  Given two orthornormal bases A and B, the basic form of the entropic
uncertainty principle is stated in terms of the sum of the Shannon entropies of
the probabilities of measuring A and B onto a given quantum state. State
independent lower bounds for this sum encapsulate the degree of incompatibility
of the observables diagonal in the A and B bases, and are usually derived by
extracting as much information as possible from the unitary operator U
connecting the two bases. Here we show a strategy to derive sequences of lower
bounds based on alternating sequences of measurements onto A and B. The problem
can be mapped into the multiple application of bistochastic processes that can
be described by the powers of the unistochastic matrices directly derivable
from U. By means of several examples we study the applicability of the method.
The results obtained show that the strategy can allow for an advantage both in
the pure state and in the mixed state scenario. The sequence of lower bounds is
obtained with resources which are polynomial in the dimension of the underlying
Hilbert space, and it is thus suitable for studying high dimensional cases.
