  We present an information geometric analysis of entropic speeds and entropy
production rates in geodesic evolution on manifolds of parametrized quantum
states. These pure states emerge as outputs of suitable su(2; C) time-dependent
Hamiltonian operators used to describe distinct types of analog quantum search
schemes. The Riemannian metrization on the manifold is specified by the Fisher
information evaluated along the parametrized squared probability amplitudes
obtained from analysis of the temporal quantum mechanical evolution of a
spin-1/2 particle in an external time-dependent magnetic field that specifies
the su(2; C) Hamiltonian model. We employ a minimum action method to transfer a
quantum system from an initial state to a final state on the manifold in a
finite temporal interval. Furthermore, we demonstrate that the minimizing
(optimum) path is the shortest (geodesic) path between the two states, and, in
particular, minimizes also the total entropy production that occurs during the
transfer. Finally, by evaluating the entropic speed and the total entropy
production along the optimum transfer paths in a number of physical scenarios
of interest in analog quantum search problems, we show in a clear quantitative
manner that to a faster transfer there corresponds necessarily a higher entropy
production rate. Thus, we conclude that lower entropic efficiency values appear
to accompany higher entropic speed values in quantum transfer processes.
