Continuity of robustness measures in quantum resource theories

TL;DR

This paper studies the mathematical continuity of robustness measures in quantum resource theories, revealing how set geometry affects their properties and introducing new robustness measures for teleportability and discord.

Contribution

It establishes conditions for the continuity of robustness measures based on the shape of free state sets and introduces new robustness measures for specific quantum tasks.

Findings

Star-convexity ensures Lipschitz-continuity of robustness

Examples of non-continuous measures due to set shape

New robustness measures for teleportability and quantum discord

Abstract

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.