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

• URL: http://arxiv.org/abs/2108.10609v1
• Date: Tue, 24 Aug 2021 09:52:29 GMT
• Title: Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Authors: Li Gao and Cambyse Rouz\'e
• Abstract summary: We introduce the coarse Ricci curvature of a quantum channel as the contraction of non-commutative metrics on the state space. We prove that the coarse Ricci curvature lower bound and its dual gradient estimate, under suitable assumptions, imply the Poincar'e inequality and transportation cost inequalities.
• Score: 17.21921346541951
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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