Coarse Ricci Curvature of Quantum Channels

Following Ollivier's work [60], we introduce the coarse Ricci curvature of a quantum channel as the contraction coefficient of non-commutative metrics on the state space. These metrics are defined as a non-commutative transportation cost in the spirit of [41], [40], 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 Poincaré inequality (spectral gap) as well as transportation cost inequalities. Using intertwining relations, we obtain positive coarse Ricci curvature bounds of Gibbs samplers, Bosonic beam-splitters as well as Pauli channels on n-qubits.