Entropic uncertainty relations for successive measurements in the presence of a minimal length

TL;DR

This paper develops entropic uncertainty relations for successive quantum measurements considering a minimal length, revealing differences between measurement and preparation uncertainties and providing bounds dependent on measurement apparatus characteristics.

Contribution

It introduces generalized entropic uncertainty relations for successive measurements with a minimal length, including state-dependent and state-independent bounds for continuous spectra.

Findings

State-independent bounds match those in the preparation scenario.

Bounds depend explicitly on measurement acceptance functions.

Entropic relations with binning are also analyzed.

Abstract

We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the R\'{e}nyi and Tsallis types. Here, specific features of measurements of observables with continuous spectra should be taken into account. First, we formulated uncertainty relations in terms of Shannon entropies. Since such relations involve a state-dependent correction term, they generally differ from preparation uncertainty relations. This difference is revealed when position is measured by the first. In contrast, state-independent uncertainty relations in terms of R\'{e}nyi and Tsallis entropies are obtained with the same lower bounds as in the preparation scenario. These bounds are explicitly dependent on the acceptance function of apparatuses in momentum measurements. Entropic uncertainty relations with binning are discussed as well.