  We investigate classical and quantum geometric information flow theories
(respectively, GIFs and QGIFs) when the geometric flow evolution and field
equations for nonholonomic Einstein systems, NES, are derived from
Perelman-Lyapunov type entropic type functionals. In this work, the term NES
encodes models of gravitational and matter fields interactions and their
geometric flow evolution subjected to nonholonomic (equivalently,
non-integrable, anholonomic) constraints. There are used canonical geometric
variables which allow a general decoupling and integration of systems of
nonlinear partial differential equations describing GIFs and QGIFs and (for
self-similar geometric flows) Ricci soliton type configurations. Our approach
is different from the methods and constructions elaborated for special classes
of solutions characterized by area--hypersurface entropy, related holographic
and dual gauge--gravity models, and conformal field theories, involving
generalizations of the Bekenstein-Hawking entropy and black hole
thermodynamics. We formulate the theory of QGIFs which in certain
quasi-classical limits encodes GIFs and models with flow evolution of NES.
There are analysed the most important properties (inequalities) for NES and
defined and computed QGIF versions of the von Neumann, relative and conditional
entropy; mutual information, (modified) entanglement and R\'{e}nyi entropy. We
construct explicit examples of generic off-diagonal exact and parametric
solutions describing stationary solitonic gravitational hierarchies and
deformations of black hole configurations. Finally, we show how Perelman's
entropy and geometric thermodynamic values, and extensions to GIF and QGIF
models can be computed for various new classes of exact solutions which cannot
be described following the Bekenstein-Hawking approach.
