On entropic uncertainty relations in the presence of a minimal length

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 Rényi entropies and the Tsallis entropies with binning. Such relations allow one to take into account a finiteness of measurement resolution.

10 pages, no figures. With final corrections, it matches the journal version

References in corpus (15)

• Heisenberg's Uncertainty Principle
• Stronger uncertainty relations for the sum of variances
• Formulation of the uncertainty relations in terms of the Renyi entropies
• GRBs Neutrinos as a Tool to Explore Quantum Gravity induced Lorentz Violation
• Majorization approach to entropic uncertainty relations for coarse-grained observables
• The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle
• Incorporation of Generalized Uncertainty Principle into Lifshitz Field Theories
• Supersymmetry Breaking as a new source for the Generalized Uncertainty Principle
• A revision of the Generalized Uncertainty Principle
• Entropic uncertainty relations for successive measurements of canonically conjugate observables
• Unsharpness of generalized measurement and its effects in entropic uncertainty relations
• Optimal uncertainty relations in a modified Heisenberg algebra
• Information Content of Gravitational Radiation and the Vacuum
• The minimal length and the Shannon entropic uncertainty relation
• Classical and Quantum Szilard Engine under Generalized Uncertainty Principle Effect

Cited by in corpus (6)

• Study of coherence and mixedness in meson and neutrino systems
• Uncertainty relations for quantum coherence with respect to mutually unbiased bases
• Uncertainty Relations for coarse-grained measurements: an overview
• Differential-escort transformations and the monotonicity of the LMC-Rényi complexity measure
• Entropic uncertainty relations for successive measurements in the presence of a minimal length
• Entropic uncertainty relation based on generalized uncertainty principle