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Entanglement Purification Beyond Identical Input

Allen Zang

Published Aug 1, 2026• 0 citations
Publisher

Abstract

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 $n$-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.

DOI
10.6082/13py0-b9466

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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.
While the concept of entanglement purification protocols (EPPs) is straightforward, the integration of EPPs in network architectures requires careful performance evaluations and optimizations that take into account realistic conditions and imperfections, especially probabilistic entanglement generation and quantum memory decoherence. It is important to understand what is guaranteed to be improved from successful EPP with arbitrary non-identical input, which determines whether we want to perform the EPP at all. When successful EPP can offer improvement, the time to perform the EPP should also be optimized to maximize the improvement. In this work, we study the guaranteed improvement and optimal time for the CNOT-based recurrence EPP, previously shown to be optimal in various scenarios. We firstly prove guaranteed improvement for multiple figures of merit, including fidelity and several entanglement measures when compared to practical baselines as functions of input states. However, it is noteworthy that the guaranteed improvement we prove does not imply the universality of the EPP as introduced in arXiv:2407.21760. Then we prove robust, parameter-independent optimal time for typical error models and figures of merit. We further explore memory decoherence described by continuous-time Pauli channels, and demonstrate the phenomenon of optimal time transition when the memory decoherence error pattern changes. Our work deepens the understanding of EPP performance in realistic scenarios and offers insights into optimizing quantum networks that integrate EPPs.
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 with two entangled resource pairs is widely employed in the literature on quantum repeater networks to counteract fidelity degradation introduced by noisy quantum memories and entanglement swapping across multiple hops. Standard purification protocols assume both resource pairs carry identical fidelity. In practice, entanglement generation is stochastic, the two resource pairs are heralded at different times, and so the first pair decoheres in memory while the second is being generated. Thus a fidelity asymmetry is a structural feature of any network operating under realistic memory conditions, leading to the question: when is it beneficial to perform purification? We derive a closed-form fidelity asymmetry tolerance delta(F) that governs whether a purification attempt is beneficial. We determine a universal upper bound delta_max of approximately 0.076 beyond which purification is always counterproductive. Our simulations show that with exponential memory decoherence, purification yields benefits in only approximately 14% of purification attempts on two resource pairs in a two-hop repeater chain. We define three network objectives: fidelity only, time only, and a combination of time and fidelity, to deliver end-to-end entanglement. We show that when the application fidelity requirement is achievable through swapping alone, no-purification is the superior policy, with its advantage increasing with the number of hops. When the fidelity requirement cannot be met with swapping alone and purification is necessary, to be effective, it must be conditioned on delta(F) between resource pairs. We introduce DeltaPurify, a policy that conditions purification decisions on local fidelity information, and show it reduces time-to-serve relative to both naive purification and no-purification across several fidelity thresholds and hops of a repeater chain.
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.
High quality time-bin entanglement is widely exploited to achieve the purposes of fundamental tests of physics and the implementation of quantum communication protocols both in free space and optical fiber propagation. However, the imperfect approaches of generating time-bin entangled state will degrade its quality and limit its practical application. Entanglement purification is to distill high quality entangled states from low quality entangled states. In this paper, we present the first entanglement purification protocol (EPP) for time-bin entanglement. We first explain this EPP for two-photon time-bin entangled state and then extend it to the system of multi-photon time-bin entangled state. We also design a possible realization of this EPP with practical spontaneous parametric down conversion (SPDC) source. Differ from the conventional EPPs, this EPP does not require the sophisticated controlled-not (CNOT) gate or similar operations, and it uses the feasible sum-frequency generation (SFG) to perform the purification. Moreover, the double-pair noise emitted from the SPDC source can be eliminated automatically which is the other advantage of this EPP. If we combine with the faithful entanglement swapping, this EPP may have potential to be a part of full quantum repeaters.