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

Abstract

A fundamental challenge in quantum resource theory is to establish operational interpretations by quantifying the advantage that quantum resources provide in specific tasks. Conventional resource theories, however, have inherent limitations in characterizing such advantages for certain quantum operations. We overcome this by introducing a novel approach that defines the resource deficiency of a state relative to maximal resource sets. This extension broadens the scope of resource theories, delivers more complete operational interpretations, and yields broad insights for classifying mixed resource states--including those whose resource properties remain inactive in given tasks--that escape conventional descriptions. We also show that a geometric measure satisfying the deficiency-based framework's requirements for coherence and entanglement captures the operational disadvantage of arbitrary states compared to maximal resource states in subchannel discrimination. In parallel, we present a practical methodology that links deficiency measures with experimental estimation of quantum gate noise constants, illustrated for Hadamard gates. The methodology is extensible to general gates, and the results demonstrate that deficiency measures can serve as key indicators for determining quantum-error-correction thresholds and predicting algorithm performance.