  A new approach extending the concept of geometric phases to adiabatic open
quantum systems described by density matrices (mixed states) is proposed. This
new approach is based on an analogy between open quantum systems and
dissipative quantum systems which uses a $C^*$-module structure. The gauge
theory associated with these new geometric phases does not take place in an
usual principal bundle structure but in an higher structure, a categorical
principal bundle (so-called principal 2-bundle or non-abelian bundle gerbes)
which is sometimes a non-abelian twisted bundle. This higher degree in the
gauge theory is a geometrical manifestation of the decoherence induced by the
environment on the quantum system.
