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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.
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Sur les inégalités de Sobolev logarithmiques
C. Ané, S. Blachère, D. Chafai, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer · 2000
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A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
J.-D. Benamou and Y. Brenier · 2000
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Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
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Derivations as square roots of Dirichlet forms
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The Concentration of Measure Phenomenon
M. Ledoux · 2005
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Strong converse and Stein’s lemma in quantum hypothesis testing
T. Ogawa and H. Nagaoka · 2005
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A large deviation approach to some transportation cost inequalities
N. Gozlan and C. Léonard · 2006
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On the geometry of metric measure spaces
K.-T. Sturm · 2006
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Transportation-information inequalities for Markov processes
A. Guillin, C. Léonard, L. Wu, and N. Yao · 2008
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Ricci curvature for metric-measure spaces via optimal transport
J. Lott and C. Villani · 2009
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Ricci curvature of Markov chains on metric spaces
Y. Ollivier · 2009
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Optimal transport: old and new
C. Villani · 2009
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Gradient flows of the entropy for finite Markov chains
J. Maas · 2011
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A user’s guide to optimal transport
M. Wirth · 2018
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I. Bardet, M. Junge, N. LaRacuente, C. Rouzé, and D. S. França · 2019
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On period, cycles and fixed points of a quantum channel
R. Carbone and A. Jenčová · 2019
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Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
E. A. Carlen and J. Maas · 2019
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Quantum systems, channels, information
A. S. Holevo · 2019
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L. Ambrosio and N. Gigli · 2012
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Ricci curvature of finite Markov chains via convexity of the entropy
M. Erbar and J. Maas · 2012
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Analysis and Geometry of Markov Diffusion Operators
D. Bakry, I. Gentil, and M. Ledoux · 2014
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An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker–Planck equation is gradient flow for the entropy
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M. Junge, E. Ricard, and D. Shlyahktenko · 2014
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Concentration of quantum states from quantum functional and transportation cost inequalities
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Complete logarithmic Sobolev inequalities via Ricci curvature bounded below I
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Relating relative entropy, optimal transport and Fisher information: a quantum HWI inequality
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Fisher information and logarithmic Sobolev inequality for matrix-valued functions
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H. Li, M. Junge, and N. LaRacuente · 2020
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Complete gradient estimates of quantum Markov semigroups
M. Wirth and H. Zhang · 2020
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