  Holevo's just-as-good fidelity is a similarity measure for quantum states
that has found several applications. One of its critical properties is that it
obeys a data processing inequality: the measure does not decrease under the
action of a quantum channel on the underlying states. In this paper, I prove a
refinement of this data processing inequality that includes an additional term
related to recoverability. That is, if the increase in the measure is small
after the action of a partial trace, then one of the states can be nearly
recovered by the Petz recovery channel, while the other state is perfectly
recovered by the same channel. The refinement is given in terms of the trace
distance of one of the states to its recovered version and also depends on the
minimum eigenvalue of the other state. As such, the refinement is universal, in
the sense that the recovery channel depends only on one of the states, and it
is explicit, given by the Petz recovery channel. The appendix contains a
generalization of the aforementioned result to arbitrary quantum channels.
