Quantum Defect Analysis and Target-Orbit-Directed Correction via the Purpose-Oriented Framework: Applications to Entanglement Repairability and Teleportation Activation

Abstract

Quantum resource theories quantify advantages but suffer from ambiguous free sets, limiting target-oriented missions. We improve upon this by introducing a purpose-oriented framework, shifting reference from ``free'' to ``maximally efficient.'' Its deficiency measure quantifies loss, classifies mixed states, estimates noise, and signals quantum-error-correction thresholds and Hadamard gate performance. We define repair capacity to show the measure metrics mission success under free-set ambiguity. This capacity equals a one-sided spectral separation of pulled-back effects, exposing the mixed-unitary boundary. We derive affine Bloch translation for qubits and odd-even separation for Werner-Holevo channels. For one-sided qubit generalized amplitude damping, optimizing below the classical teleportation limit yields finite-temperature onset $z_{\beta }=1/\sqrt{2}$ and optimal interaction time. Thus, establishing the purpose-oriented framework is not merely a mathematical alternative but systematically integrates non-random-unitary dynamics, algorithmic calibration, and dissipative task activation, advancing practical quantum resource utilization.

32 pages,5 figures, 1 table