Projective robustness for quantum channels and measurements and their operational significance

Recently, the projective robustness of quantum states has been introduced in [arXiv:2109.04481(2021)]. It shows that the projective robustness is a useful resource monotone and can comprehensively characterize capabilities and limitations of probabilistic protocols manipulating quantum resources deterministically. In this paper, we will extend the projective robustness to any convex resource theories of quantum channels and measurements. First, We introduce the projective robustness of quantum channels and prove that it satisfies some good properties, especially sub- or supermultiplicativity under any free quantum process. Moreover, we use the projective robustness of channels to give lower bounds on the errors and overheads in any channel resource distillation. Meanwhile, we show that the projective robustness of channels quantifies the maximal advantage that a given channel outperforms all free channels in simultaneous discrimination and exclusion of a fixed state ensemble. Second, we define the projective robustness of quantum measurements and prove that it exactly quantifies the maximal advantage that a given measurement provides over all free measurements in simultaneous discrimination and exclusion of two fixed state ensembles. Finally, within a specific channel resource setting based on measurement incompatibility, we show that the projective robustness of quantum channels coincides with the projective robustness of measurement incompatibility.