Curvature-dimension conditions for symmetric quantum Markov semigroups

DOI10.1007/S00023-022-01220-XarXiv2105.08303MaRDI QIDQ6367897

Publication date: 18 May 2021

Abstract: Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet-Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras and depolarizing semigroups.

Mathematics Subject Classification ID

Finite-dimensional groups and algebras motivated by physics and their representations (81R05) Commutation relations and statistics as related to quantum mechanics (general) (81S05) Measures of information, entropy (94A17) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Derivations, dissipations and positive semigroups in (C^*)-algebras (46L57) Linear operators in algebras (47C05) Optimal transportation (49Q22)

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