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We introduce optimal transport of states on AF-$C^{*}$-algebras, using adynamic formulation of the metric in the spirit of Benamou-Brenier. Essentialfor having energy functionals is an extension of quasi-entropies after Hiai andPetz to the AF-setting. The induced metric geometry on states is a limitgeometry of an ascending chain of its restrictions to states on generatingfinite-dimensional $C^{*}$-subalgebras. We prove equivalence of lower Riccibounds and a quantum Bakry-\'Emery gradient condition. As we do not assumeergodicity of the heat semigroup, Fermionic quantisation of spectral triplesinduces examples. This establishes $\textrm{Ric}\geq 0$ for quantumthermodynamic evolutions.

Quantum optimal transport for approximately finite-dimensional $C^{*}$-algebras