Entropic uncertainty relations for successive measurements of canonically conjugate observables

TL;DR

This paper develops entropic uncertainty relations for successive measurements of conjugate variables like position and momentum, incorporating detector resolution and using generalized entropies such as Rényi and Tsallis.

Contribution

It introduces a new reformulation of measurement scenarios and derives entropic uncertainty relations for conjugate observables considering unbounded operators and finite detector resolution.

Findings

Derived Rényi and Tsallis entropic uncertainty relations for successive measurements.

Reformulated measurement scenarios to account for detector resolution effects.

Discussed uncertainty relations specifically for position and momentum measurements.

Abstract

Uncertainties in successive measurements of general canonically conjugate variables are examined. Such operators are approached within a limiting procedure of the Pegg-Barnett type. Dealing with unbounded observables, we should take into account a finiteness of detector resolution. An appropriate reformulation of two scenarios of successive measurements is proposed and motivated. Uncertainties are characterized by means of generalized entropies of both the R\'{e}nyi and Tsallis types. The R\'{e}nyi and Tsallis formulations of uncertainty relations are obtained for both the scenarios of successive measurements of canonically conjugate operators. Entropic uncertainty relations for the case of position and momentum are separately discussed.