  We show that complete positivity is not only sufficient but also necessary
for the validity of the quantum data-processing inequality. As a consequence,
the reduced dynamics of a quantum system are completely positive, even in the
presence of initial correlations with its surrounding environment, if and only
if such correlations do not allow any anomalous backward flow of information
from the environment to the system. Our approach provides an intuitive
information-theoretic framework to unify and extend a number of previous
results.
