Non commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems
We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite dimensional $C^ C 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.
Journal of statistical physics