On Fidelity-Preserving Entanglement Purification Protocols

Entanglement purification protocols (EPPs) are crucial to suppress inevitable noise in quantum entanglement, and therefore it is important to understand their fundamental limits. Here, we introduce the fidelity-preserving (FP) property for EPPs where the output fidelity is at least as high as the highest input fidelity. This property is desirable for near-term quantum networks and distributed quantum computing architectures with probabilistic entanglement generation and noisy quantum memories. We examine the entire family of n-to-1 bilocal Clifford EPPs (BiCEPs) for Bell diagonal states (BDS), prove the necessary and sufficient condition of FP for biCEPs, and conclude that an FP biCEP for arbitrary BDS does not exist. We also prove a second no-go theorem – even with fidelity information (FI), biCEP can only achieve trivial FP for arbitrary BDS. As a direct corollary, we show that biCEP still cannot be FP without FI, and can only be trivially FP with FI, even with an arbitrary number of pure Bell pairs as catalyst. Finally, we demonstrate that the recurrence protocol satisfies a relaxed version of FP when the input fidelity has an upper bound below unity for Werner states, while the allowed fidelity region inevitably shrinks for higher upper bound. Our results provide fundamental insights on EPPs and quantum information processing (QIP) beyond the independent and identically distributed (i.i.d.) setting, and also reveal the practical limitations of integrating EPPs into real-world QIP architectures.