Epsilon measures of state-based quantum resource theory

The quantification of state-based quantum ``resources'' such as entanglement, coherence, and nonstabilizer states lies at the heart of quantum science and technology, providing potential advantages over classical methods. In a realistic scenario, due to the imperfections and uncertainties in physical devices, we are unable to perfectly prepare or detect the true quantum states. Consequently, it is necessary to study the quantification of quantum resources under such circumstances. In this work, by focusing on the state-based quantum resource theory, we introduce a family of resource measures called $\ensuremath{\epsilon}$ measure that relies on a precision parameter to address this issue. This family of resource measures inherits the fundamental properties of the original resource measure, such as weak monotonicity, convexity, monogamy, and so forth. Furthermore, the $\ensuremath{\epsilon}$ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the $\ensuremath{\epsilon}$ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the $\ensuremath{\epsilon}$ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the $\ensuremath{\epsilon}$ measure of the resource rank. Finally, we outline how this work can be extended to channel-based resources among others.