Comparing quantum incompatibility of device sets from an operational perspective

To effectively utilize quantum incompatibility as a resource in quantum information processing, it is crucial to evaluate how incompatible a set of devices is. In this study, we propose an ordering to compare incompatibility and reveal its various properties based on the operational intuition that larger incompatibility can be detected with fewer states. We especially focus on the typical class of incompatibility exhibited by mutually unbiased qubit observables and numerically demonstrate that the ordering yields new classifications among sets of devices. Moreover, the equivalence relation induced by this ordering is proved to uniquely characterize mutually unbiased qubit observables among all pairs of unbiased qubit observables. The operational ordering also has a direct implication for a specific protocol called distributed sampling.