  Quantum channels depending on a number of classical control parameters are
considered. Assuming the stochastic fluctuations of the control parameters in
the small errors limit it is shown that the channel fidelity is equal to the
average value of the channel purities calculated in the cases of the control
errors presence and absence respectively. The result validity is demonstrated
on two particular examples, namely, one-qubit quantum gates implemented in the
framework of the ion trap setup and the single qubit anisotropic depolarizing
quantum channel.
