Positive and non-positive measurements in energy distillation from quantum batteries

We investigate energy distillation from quantum batteries within the framework of generalized quantum measurements, including both positive operator-valued measurements (POVMs) and physically realizable non-positive operator-valued measurements (NPOVMs) performed on an auxiliary system coupled to the battery. Two classes of NPOVMs, namely type-1 and type-2, are analyzed in the presence of environmental noise acting on the auxiliary system. We derive general expressions for the distillable energy corresponding to positive and non-positive measurements and show that the distillable energy obtained via NPOVMs remains robust against environmental noise. For a specific model, we demonstrate that the energy extracted using type-1 NPOVMs exceeds or equals that obtained via POVMs under amplitude-damping, dephasing, and bit-flip noise, while type-2 NPOVMs outperform POVMs under amplitude-damping noise. These results establish a clear operational advantage of NPOVMs for energy extraction. We also analyze restricted measurement settings and compare the accessible distillable energy for constrained positive and type-1 non-positive measurements.