Entropic Uncertainty Relations for Successive Measurements in the Presence of a Minimal Length.

Alexey E Rastegin

doi:10.3390/e20050354

Abstract

We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the Rényi 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 the position is measured by the first. In contrast, state-independent uncertainty relations in terms of Rényi 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.

Highlights

The Heisenberg uncertainty principle [1] is avowed as a fundamental scientific concept.Heisenberg examined his thought experiment rather qualitatively
Studies of scenarios with successive measurements allow us to understand whether preparation uncertainty relations are applicable to one or another question
We briefly review entropic uncertainty relations for successive projective measurements

Summary

Introduction

The Heisenberg uncertainty principle [1] is avowed as a fundamental scientific concept.Heisenberg examined his thought experiment rather qualitatively. An explicit formal derivation appeared in [2] This approach was later extended to arbitrary pairs of observables [3]. These traditional formulations are treated as preparation uncertainty relations [4], since repeated trials with the same quantum state are assumed here. This simple scenario differs from the situations typical in quantum information science. Basic developments within the entropic approach to quantum uncertainty are reviewed in [6,7,8]. Studies of scenarios with successive measurements allow us to understand whether preparation uncertainty relations are applicable to one or another question

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Conclusion