On entropic uncertainty relations in the presence of a minimal length

Alexey E. Rastegin

Published 2016-07-28Version 1

We study entropic uncertainty relations for the position and momentum within the generalized uncertainty principle. This principle is motivated by the existence of a minimal observable length. Then the position and momentum operators satisfy the modified commutation relation, for which more than one algebraic representation is known. One of them is described by auxiliary momentum so that the momentum and coordinate wave functions are connected by the Fourier transform. However, the probability density functions of the physically true and auxiliary momenta are different. This fact results in changes of related entropies and, therefore, in a modification of entropic bounds. As measures of uncertainties, the Shannon entropy and generalized entropic functions are used. Using differential Shannon entropies, we give a state-dependent formulation with correction term. State-independent uncertainty relations are obtained in terms of the R\'{e}nyi entropies and, with binning, the Tsallis entropies. Also, these relations take into account a finiteness of measurement resolution.