  The relative entropy is a principal measure of distinguishability in quantum
information theory, with its most important property being that it is
non-increasing with respect to noisy quantum operations. Here, we establish a
remainder term for this inequality that quantifies how well one can recover
from a loss of information by employing a rotated Petz recovery map. The main
approach for proving this refinement is to combine the methods of [Fawzi and
Renner, arXiv:1410.0664] with the notion of a relative typical subspace from
[Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper
constitutes partial progress towards a remainder term which features just the
Petz recovery map (not a rotated Petz map), a conjecture which would have many
consequences in quantum information theory.
  A well known result states that the monotonicity of relative entropy with
respect to quantum operations is equivalent to each of the following
inequalities: strong subadditivity of entropy, concavity of conditional
entropy, joint convexity of relative entropy, and monotonicity of relative
entropy with respect to partial trace. We show that this equivalence holds true
for refinements of all these inequalities in terms of the Petz recovery map. So
either all of these refinements are true or all are false.
