  We study dynamical optimal transport metrics between density matrices
associated to symmetric Dirichlet forms on finite-dimensional $C^*$-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.
