  This article elaborates on entanglement entropy and quantum information
theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical
systems. A set of basic geometric and quantum mechanics and probability
concepts together with methods of computation are developed in general
covariant form for curved phase spaces modelled as cotangent Lorentz bundles.
The constructions are based on ideas relating the Grigory Perelman's entropy
for geometric flows and associated statistical thermodynamic systems to the
quantum von Neumann entropy, classical and quantum relative and conditional
entropy, mutual information etc. We formulate the concept of the entanglement
entropy of quantum geometric information flows and study properties and
inequalities for quantum, thermodynamic and geometric entropies characterising
such systems.
