All non-Gaussian states are advantageous for channel discrimination: Robustness of non-convex continuous variable quantum resources

Abstract: Quantum resource theories provide a mathematical framework for quantifying the advantage given by quantum phenomena in various tasks. The generalized robustness is one such quantifier, and enjoys an operational interpretation in the setting of channel discrimination. It is a well studied resource monotone in finite-dimensional or convex resource theories, however as of yet it has not been studied in the setting of infinite-dimensional resource theories with a non-convex set of free states. In this work, we define the generalized robustness for an arbitrary resource theory, without restricting to convex sets or to finite dimensions. We show it has two operational interpretations: firstly, it provides an upper bound on the maximal advantage in a multi-copy channel discrimination task. Secondly, in many relevant theories, it quantifies the worst-case advantage in single-copy channel discrimination when considering a decomposition of the free states into convex subsets. Finally, we apply our results to the resource theory of non-Gaussianity, thus showing that all non-Gaussian states can provide an advantage in some channel discrimination task, even those that are simply mixtures of Gaussian states. To illustrate our findings, we provide exact formulas for the robustness of non-Gaussianity of Fock states, along with an analysis of the robustness for a family of non-Gaussian states within the convex hull of Gaussian states.