Data processing makes POVMs coarser and observational entropies larger

TL;DR

This paper introduces a criterion to compare POVM measurements based on their information extraction capabilities, showing that coarser measurements always yield larger observational entropies and thus less information.

Contribution

It generalizes the concept of coarser POVMs and links measurement coarseness to increased observational entropy, providing a unified framework for understanding measurement informativeness.

Findings

Coarser POVMs extract less information from systems.

Observational entropy increases with measurement coarseness.

The criterion unifies and extends previous results on measurement entropy.

Abstract

We find a criterion to compare POVM measurements and decide which ones can extract more information from physical systems, with coarser POVMs always extracting less information. This criteria generalizes the previous definition of coarser POVM, and is motivated by the idea that information cannot be gained by processing the measurement outcomes. The information that a measurement cannot extract is quantified by observational entropy or coarse-grained entropy. Adequately, coarser measurements have larger observational entropies. Moreover, the characterization and properties of coarser measurements that we provide allow to straightforwardly derive several previously known results about observational entropy.