  In this paper, we construct the metric tensor and volume for the manifold of
purifications associated with an arbitrary reduced density operator $\rho_S$.
We also define a quantum coarse-graining (CG) to study the volume where
macrostates are the manifolds of purifications, which we call surfaces of
ignorance (SOI), and microstates are the purifications of $\rho_S$. In this
context, the volume functions as a multiplicity of the macrostates that
quantifies the amount of information missing from $\rho_S$. Using examples
where the SOI are generated using representations of $SU(2)$, $SO(3)$, and
$SO(N)$, we show two features of the CG. (1) A system beginning in an atypical
macrostate of smaller volume evolves to macrostates of greater volume until it
reaches the equilibrium macrostate in a process in which the system and
environment become strictly more entangled, and (2) the equilibrium macrostate
takes up the vast majority of the coarse-grainied space especially as the
dimension of the total system becomes large. Here, the equilibrium macrostate
corresponds to maximum entanglement between system and environment. To
demonstrate feature (1) for the examples considered, we show that the volume
behaves like the von Neumann entropy in that it is zero for pure states,
maximal for maximally mixed states, and is a concave function w.r.t the purity
of $\rho_S$. These two features are essential to typicality arguments regarding
thermalization and Boltzmann's original CG.
