  Based on a generic quantum open system model, we study the geometric nature
of decoherence by defining a complex-valued geometric phase through stochastic
pure states describing non-unitary, non-cyclic and non-adiabatic evolutions.
The ensemble average of the complex geometric phases for the pure stochastic
states yields a conventional geometric phase together with an amplitude factor.
We show that the decoherence process described by the decaying amplitude can be
a geometric quantity independent of the system's dynamics. It is a remarkable
fact that the geometric phase of a quantum system can serve as an ideal
realisation of quantum gates due to its robustness against dynamical errors,
however, in this paper we show that, for some open quantum systems, a desirable
geometric phase may be accompanied by an unwanted robust geometric decoherence
factor. Two exactly solvable models are studied to demonstrate that, while the
decoherence is a purely dynamical effect for a dephasing two-level model, the
decoherence in a dissipative two-level model can be a geometric process.
Finally, we show that such a geometric decoherence effect may be eliminated by
a non-perturbative control scheme.
