  Quantum entropies and state distances are analyzed in polaronic systems with
short range (Holstein model) and long range (Fr$\ddot{o}$hlich model)
electron-phonon coupling. These quantities are extracted by a variational wave
function which describes very accurately polaron systems with arbitrary size in
all the relevant parameter regimes. With the use of quantum information tools,
the crossover region from weak to strong coupling regime can be characterized
with high precision. Then, the linear entropy is found to be very sensitive to
the range of the electron-phonon coupling and the adiabatic ratio. Finally, the
entanglement entropy is studied as a function of the system size pointing out
that it not bounded, but scales as the logarithm of the size either for weak
electron-phonon coupling or for short range interaction. This behavior is
ascribed to the peculiar coupling induced by the single electron itinerant
dynamics on the phonon subsystem.
