Entropic uncertainty relations for successive generalized measurements

TL;DR

This paper establishes entropic uncertainty relations for successive generalized quantum measurements, highlighting how measurement unsharpness and observable incompatibility limit measurement precision in different operational scenarios.

Contribution

It introduces a framework for entropic uncertainty relations in successive measurements, considering composite and marginal measurement schemes, and links unsharpness to measurement disturbance and observable incompatibility.

Findings

Unsharpness limits measurement precision in successive schemes.

Disturbance from the first measurement affects the second measurement.

Incompatibility of observables contributes to measurement unsharpness.

Abstract

We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two {distinctive operational} scenarios. In the first scenario, by merging {two successive measurements} into one we consider successive measurement scheme as a method to perform an overall {composite} measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one's ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the L\"{u}ders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement.