Classifying measurement incompatibility under classical pre- and post-processing operations

Abstract Measurement incompatibility has proved to be an important resource for quantum information processing. In this work, we present an operational approach that leverages classical operations on the inputs (pre-processing) and outputs (post-processing) of measurement devices to explore different layers of incompatibility among the measurements performed by the device. We study classifications of measurement incompatibility with respect to (w.r.t.) these two types of classical operations, viz., post-processing or coarse-graining (CG) of measurement outcomes and pre-processing or convex-mixing of different measurements. We derive analytical criteria for determining when a set of projective measurements is fully incompatible w.r.t. CG or convex-mixing. Robustness against white noise for different layers of incompatibility for mutually unbiased bases (MUBs) is investigated. Furthermore, we study operational witnesses for incompatibility subject to these classical operations, using the input–output statistics of Bell-type experiments as well as experiments in the prepare-and-measure scenario.