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

Projective robustness for quantum channels and measurements and their operational significance