  Twisted classical solutions to the $\mathbb{C}P^{N-1}$ model play a key role
in the analysis of such models on the spatially compactified cylinder
$\mathbb{S}_L^1 \times {\mathbb{R}^1}$ and have recently been shown to be
important for the resurgent structure of this quantum field theory. Instantons
and non-self-dual solutions both fractionalize, and domain walls formed by such
topological solutions can be associated with $N$-vacua having maximally
repulsive energy eigenvalues. The purpose of this paper is to reinforce this
view through the investigation of a number of parallels between the
$\mathbb{C}P^{N-1}$ model and $N$-level quantum mechanical density matrices.
Specifically, we demonstrate the existence of a time-evolution equation for the
$\mathbb{C}P^{N-1}$ instanton projector analogous to the Liouville-von Neumann
equation in the quantum mechanical formalism. The group theoretical analysis of
density matrices and the $\mathbb{C}P^{N-1}$ model are also closely related.
Finally, we explore the emergence of geometrical (Berry) phases in both systems
and their interrelationship.
