  For a unital completely positive map $\Phi$ ("quantum channel") governing the
time propagation of a quantum system, the Stinespring representation gives an
enlarged system evolving unitarily. We argue that the Stinespring
representations of each power $\Phi^m$ of the single map together encode the
structure of the original quantum channel and provides an interaction-dependent
model for the bath. The same bath model gives a "classical limit" at infinite
time $m\to\infty$ in the form of a noncommutative "manifold" determined by the
channel. In this way a simplified analysis of the system can be performed by
making the large-$m$ approximation. These constructions are based on a
noncommutative generalization of Berezin quantization. The latter is shown to
involve very fundamental aspects of quantum-information theory, which are
thereby put in a completely new light.
