Ricci curvature of quantum channels on non-commutative transportation metric spaces

TL;DR

This paper introduces a quantum analogue of Ricci curvature based on non-commutative transportation metrics, linking it to functional inequalities and providing bounds for various quantum channels.

Contribution

It defines the coarse Ricci curvature for quantum channels using non-commutative transportation costs and establishes its implications for quantum functional inequalities.

Findings

Lower bounds on Ricci curvature imply Poincaré and transportation inequalities.

Positive Ricci curvature bounds are obtained for Gibbs samplers and quantum channels.

The approach unifies different quantum Wasserstein distances in the literature.

Abstract

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.