  Following Ollivier's work, we introduce the coarse Ricci curvature of a
quantum channel as the contraction of non-commutative metrics on the state
space. These metrics are defined as a non-commutative transportation cost in
the spirit of [N. Gozlan and C. L\'{e}onard. 2006], which gives a unified
approach to different quantum Wasserstein distances in the literature. We prove
that the coarse Ricci curvature lower bound and its dual gradient estimate,
under suitable assumptions, imply the Poincar\'{e} inequality (spectral gap) as
well as transportation cost inequalities. Using intertwining relations, we
obtain positive bounds on the coarse Ricci curvature of Gibbs samplers, Bosonic
and Fermionic beam-splitters as well as Pauli channels on n-qubits.
