Continuity of robustness measures in quantum resource theories

Robustness measures are increasingly prominent resource quantifiers that have been introduced for quantum resource theories such as entanglement and coherence. Despite the generality of these measures, their usefulness is hindered by the fact that some of their mathematical properties remain unclear, especially when the set of resource-free states is non-convex. In this paper, we investigate continuity properties of different robustness functions. We show that their continuity depends on the shape of the set of free states. In particular, we demonstrate that in many cases, star-convexity is sufficient for Lipschitz-continuity of the robustness, and we provide specific examples of sets leading to non-continuous measures. Finally, we illustrate the applicability of our results by defining a robustness of teleportability and of quantum discord.

Automated summary

Robustness measures are quantifiers of quantum resources such as entanglement and coherence. However, their usefulness is hindered by unclear mathematical properties, especially when the set of resource-free states is non-convex. This paper investigates the continuity properties of different robustness functions and shows that their continuity depends on the shape of the set of free states. Specific examples of non-continuous measures are provided. The applicability of the results is illustrated by defining a robustness of teleportability and quantum discord.