Witnessing measurement incompatibility via communication tasks

Quantum theory offers measurement incompatibility, that is, the existence of quantum measurements that cannot be carried out simultaneously on single systems. Measurement incompatibility is essential for probing many aspects of quantum correlations and quantum information processing. However, its fundamental and generic link with nonclassical correlations observed in the simplest prepare-and-measure scenario is still untold. In the prepare-and-measure scenario, we uncover that d-dimensional classical systems assisted with shared randomness reproduce all the input-output statistics obtained from any set of d-dimensional compatible quantum measurements. Thus, any quantum advantage in one-way communication tasks with d-dimensional systems witnesses incompatibility of the measurements on the receiver's end in a semi-device-independent way. To witness incompatibility of an arbitrary number of quantum measurements acting on an arbitrary dimension, wherein different measurements have different outcomes, we introduce a class of communication tasks - a general version of random access codes. We provide generic upper bounds on the success metric of these tasks for compatible measurements. These bounds are tight whenever the dimension on which the measurements act is not larger than the number of outcomes of any of the measurements.