Curvature-dimension conditions for symmetric quantum Markov semigroups

Wirth, Melchior; Zhang, Haonan

doi:10.48550/arxiv.2105.08303

Cited by 1 publication

(3 citation statements)

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“…We can also discuss the approximate tensorization property of F p,σ similarly to [53,Theorem 5.1] and the stability of the data processing inequality for the divergence F p,σ similarly to [59,Proposition 5.1]. Third, in view of [97], it is straightforward to consider the curvature-dimension conditions for quantum systems and investigate the finite-dimension version of the quantum Beckner's inequality. The details and refinements of these results would be worth being reported elsewhere.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…This helps us to obtain concentration inequalities in a similar manner as [60]. We first recall the carré du champ operator (gradient form) associated with P t = e tL [60,97]:…”

Section: Moment Estimates and Concentration Inequalitiesmentioning

confidence: 99%

“…Based on Carlen and Maas's results, Datta and Rouzé [40,86] considered the Ricci curvature of a QMS, and obtained some quantum Sobolev and concentration inequalities, generalizing the results in the classical regime [46,47,79]. Wirth and Zhang [97] further introduced noncommutative curvature-dimension conditions and derived some dimension-dependent quantum functional inequalities.…”

Section: Introductionmentioning

confidence: 99%

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Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Li¹,

Lu²

2022

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In this work, we define the quantum p-divergences and introduce the Beckner's inequalities for the primitive quantum Markov semigroups with σ-detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative Lp-norm. We obtain a number of implications between Beckner's inequalities and other quantum functional inequalities. In particular, we show that the quantum Beckner's inequalities interpolate between the Sobolev-type inequalities and the Poincaré inequality in a sharp way. We provide a uniform lower bound for the Beckner constants αp in terms of the spectral gap, and establish the stability of αp with respect to the invariant state. As applications of Beckner's inequalities, we obtain an improved bound for the mixing time. For the symmetric quantum Markov semigroups, we derive moment estimates, which further implies a concentration inequality.We introduce a new class of quantum transport distances W 2,p interpolating the quantum 2-Wasserstein distance by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869] and a noncommutative Ḣ−1 Sobolev distance. We show that the quantum Markov semigroup with σ-detailed balance is the gradient flow of the quantum p-divergence with respect to the metric W 2,p . We prove that the set of quantum states equipped with W 2,p is a complete geodesic space. We then consider the associated entropic Ricci curvature lower bound via the geodesic convexity of p-divergence, and obtain an HWI-type interpolation inequality. This enables us to prove that the positive Ricci curvature implies the quantum Beckner's inequality, from which a transport cost and the Poincaré inequalities can follow.

show abstract

Section: Conclusion and Discussionmentioning

confidence: 99%

“…This helps us to obtain concentration inequalities in a similar manner as [60]. We first recall the carré du champ operator (gradient form) associated with P t = e tL [60,97]:…”

Section: Moment Estimates and Concentration Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Li¹,

Lu²

2022

Preprint

View full text Add to dashboard Cite

show abstract

Curvature-dimension conditions for symmetric quantum Markov semigroups

Cited by 1 publication

References 36 publications

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

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