Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 178, Issue 2, Pages 319-378Publisher
SPRINGER
DOI: 10.1007/s10955-019-02434-w
Keywords
Non-commutative optimal transport; Functional inequalities; Lindblad equation; Gradient flow
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Funding
- Institute of Science and Technology (IST Austria)
- NSF [DMS-174625]
- European Research Council (ERC) under the European Union [716117]
- Austrian Science Fund (FWF) [SFB F65]
- European Research Council (ERC) [716117] Funding Source: European Research Council (ERC)
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We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein-Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.
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