  We study a kind of geometric phases for entangled quantum systems, and
particularly a spin driven by a magnetic field and entangled with another spin.
The new kind of geometric phase is based on an analogy between open quantum
systems and dissipative quantum systems which uses a C\ast-module structure. We
show that the system presents from the viewpoint of their geometric phases, two
behaviours. The first one is identical to the behaviour of an isolated spin
driven by a magnetic field, as the problem originally treated by Berry. The
second one is specific to the decoherence process. The gauge structures induced
by these geometric phases are then similar to a magnetic monopole gauge
structure for the first case, and can be viewed as a kind of instanton gauge
structure for the second case. We study the role of these geometric phases in
the evolution of a mixed state, particularly by focusing on the evolution of
the density matrix coherence. We investigate also the relation between the
geometric phase of the mixed state of one of the entangled systems and the
geometric phase of the bipartite system.
