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There are several inequalities in physics which limit how well we can processphysical systems to achieve some intended goal, including the second law ofthermodynamics, entropy bounds in quantum information theory, and theuncertainty principle of quantum mechanics. Recent results provide physicallymeaningful enhancements of these limiting statements, determining how well onecan attempt to reverse an irreversible process. In this paper, we apply andextend these results to give strong enhancements to several entropyinequalities, having to do with entropy gain, information gain, entropicdisturbance, and complete positivity of open quantum systems dynamics. Ourfirst result is a remainder term for the entropy gain of a quantum channel.This result implies that a small increase in entropy under the action of asubunital channel is a witness to the fact that the channel's adjoint can beused as a recovery map to undo the action of the original channel. Our secondresult regards the information gain of a quantum measurement, both without andwith quantum side information. We find here that a small information gainimplies that it is possible to undo the action of the original measurement ifit is efficient. The result also has operational ramifications for theinformation-theoretic tasks known as measurement compression without and withquantum side information. Our third result shows that the loss of Holevoinformation caused by the action of a noisy channel on an input ensemble ofquantum states is small if and only if the noise can be approximately correctedon average. We finally establish that the reduced dynamics of asystem-environment interaction are approximately completely positive andtrace-preserving if and only if the data processing inequality holdsapproximately.

Approximate reversibility in the context of entropy gain, information gain, and complete positivity