A Deficiency-Based Framework for the Operational Interpretation of Quantum Resources with Applications

A fundamental challenge in quantum resource theory lies in establishing operational interpretations by quantifying the distinct advantages that quantum resources provide over classical resources in specific physical tasks. However, conventional quantum resource theories have inherent limitations in characterizing operational advantages for certain quantum tasks. To overcome these limitations, we propose a novel framework that defines the resource deficiency of a given state relative to the set of maximal resource states in physical tasks. This extension not only broadens the scope of quantum resource theories and provides more comprehensive operational interpretations, but also delivers crucial insights for classifying and interpreting mixed resource states -- specifically those with inactive resource properties in certain tasks -- that have remained uncharacterized in conventional quantum resource theories. Moreover, we further demonstrate that the proposed geometric measure satisfies the framework's requirements for both quantum coherence and entanglement, while also demonstrating its ability to characterize the operational disadvantage of arbitrary states compared to maximal resource states in subchannel discrimination tasks under specific conditions.