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Published on: August 2, 2019
Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems
1Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA.
Abstract:
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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