  We present an information geometric analysis of both entropic speeds and
entropy production rates arising from geodesic evolution on manifolds
parametrized by pure quantum states. In particular, we employ pure states that
emerge as outputs of suitably chosen su(2; C) time-dependent Hamiltonian
operators that characterize analog quantum search algorithms of specific types.
The su(2; C) Hamiltonian models under consideration are specified by external
time-dependent magnetic fields within which spin-1/2 test particles are
immersed. The positive definite Riemannian metrization of the parameter
manifold is furnished by the Fisher information function. The Fisher
information function is evaluated along parametrized squared probability
amplitudes obtained from the temporal evolution of these spin-1/2 test
particles. A minimum action approach is then utilized to induce the transfer of
the quantum system from its initial state to its final state on the parameter
manifold over a finite temporal interval. We demonstrate in an explicit manner
that the minimal (that is, optimum) path corresponds to the shortest (that is,
geodesic) path between the initial and final states. Furthermore, we show that
the minimal path serves also to minimize the total entropy production occurring
during the transfer of states. Finally, upon evaluating the entropic speed as
well as the total entropy production along optimal transfer paths within
several scenarios of physical interest in analog quantum searching algorithms,
we demonstrate in a transparent quantitative manner a correspondence between a
faster transfer and a higher rate of entropy production. We therefore conclude
that higher entropic speed is associated with lower entropic efficiency within
the context of quantum state transfer.
