  We study holographic subregion complexity, and its possible connection to
purification complexity suggested recently by Ag\'on et al. In particular, we
study the conjecture that subregion complexity is the purification complexity
by considering holographic purifications of a holographic mixed state. We argue
that these include states with any amount of coarse-graining consistent with
being a purification of the mixed state in question, corresponding
holographically to different choices of the cutoff surface. We find that within
the complexity = volume and complexity = spacetime volume conjectures, the
subregion complexity is equal to the holographic purification complexity. For
complexity = action, the subregion complexity seems to provide an upper bound
on the holographic purification complexity, though we show cases where this
bound is not saturated. One such example is provided by black holes with a
large genus behind the horizon, which were studied by Fu et al. As such, one
must conclude that these offending geometries are not holographic, that CA must
be modified, or else that holographic subregion complexity in CA is not dual to
the purification complexity of the corresponding reduced state.
