On entropic uncertainty relations in the presence of a minimal length

TL;DR

This paper investigates how entropic uncertainty relations are modified under the generalized uncertainty principle that includes a minimal length, affecting the connection between wave functions and the form of uncertainty bounds.

Contribution

It derives new entropic uncertainty relations incorporating minimal length effects using differential Shannon, Rényi, and Tsallis entropies, accounting for measurement resolution.

Findings

State-dependent entropic uncertainty relations with correction terms.

State-independent bounds using Rényi and Tsallis entropies.

Relations account for finite measurement resolution.

Abstract

Entropic uncertainty relations for the position and momentum within the generalized uncertainty principle are examined. Studies of this principle are 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. As the corresponding entropies differ, known entropic uncertainty relations are changed. 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 the Tsallis entropies with binning. Such relations allow one to take into account a finiteness of measurement resolution.