  There are several important bipartite systems of great interest in condensed
matter physics and in quantum information science. In condensed matter systems,
the subsystems are examined traditionally by using the Green function and
mean-field-like methods based on the Heisenberg representation. In quantum
information science, the subsystems are handled by composite density matrix,
its marginals describing the subsystems, and the Kraus representation to
elucidate the subsystem properties. In this work, a relationship is first
established between the two techniques which appear to be distinct at first
sight. This will be illustrated in detail by presenting the two methods in the
case of the celebrated exactly soluable Jaynes - Cummings model (1963) of a
two-state atom interacting with a one-mode quantized electromagnetic field. The
dynamics of this system was treated in the Heisenberg representation by
Ackerhalt and Rzazewski (1975). We present here the corresponding subdynamics
using Kraus representation. A relationship between the two approaches is
established and the relative merits of the two techniques are discussed in
elucidating the distinctive features of the subdynamics. The striking effects
of interaction and entanglement are made transparent in two illustrative
examples: the transformations that manifest in non-interacting number and spin
representations due to interactions in their respective subspaces.
