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arXiv · 2108.10609

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

Li Gao·Cambyse Rouzé
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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.

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Li Gao, Cambyse Rouzé. 2021-08-24. Ricci curvature of quantum channels on non-commutative transportation metric spaces. https://arxiv.org/abs/2108.10609

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