Shannon entropy as a measure of uncertainty in positions and momenta

TL;DR

This paper explores Shannon entropy as a measure of uncertainty in quantum position and momentum measurements, deriving a generalized uncertainty relation for finite detector counts that extends previous infinite-detector models.

Contribution

It introduces a new uncertainty relation for Shannon entropies with finite detectors, broadening the applicability of entropic uncertainty principles.

Findings

Derived a generalized Shannon entropy uncertainty relation for finite detectors

Extended previous models based on infinite detector assumptions

Provided insights into measurement uncertainty in quantum systems

Abstract

This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguacu (Brazil) from 2 to 6 May 2011. In the first part I briefy report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of fnite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infnite number of detectors (bins).