  Quantum technology is progressing towards fast quantum control over systems
interacting with small environments. Hence such technologies are operating in a
regime where the environment remembers the system's past, and the applicability
of complete-positive trace preserving maps is no longer valid. The departure
from complete positivity means many useful bounds, like entropy production,
Holevo, and data processing inequality are no longer applicable to such
systems. We address these issues by deriving a generalized bound for entropy
valid for quantum dynamics with arbitrary system-environment correlations. We
employ superchannels, which map quantum operations performed by the
experimenter, represented in terms of completely positive maps, to states. Our
bound has information-theoretic applications, as it generalizes the data
processing inequality and the Holevo bound. We prove that both data processing
inequality and the Holevo are valid even when system is correlated with the
environment.
