In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operatorρS. 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 ofρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing fromρS. Using examples where the SOI are generated using representations ofSU(2),SO(3), andSO(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-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the 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 with respect to the purity ofρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann’s original CG.