For a unital completely positive map Φ ("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 Φm 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→∞ 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