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 protocols critical for error correction but inherently limited in computational power. In this work, 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 purification of two-copy noisy states for qubit systems and all odd dimensions. We further extend this no-go result of state purification using three and four copies via numerical computations of semidefinite programs. Our findings highlight the indispensable role of non-stabilizer 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 more robust quantum information processing.