Approximate recoverability and relative entropy II: 2-positive channels of general von Neumann algebras

We generalize our results in paper I in this series to quantum channels between general von Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two von Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.