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We investigate classical and quantum geometric information flow theories(respectively, GIFs and QGIFs) when the geometric flow evolution and fieldequations for nonholonomic Einstein systems, NES, are derived fromPerelman-Lyapunov type entropic type functionals. In this work, the term NESencodes models of gravitational and matter fields interactions and theirgeometric flow evolution subjected to nonholonomic (equivalently,non-integrable, anholonomic) constraints. There are used canonical geometricvariables which allow a general decoupling and integration of systems ofnonlinear partial differential equations describing GIFs and QGIFs and (forself-similar geometric flows) Ricci soliton type configurations. Our approachis different from the methods and constructions elaborated for special classesof solutions characterized by area--hypersurface entropy, related holographicand dual gauge--gravity models, and conformal field theories, involvinggeneralizations of the Bekenstein-Hawking entropy and black holethermodynamics. We formulate the theory of QGIFs which in certainquasi-classical limits encodes GIFs and models with flow evolution of NES.There are analysed the most important properties (inequalities) for NES anddefined and computed QGIF versions of the von Neumann, relative and conditionalentropy; mutual information, (modified) entanglement and R\'{e}nyi entropy. Weconstruct explicit examples of generic off-diagonal exact and parametricsolutions describing stationary solitonic gravitational hierarchies anddeformations of black hole configurations. Finally, we show how Perelman'sentropy and geometric thermodynamic values, and extensions to GIF and QGIFmodels can be computed for various new classes of exact solutions which cannotbe described following the Bekenstein-Hawking approach.

quantum information processing

Quantum geometric information flows and relativistic generalizations of  G. Perelman thermodynamics for nonholonomic Einstein systems with black holes  and stationary solitonic hierarchies