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Curvature-dimension conditions for symmetric quantum Markov semigroups (2105.08303v2)
Published 18 May 2021 in math.OA, cs.IT, math-ph, math.FA, math.IT, and math.MP
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.