No-Go Theorems for Universal Quantum State Purification via Classically Simulable Operations

Quantum state purification, a process that aims to recover a state closer to a system's principal eigenstate from multiple copies of an unknown noisy quantum state, is crucial for restoring noisy states to a more useful form in quantum information processing. Fault-tolerant quantum computation relies on stabilizer operations, which are classically simulable operations critical for error correction but inherently limited in computational power. In this Letter, we investigate the limitations of classically simulable operations for quantum state purification. We demonstrate that while certain classically simulable operations can enhance fidelity for specific noisy state ensembles, they cannot achieve universal purification. We prove that neither deterministic nor probabilistic protocols using only classically simulable operations can achieve universal two-to-one purification for qubit systems or systems of any odd dimension. We further extend this no-go result to state purification using three and four copies via semidefinite programming and numerical evidence. Our findings highlight the indispensable role of nonstabilizer resources and the inherent limitations of classically simulable operations in quantum state purification, emphasizing the necessity of harnessing the full power of quantum operations for robust quantum information processing.