  The manifold of ground states of a family of quantum Hamiltonians can be
endowed with a quantum geometric tensor whose singularities signal quantum
phase transitions and give a general way to define quantum phases. In this
paper, we show that the same information-theoretic and geometrical approach can
be used to describe the geometry of quantum states away from equilibrium. We
construct the quantum geometric tensor $Q_{\mu\nu}$ for ensembles of states
that evolve in time and study its phase diagram and equilibration properties.
If the initial ensemble is the manifold of ground states, we show that the
phase diagram is conserved, that the geometric tensor equilibrates after a
quantum quench, and that its time behavior is governed by out-of-time-order
commutators (OTOCs). We finally demonstrate our results in the exactly solvable
Cluster-XY model.
