  The study of quantum systems evolving from initial states to distinguishable,
orthogonal final states is important for information processing applications
such as quantum computing and quantum metrology. However, for most unitary
evolutions and initial states the system does not evolve to an orthogonal
quantum state. Here, we ask what proportion of quantum states evolves to nearly
orthogonal systems as a function of the dimensionality of the Hilbert space of
the system, and numerically study the evolution of quantum states in
low-dimensional Hilbert spaces. We find that, as well as the speed of dynamical
evolution, the level of maximum distinguishability depends critically on the
Hamiltonian of the system.
