Gradient Flows and Optimal Transport in Discrete and Quantum Systems
summary
Shows that Markov chain evolution equals the gradient flow of relative entropy for a new Riemannian transport metric built from a discrete Benamou–Brenier formula using the logarithmic mean, proving a JKO analogue and giving a Ricci lower bound notion via entropy convexity on graphs. Introduces a quantum version: Lindblad generators, a non-commutative chain rule and a matrix-valued Benamou–Brenier distance on density matrices, yielding log-Sobolev/Talagrand inequalities and exponential entropy decay for quantum Ornstein–Uhlenbeck semigroups.
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chapters (14)
- 0:00Introduction to the topic
- 1:42Optimal transport basics
- 4:38Gradient flows in metric spaces
- 8:16Extensions to general spaces
- 11:42Geometry and Ricci curvature
- 14:40Ricci curvature in metric spaces
- 19:22Challenges in discrete spaces
- 23:09Discrete optimal transport
- 29:12Dynamical formulations
- 33:37Discrete gradient flows
- 37:34Curvature and convergence
- 42:02Quantum optimal transport
- 46:43Gradient flows in quantum systems
- 1:03:30Q&A session
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