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where the environment remembers the system's past, and the applicabilityof complete-positive trace preserving maps is no longer valid.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complete positivity means many useful bounds, like entropy production,Holevo, and data processing inequality are no longer applicable to suchsystems.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

Entropy bounds for quantum processes with initial correlations