  We consider a special iterated quantum protocol with measurement-induced
nonlinearity for qubits, where all pure initial states on the Bloch sphere can
be considered chaotic. The dynamics is ergodic with no attractive fixed cycles.
We show that initial noise radically changes this behavior. The completely
mixed state is an attractive fixed point of the dynamics induced by the
protocol. Our numerical simulations strongly indicate that initially mixed
states all converge to the completely mixed state. The presented protocol is an
example, where gaining information from measurements and employing it to
control an ensemble of quantum systems enables us to create ergodicity, which
in turn is destroyed by any initial noise.
