Ángela Capel siiting on the stairs of the LMU Physics foyer.

MCQST Distinguished PostDoc 2020

Technical University of Munich

Mathematical Physics | M5

Boltzmannstrasse 3

85748 Garching

Tel. +49 89 289 18321



I love the beauty of the particular types of abstract patterns involved in my research, feel inspired by the flash of enlightenment I experience when a new idea connects previously unconnected ones and fulfil my expectations by applying them to solve non-trivial problems.


Research focus: Study of the speed of convergence of quantum dissipative evolutions via quantum functional inequalities.

My envisaged project at MCQST concerns the question of how fast a quantum thermal dissipative evolution converges to its equilibrium state. This velocity of thermalization is given by the time that it takes for every initial physical state undergoing a dissipative evolution to be almost indistinguishable from the thermal equilibrium, which can be studied via the optimal constants associated to some quantum functional inequalities. In particular, we are interested in physical systems for which this convergence is fast enough, and thus have nice properties such as stability against external perturbations and satisfy an area law. My goal is to design and implement the necessary mathematical tools to obtain conditions on physical systems that imply rapid mixing on quantum systems and that can be used to estimate the speed of convergence of an algorithm or the noise generated in a quantum circuit.



Approximate tensorization of the relative entropy for noncommuting conditional expectations

I. Bardet, A. Capel, C. Rouzé

Annales Henri Poincare 23, 101–140 (2022).

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In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

DOI: 10.1007/s00023-021-01088-3

Weak Quasi-Factorization for the Belavkin-Staszewski Relative Entropy

A. Bluhm, A. Capel, A. Perez-Hernandez

IEEE International Symposium on Information Theory (ISIT) 118-123 (2021).

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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.

DOI: 10.1109/isit45174.2021.9517893

On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

I. Bardet, A. Capel, A. Lucia, D. Pérez-García, C. Rouzé

Journal of Mathematical Physics 62, 61901 (2021).

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The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated with the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular, we show that for the heat-bath dynamics of 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

DOI: 10.1063/1.5142186

The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions

Ángela Capel, Cambyse Rouzé, Daniel Stilck França


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Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the system to equilibrium under a condition of spatial mixing of the stationary Gibbs states and the rapid decay of the relative entropy on finite-size blocks. Our result leads to the first examples of the positivity of the modified logarithmic Sobolev inequality for quantum lattice spin systems independently of the system size. Moreover, we show that our notion of spatial mixing is a consequence of the recent quantum generalization of Dobrushin and Shlosman's complete analyticity of the free-energy at equilibrium. The latter typically holds above a critical temperature Tc. Our results have wide-ranging applications in quantum information. As an illustration, we discuss four of them: first, using techniques of quantum optimal transport, we show that a quantum annealer subject to a finite range classical noise will output an energy close to that of the fixed point after constant annealing time. Second, we prove Gaussian concentration inequalities for Lipschitz observables and show that the eigenstate thermalization hypothesis holds for certain high-temperture Gibbs states. Third, we prove a finite blocklength refinement of the quantum Stein lemma for the task of asymmetric discrimination of two Gibbs states of commuting Hamiltonians satisfying our conditions. Fourth, in the same setting, our results imply the existence of a local quantum circuit of logarithmic depth to prepare Gibbs states of a class of commuting Hamiltonians.


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