Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

Erbar, Matthias; Fathi, Max

doi:10.1016/j.jfa.2018.03.011

Cited by 36 publications

(40 citation statements)

References 44 publications

(91 reference statements)

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“…one which has the completely mixed state as its unique invariant state), we show that it also implies MLSI(c 2 D −2 ) for some universal positive constant c 2 (Theorem 7). We hence extend the results of [11] to the quantum regime.…”

Section: Ricci Lower Bound (Quantum Setting)mentioning

confidence: 53%

“…In [11,12,19], a modified version of the Ricci lower bound was defined for Markov processes on finite sets, which led to the unification of the previously discussed functional and concentration inequalities in this discrete framework. In particular, it was proved in [11] that one can recover the Poincaré and modified log-Sobolev inequalities from the Ricci lower bound, provided the diameter of P(M), with respect to the Wasserstein distance, W 2 , is bounded.…”

Section: Ricci Curvature and Ricci Lower Bound (Classical Setting)mentioning

confidence: 99%

“…Another straightforward consequence of the κ-displacement convexity of the quantum relative entropy for κ > 0 is the following estimate on the diameter Diam L (D(H)), which is a quantum analogue of the Bonnet-Myers theorem (see Proposition 7.3 of [11]).…”

Section: A Quantum Hwi Inequalitymentioning

confidence: 99%

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Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality

Datta

Rouzé

2020

Ann. Henri Poincaré

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Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one. 1 Here we assume that the QMS is primitive, i.e. it has a unique invariant state. 2 Given a metric space (X , d), a probability measure µ is said to satisfy Gaussian (resp. exponential) concentration on it if there exist positive constants a, b such that for any A ⊆ X , and r > 0, µ(A) ≥ 1/2 =⇒ µ(Ar) ≥ 1 − ae −bf (r) .

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Section: Ricci Lower Bound (Quantum Setting)mentioning

confidence: 53%

Section: Ricci Curvature and Ricci Lower Bound (Classical Setting)mentioning

confidence: 99%

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Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality

Datta

Rouzé

2020

Ann. Henri Poincaré

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“…It is possible to characterize Ricci curvature in terms of a gradient estimate in the spirit of Bakry-Émery; see [17] for the corresponding statement in the setting of finite Markov chains and [46] for an implementation in the Lindblad setting.…”

Section: Geodesic Convexity Of the Entropymentioning

confidence: 99%

Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

Cárlen

Maas

2019

J Stat Phys

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We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional C *-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein-Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transportentropy inequalities, and spectral gap estimates.

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“…Curvature-dimension inequalities in the context of optimal transport in the discrete setting have been studied in [Maa11], [EM12], [Erb14], [EKS15], [FS18], [EF18].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Curvature-dimension inequalities for non-local operators in the discrete setting

2019

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bstract. We study Bakry-Émery curvature-dimension inequalities for non-local operators on the one-dimensional lattice and prove that operators with finite second moment have finite dimension. Moreover, we show that a class of operators related to the fractional Laplacian fails to have finite dimension and establish both positive and negative results for operators with sparsely supported kernels. Furthermore, a large class of operators is shown to have no positive curvature. The results correspond to CD inequalities on locally infinite graphs.

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Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

Cited by 36 publications

References 44 publications

Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality

Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality

Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

Curvature-dimension inequalities for non-local operators in the discrete setting

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