  We propose a category-theoretic definition of retrodiction and use it to
exhibit a time-reversal symmetry for all quantum channels. We do this by
introducing retrodiction families and functors, which capture many intuitive
properties that retrodiction should satisfy and are general enough to encompass
both classical and quantum theories alike. Classical Bayesian inversion and all
rotated and averaged Petz recovery maps define retrodiction families in our
sense. However, averaged rotated Petz recovery maps, including the universal
recovery map of Junge-Renner-Sutter-Wilde-Winter, do not define retrodiction
functors, since they fail to satisfy some compositionality properties. Among
all the examples we found of retrodiction families, the original Petz recovery
map is the only one that defines a retrodiction functor. In addition,
retrodiction functors exhibit an inferential time-reversal symmetry consistent
with the standard formulation of quantum theory. The existence of such a
retrodiction functor seems to be in stark contrast to the many no-go results on
time-reversal symmetry for quantum channels. One of the main reasons is because
such works defined time-reversal symmetry on the category of quantum channels
alone, whereas we define it on the category of quantum channels and quantum
states. This fact further illustrates the importance of a prior in
time-reversal symmetry.
