We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy. We find that for small full width at half maximum (FWHM) of the position entropy density, the FWHM of the momentum entropy density is large and vice versa. By increasing the confined potential depth, the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases. By increasing the potential depth, the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases. By increasing the number of wells, the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well. As an example, we might localize the ground state as well as the position entropy densities of the 1st, 2nd, and 6th excited states for a four-well quantum system. Also, we verify the Bialynicki–Birula–Mycieslki (BBM) inequality.