  A quantum system's state is identified with a density matrix. Though their
probabilistic interpretation is rooted in ensemble theory, density matrices
embody a known shortcoming. They do not completely express an ensemble's
physical realization. Conveniently, when working only with the statistical
outcomes of projective and positive operator-valued measurements this is not a
hindrance. To track ensemble realizations and so remove the shortcoming, we
explore geometric quantum states and explain their physical significance. We
emphasize two main consequences: one in quantum state manipulation and one in
quantum thermodynamics.
