Quantum-Affinity-Based Measure of Resource Deficiency and Its Applications

Since quantum resource theory provides a unified framework to characterize the valuable properties of quantum states, many physically meaningful free sets in this theory often lack convexity, which limits the applicability of conventional convex resource quantifiers. It is therefore essential to develop new resource measures suitable for nonconvex free sets. We propose and systematically study a quantum resource deficiency measure DA based on quantum affinity. We verify that DA satisfies the standard axiomatic constraints and is universally valid for both coherence and entanglement. By virtue of the convex roof construction, strong monotonicity is strictly proven for coherence and entanglement. For low-dimensional systems, analytical expressions are given for coherence deficiency. For entanglement deficiency, we further provide an alternative proof which is independent of the convex roof extension. Effective upper bounds are presented for high-dimensional systems by using mixed permutation channels and maximally coherent mixed states (MCMSs). It is further proven that the deficiency of coherence of any quantum state can be estimated by the corresponding MCMS. Moreover, we reveal a general trade-off between the resource deficiency and the resource abundance (coherence or entanglement). As applications, we show that DA quantifies the operational disadvantage in subchannel discrimination and establish an uncertainty relation between the logical error probability and the deficiency of quantum states in quantum error correction, offering quantitative constraints for the preparation of logical states and the tolerance to faults.