  This paper introduces assignment flows for density matrices as state spaces
for representing and analyzing data associated with vertices of an underlying
weighted graph. Determining an assignment flow by geometric integration of the
defining dynamical system causes an interaction of the non-commuting states
across the graph, and the assignment of a pure (rank-one) state to each vertex
after convergence. Adopting the Riemannian Bogoliubov-Kubo-Mori metric from
information geometry leads to closed-form local expressions which can be
computed efficiently and implemented in a fine-grained parallel manner.
  Restriction to the submanifold of commuting density matrices recovers the
assignment flows for categorial probability distributions, which merely assign
labels from a finite set to each data point. As shown for these flows in our
prior work, the novel class of quantum state assignment flows can also be
characterized as Riemannian gradient flows with respect to a non-local
non-convex potential, after proper reparametrization and under mild conditions
on the underlying weight function. This weight function generates the
parameters of the layers of a neural network, corresponding to and generated by
each step of the geometric integration scheme.
  Numerical results indicates and illustrate the potential of the novel
approach for data representation and analysis, including the representation of
correlations of data across the graph by entanglement and tensorization.
