  We prove a version of the data-processing inequality for the relative entropy
for general von Neumann algebras with an explicit lower bound involving the
measured relative entropy. The inequality, which generalizes previous work by
Sutter et al. on finite dimensional density matrices, yields a bound how well a
quantum state can be recovered after it has been passed through a channel. The
natural applications of our results are in quantum field theory where the von
Neumann algebras are known to be of type III. Along the way we generalize
various multi-trace inequalities to general von Neumann algebras.
