  Quantum information processing exploits all the features quantum mechanics
offers. Among them there is the possibility to induce nonlinear maps on a
quantum system by involving two or more identical copies of the given system in
the same state. Such maps play a central role in distillation protocols used
for quantum key distribution. We determine that such protocols may exhibit
sensitive, quasi-chaotic evolution not only for pure initial states but also
for mixed states, i.e. the complex dynamical behavior is not destroyed by small
initial uncertainty. We show that the appearance of sensitive, complex dynamics
associated with a fractal structure in the parameter space of the system has
the character of a first order phase transition. The purity of the initial
state plays the role of the control parameter and the dimension of the fractal
structure is independent of the purity value after passing the phase transition
point. The critical purity coincides with the purity of a repelling fixed point
of the dynamics and we show that all the pre-images of states from the close
neighborhood of pure chaotic initial states have purity larger than this.
Initial states from this set can be considered as quasi-chaotic.
