Universal quantum state purification with energy-preserving operations

Abstract: Quantum state purification, which operates not by identifying and correcting specific errors but by repeatedly projecting multiple noisy copies onto special subspaces, provides a syndrome-free alternative to quantum error correction. Existing purification protocols, however, generally assume unconstrained operations and thus overlook the energetic restrictions inherent in realistic quantum devices. Here, we establish a general framework for universal state purification under energy-conservation constraints for depolarizing noise. We derive a necessary and sufficient condition for the nonexistence of universal energy-preserving purification and, whenever such purification is feasible, analytically determine the optimal performance and the corresponding protocols. We further show how the optimal protocols can be systematically implemented using only energy-preserving operations. Numerical results confirm the effectiveness of the proposed scheme. Our framework recovers the standard purification setting as a special case and naturally extends to scenarios assisted by external energy resources. These results identify fundamental physical limits on state distillation and provide an energy-efficient route to quantum error mitigation.