Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups

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...

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Published inAnnales Henri Poincaré Vol. 24; no. 3; pp. 717 - 750
Main Authors Wirth, Melchior, Zhang, Haonan
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 2023
Springer Nature B.V
Subjects
Algebra
Classical and Quantum Gravitation
Concavity
Curvature
Dynamical Systems and Ergodic Theory
Elementary Particles
Group theory
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Matrices (mathematics)
Multipliers
Original Paper
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Relativity Theory
Semigroups
Theoretical
Primary 81S22
Secondary 46L57
81R05
49Q22
47C99
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Summary: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 generalized depolarizing semigroups.
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content type line 23
Communicated by Alain Joye.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-022-01220-x