026

Modified logarithmic Sobolev inequalities for Abelian quantum double models

We establish rapid mixing for Davies Markov semigroups associated with 2D Abelian quantum double models at any positive temperature. A condition of Dobrushin-Shlosman (DS) type holds at any temperature, and we show that the latter implies a modified logarithmic Sobolev inequality for the Davies Lindbladian. A key step in the argument is to verify a strong martingale condition for the local conditional expectations of the Davies semigroup in the regime of validity of the DS condition.

quant-ph↗

arXiv2025

Modified logarithmic Sobolev inequalities for CSS codes

We consider the class of Davies quantum semigroups modelling thermalization for translation-invariant Calderbank-Shor-Steane (CSS) codes in D dimensions. We prove that conditions of Dobrushin-Shlosman-type on the quantum Gibbs state imply a modified logarithmic Sobolev inequality with a constant that is uniform in the system's size. This is accomplished by generalizing parts of the classical results on thermalization by Stroock, Zegarlinski, Martinelli, and Olivieri to the CSS quantum setting. The results in particular imply the rapid thermalization at any positive temperature of the toric code in 2D and the star part of the toric code in 3D, implying a rapid loss of stored quantum information for these models.

quant-ph↗

arXiv2025

Lipschitz vs Linear Numerical Index in certain Banach spaces

We show that for real Banach spaces that are either separable or dual spaces, the Lipschitz numerical index coincides with the classical (linear) numerical index. This result provides partial evidence toward the question posed by Wang, Huang, and Tan (2014) of whether these two quantities coincide for every real Banach space. Our approach relies on two standard linearization techniques for Lipschitz maps: differentiation via convolution with Gaussian probability measures, and invariant means.

math.FA↗

arXiv2025

Spectral Gap Bounds for Quantum Markov Semigroups via Correlation Decay

Starting from an arbitrary full-rank state of a lattice quantum spin system, we define a "canonical purified Hamiltonian" and characterize its spectral gap in terms of a spatial mixing condition (or correlation decay) of the state. When the state considered is a Gibbs state of a local, commuting Hamiltonian at positive temperature, we show that the spectral gap of the canonical purified Hamiltonian provides a lower bound to the spectral gap of a large class of reversible generators of quantum Markov semigroup, including local and ergodic Davies generators. As an application of our construction, we show that the mixing condition is always satisfied for any finite-range 1D model, as well as by Kitaev's quantum double models.

quant-ph↗

arXiv2024

Strong decay of correlations for Gibbs states in any dimension

Quantum 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 $ρ_{AC}$ on these regions to the product of its marginals, $$\| ρ_{AC} ρ_A^{-1} \otimes ρ_C^{-1} - 1_{AC}\|\, ,$$ 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 of the system is commuting and also commutes with every marginal of the Gibbs state. The proof of these results employs a variety of tools such as Araki's expansionals, quantum belief propagation and cluster expansions.

quant-ph↗

arXiv2024

General Continuity Bounds for Quantum Relative Entropies

In this article, we generalize a proof technique by Alicki, Fannes and Winter and introduce a method to prove continuity bounds for entropic quantities derived from different quantum relative entropies. For the Umegaki relative entropy, we mostly recover known almost optimal bounds, whereas, for the Belavkin-Staszewski relative entropy, our bounds are new. Finally, we use these continuity bounds to derive a new entropic uncertainty relation.

quant-ph↗

arXiv2024

Continuity of quantum entropic quantities via almost convexity

Based 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.

quant-ph↗

arXiv2023

Thermalization in Kitaev's quantum double models via Tensor Network techniques

We show that the Davies generator associated to any 2D Kitaev's quantum double model has a non-vanishing 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.

quant-ph↗

arXiv2022

Exponential decay of mutual information for Gibbs states of local Hamiltonians

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.

quant-ph↗

arXiv2021

Weak quasi-factorization for the Belavkin-Staszewski relative entropy

Quasi-factorization-type inequalities for the relative entropy have recently proven to be fundamental in modern proofs of modified logarithmic Sobolev inequalities for quantum spin systems. In this paper, we show some results of weak quasi-factorization for the Belavkin-Staszewski relative entropy, i.e. upper bounds for the BS-entropy between two bipartite states in terms of the sum of two conditional BS-entropies, up to some multiplicative and additive factors.

math-ph↗

arXiv2020

Locality estimes for complex time evolution in 1D

It 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 translational 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 analiticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the abscence 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 time-evolution 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.

math-ph↗