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We find a criterion to compare POVM measurements and decide which ones canextract more information from physical systems, with coarser POVMs alwaysextracting less information. This criteria generalizes the previous definitionof coarser POVM, and is motivated by the idea that information cannot be gainedby processing the measurement outcomes. The information that a measurementcannot extract is quantified by observational entropy or coarse-grainedentropy. Adequately, coarser measurements have larger observational entropies.Moreover, the characterization and properties of coarser measurements that weprovide allow to straightforwardly derive several previously known resultsabout observational entropy.

Data processing makes POVMs coarser and observational entropies larger