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arXiv:2407.21760 · Physical Review Letters

No-Go Theorems for Universal Entanglement Purification

Allen Zang, X. Chen, Eric Chitambar, Martin Suchara, Tian Zhong

Published May 13, 2025• 14 citations
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Abstract

An entanglement purification protocol (EPP) aims to transform multiple noisy entangled states into a single entangled state with higher fidelity. In this work we consider input-independent EPPs that always yield an output fidelity no worse than each of the original noisy states, a property we call universality. We prove there is no n-to-1 EPP implementable by local operations and classical communication that is universal for all two-qubit entangled states, whereas such an EPP is possible using more general positive partial transpose-preserving (PPT) operations. We also show that universality is impossible by any bilocal Clifford EPP even when restricted to states with fidelities above an arbitrarily high threshold.

DOI
10.1103/physrevlett.134.190803

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An entanglement purification protocol (EPP) aims to transform multiple noisy entangled states into a single entangled state with higher fidelity. In this work we consider input-independent EPPs that always yield an output fidelity no worse than each of the original noisy states, a property we call universality. We prove there is no n-to-1 EPP implementable by local operations and classical communication that is universal for all two-qubit entangled states, whereas such an EPP is possible using more general positive partial transpose-preserving (PPT) operations. We also show that universality is impossible by any bilocal Clifford EPP even when restricted to states with fidelities above an arbitrarily high threshold.
Entanglement purification protocols (EPPs) are essential for quantum communication networks, which are the backbone of fault-tolerant, scalable distributed quantum information processing. While simplifying theoretical studies, the common assumption of identical input is almost never justified in reality. This dissertation investigates entanglement purification with non-identical input from three perspectives: what is fundamentally restricted, what is guaranteed, and how to enhance EPP performance. We first introduce the concept of input-independent universal EPPs and prove no-go theorems for their existence. Despite the no-go theorems, we prove guaranteed improvements that EPPs can achieve, with the 2-to-1 CNOT-based EPP as a canonical example. We also prove the optimal time to apply this EPP in realistic quantum repeater scenarios for typical error models and figures of merit. We then prove that shared randomness, a classical resource, can enhance the performance of an arbitrary nnn-to-1 bilocal Clifford entanglement purification protocol using the strategy of accumulating and shuffling in a multi-source scenario. Altogether, this dissertation offers insights into both the theoretical understanding of EPPs with non-identical input and the practical integration of EPPs into realistic quantum networks.
Entanglement purification protocols (EPPs) are essential for improving entanglement fidelity to support fault-tolerant distributed quantum information processing. Practical entanglement sources are often heterogeneous and source labels may be unavailable at the EPP layer. We show that classical shared randomness, together with buffer memories, can enhance entanglement purification when source labels are unavailable, without state characterization or EPP circuit optimization. The strategy is to accumulate multiple entanglement distribution rounds and then use shared randomness to shuffle all the stored entangled states before packaging them as inputs to the EPP. For any nnn Werner sources and any fixed nnn-to-1 bilocal Clifford EPP, we prove that accumulating and shuffling improves the expected success probability and the success-weighted output Bell fidelity over the baseline without accumulating and shuffling, for every nnn, for every finite number of accumulation rounds and in the asymptotic limit, and the improvement increases monotonically with the number of accumulation rounds.
One of the fundamental concepts of quantum information theory is that of entanglement purification; that is, the transformation of a partially entangled state into a smaller-dimensional, more completely entangled state. Of particular interest are protocols for entanglement purification (EPPs) that alternate purely local operations with one- or two-way classical communication. In the present work, we consider a more general, but simpler, class of transformations, called separable superoperators. Since every EPP is a separable superoperator, bounds on separable superoperators apply as well to EPPs; we use this fact to give a new upper bound on the rate of EPPs on Bell-diagonal states, and thus on the capacity of Bell-diagonal channels.
We investigate measurement-based entanglement purification protocols (EPP) in the presence of local noise and imperfections. We derive a universal, protocol-independent threshold for the required quality of the local resource states, where we show that local noise per particle of up to 24% is tolerable. This corresponds to an increase of the noise threshold by almost an order of magnitude, based on the joint measurement-based implementation of sequential rounds of few-particle EPP. We generalize our results to multipartite EPP, where we encounter similarly high error thresholds.
Entanglement purification is an indispensable ingredient in extended quantum communication networks and usually determines the efficiency and communication rate of quantum communication protocols. Different from all existing entanglement purification protocols (EPPs) where two or more copies of low quality mixed entangled states are selected from the same ensemble, here we describe a general and optimal EPP for arbitrary initial mixed states from different ensembles. We show that the successful operation of EPP may not obtain a higher fidelity mixed state, while the discarded source pair, which is usually regarded as a failure in existing EPPs, may have residual entanglement and can be reused to increase the yield of entanglement purification. We give the criterions of both the successful purification to obtain a higher fidelity mixed state and the existence of residual entanglement. Moreover, we reveal that entanglement purification procedure causes some entanglement loss. Finally, we provide an optimal approach to reduce the entanglement loss. This approach can also be used to increase the yield of entanglement purification. Our EPP may have potential application in long-distance quantum communications.
In this article, we present a novel idea for entanglement purification with joint measurements, the joint entanglement purification protocol(joint-EPP). In some quantum communication tasks using entangled pairs, one party holds the whole entangled pairs at the final stage, he or she is able to perform joint measurements on the pairs. In this situation the proposed joint-EPP can improve the entanglement purification by allowing a higher tolerated error rate, faster convergence which uses less steps to achieve a high fidelity, and higher production rate which produces more number of good entangled pairs from a given number of bad entangled pairs. We have established the general correspondence between linear classical codes and the joint-EPP. It has been shown that the joint-EPP can correct errors as long as the error threshod is no larger than 0.5. The joint-EPP works even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples, definitions and applications of the joint-EPP.