  We introduce quantum optimal transport of states on tracial
AF-$C^{*}$-algebras to study non-spatial transport of quantum information, and
view it as the pointwise case of a general parametrised one. We define quantum
optimal transport distances as dynamic transport distances in a tracial but
non-ergodic and infinite-dimensional quantum setting, called
AF-$C^{*}$-setting. We further extend foundational results of Carlen and Maas
to the AF-$C^{*}$-setting and develop a theory of quantum optimal transport
yielding non-spatial lower Ricci bounds suitable for meaningful geometric
analysis. Essential for our discussion is a coarse graining process arising
from the underlying metric geometry as encoding scheme of the given tracial
AF-$C^{*}$-algebra. In the logarithmic mean setting, we apply the coarse
graining process to show equivalence of the EVI$_{\lambda}$-gradient flow
property for quantum relative entropy, its strong geodesic $\lambda$-convexity,
a, possibly infinite-dimensional, Bakry-\'Emery condition, and a Hessian lower
bound condition. We then define lower Ricci bounds of our quantum gradients
using any one of said equivalent conditions, give sufficient conditions for
lower Ricci bounds of direct sum quantum gradients and, assuming lower Ricci
bounds, derive functional inequalities HWI$_{\lambda}$, MLSI$_{\lambda}$ and
TW$_{\lambda}$ in the AF-$C^{*}$-setting alongside their chain of implications.
Fundamental example classes give quantum optimal transport of normal states on
hyperfinite factors of type I and II with both non-negative and strictly
positive lower Ricci bounds. An application is given by first and second
quantisation of spectral triples.
