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Given two orthornormal bases A and B, the basic form of the entropicuncertainty principle is stated in terms of the sum of the Shannon entropies ofthe probabilities of measuring A and B onto a given quantum state. Stateindependent lower bounds for this sum encapsulate the degree of incompatibilityof the observables diagonal in the A and B bases, and are usually derived byextracting as much information as possible from the unitary operator Uconnecting the two bases. Here we show a strategy to derive sequences of lowerbounds based on alternating sequences of measurements onto A and B. The problemcan be mapped into the multiple application of bistochastic processes that canbe described by the powers of the unistochastic matrices directly derivablefrom 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 inthe pure state and in the mixed state scenario. The sequence of lower bounds isobtained with resources which are polynomial in the dimension of the underlyingHilbert space, and it is thus suitable for studying high dimensional cases.

Sequences of lower bounds for entropic uncertainty relations from bistochastic maps