  The space of K\"ahler metrics can, on the one hand, be approximated by
subspaces of algebraic metrics, while, on the other hand, can be enlarged to
finite-energy spaces arising in pluripotential theory. The latter spaces are
realized as metric completions of Finsler structures on the space of K\"ahler
metrics. The former spaces are the finite-dimensional spaces of Fubini--Study
metrics of K\"ahler quantization. The goal of this article is to draw a
connection between the two. We show that the Finsler structures on the space of
K\"ahler potentials can be quantized. More precisely, given a K\"ahler manifold
polarized by an ample line bundle we endow the space of Hermitian metrics on
powers of that line bundle with Finsler structures and show that the resulting
path length metric spaces recover the corresponding metric completions of the
Finsler structures on the space of K\"ahler potentials. This has a number of
applications, among them a new approach to the rooftop envelopes and
Pythagorean formulas of K\"ahler geometry, a new Lidskii type inequality on the
space of K\"ahler metrics, and approximation of finite energy potentials, as
well as geodesic segments by the corresponding smooth algebraic objects.
