  We generalize our results in paper I in this series to quantum channels
between general v. 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 v. Neumann algebras. Compared to the usual inequality, there is an
explicit lower bound involving the fidelity between the original state and a
recovery channel.
