Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups

2022/08/08 by Melchior Wirth, Haonan Zhang · 1 voice
Mathematics · #Advanced Operator Algebra Research #Curvature #Dimension (graph theory) #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Markov chain #Mathematics #Noncommutative geometry #Pure mathematics #Ricci curvature #Scalar curvature

paper · pdf · doi:10.1007/s00023-022-01220-x

openalex publication_date 2022/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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 generalized depolarizing semigroups.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Gradient flows of the entropy for finite Markov chains
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Geodesic convexity of the relative entropy in reversible Markov chains
• Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Relative entropy for von Neumann subalgebras
• Problemática territorial y ordenación del territorio en el país valenciano

Discussions

• AUTOMATIC REPOST OF OLD TWEETS FROM TWITTER (X) FOR ARCHIVING PURPOSES [01.11.2024] --- New publication on "Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups" by M. Wirth and H. Z [bsky, 0 points, 0 comments]

Related

• Curvature-dimension conditions for symmetric quantum Markov semigroups
• Problemática territorial y ordenación del territorio en el país valenciano
• Intertwining Curvature Bounds for Graphs and Quantum Markov Semigroups
• Geometric Approach Towards Complete Logarithmic Sobolev Inequalities
• Exponential Relative Entropy Decay Along Quantum Markov Semigroups