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arXiv:2607.21907

Entanglement purification for arbitrary multipartite high-dimensional Greenberger-Horne-Zeilinger state

Yu-Qing Kong, Jia-Ye Liu, Hui-Hui Kong, Hai-Rui Wei

quant-phSubmitted Jul 24, 2026• Announced Jul 27, 2026• 0 citations
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Abstract

High-dimensional qudit (i.e., $d$-level or $d$-state) systems outperform two-dimensional qubit (i.e., 2-level or 2-state) systems in some quantum information processing tasks. We exploit entanglement purification protocols (EPPs) for extracting a subset of high quality arbitrary $d$-dimensional $n$-partite Greenberger-Horne-Zeilinger (GHZ) states from a large set of less entangled GHZ states. In our protocols, qudit-flip and phase-flip errors can be corrected, and the fidelity of the output state can be asymptotically improved to unity by iterating the EPP process. Moreover, the schemes are immune to the number of polluted photons, the fidelity thresholds of the proposed EPPs are developed, and the spatial-based single-qudit operations can be well manipulated with a range of balanced beam splitters, and phase shifters. These features make the proposed schemes offer an alternative method for high-dimensional multipartite entanglement purification.

Journal reference
Advanced Quantum Technologies 9(7): e70364 (2026)
Comments
16 pages, 5 figures, 4 Tables

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High-dimensional qudit (i.e., ddd-level or ddd-state) systems outperform two-dimensional qubit (i.e., 2-level or 2-state) systems in some quantum information processing tasks. We exploit entanglement purification protocols (EPPs) for extracting a subset of high quality arbitrary ddd-dimensional nnn-partite Greenberger-Horne-Zeilinger (GHZ) states from a large set of less entangled GHZ states. In our protocols, qudit-flip and phase-flip errors can be corrected, and the fidelity of the output state can be asymptotically improved to unity by iterating the EPP process. Moreover, the schemes are immune to the number of polluted photons, the fidelity thresholds of the proposed EPPs are developed, and the spatial-based single-qudit operations can be well manipulated with a range of balanced beam splitters, and phase shifters. These features make the proposed schemes offer an alternative method for high-dimensional multipartite entanglement purification.
ABSTRACT High‐dimensional qudit (i.e., ‐level or ‐state) systems outperform 2D qubit (i.e., two‐level or two‐state) systems in some quantum information processing tasks. We exploit entanglement purification protocols (EPPs) for extracting a subset of high‐quality arbitrary ‐dimensional ‐partite Greenberger‐Horne‐Zeilinger (GHZ) states from a large set of less entangled GHZ states. In our protocols, qudit‐flip and phase‐flip errors can be corrected, and the fidelity of the output state can be asymptotically improved to unity by iterating the EPP process. Moreover, the schemes are immune to the number of polluted photons, the fidelity thresholds of the proposed EPPs are developed, and the spatial‐based single‐qudit operations can be well manipulated with a range of balanced beam splitters, and phase shifters. These features make the proposed schemes offer an alternative method for high‐dimensional multipartite entanglement purification.
High-dimensional quantum entanglement offers significant advantages over its two-dimensional counterparts in various quantum information processing tasks. We propose a scalable method for implementing an arbitrary high-dimensional entanglement purification protocol (HDEPP) using path superposition. By selecting appropriate single-qudit gates, qudit-flip errors and phase-flip errors can be purified. Counterintuitively, the HDEPP becomes more robust for higher-dimensional systems and exhibits enhanced purification fidelity as the dimension increases. Our HDEPP not only requires fewer operational resources but also in principle can achieve unity fidelity through multiple rounds of iteration. Furthermore, we develop feasible optical schemes to purify the frequency degree of freedom of photons suffering from qudit-flip and phase-flip errors using hyperentanglement. This work paves the way toward high-dimensional quantum communication and quantum networks.
A multiparticle entangled state is an indispensable resource in quantum information processing. However, the inherent noise in quantum channels may degrade the maximally entangled state to a mixed entangled state. An entanglement purification protocol (EPP) is a powerful tool to distill high-quality entanglement from low-quality entanglement. Conventional multiparticle EPPs (MEPPs) mainly focus on two noisy copies. In this paper, we investigate a single-copy MEPP (SCMEPP) in linear optics that requires only a pair of hyperentanglements. Specifically, we first purify the bit-flip errors for three-particle polarization Greenberger–Horne–Zeilinger (GHZ) states at the cost of time-bin entanglement, and then an analysis of correcting the phase-flip errors is carried out. Additionally, we extend this SCMEPP into purification for arbitrary multiparticle GHZ states. Moreover, we discuss the discarding components that have been regarded as failure in conventional two-copy MEPPs, and they still have entanglement if the fidelities of the initial states satisfy specific conditions. This can be reused to distill high-quality entanglement, therefore increasing the efficiency of the SCMEPP, which may have potential application to realize high-efficiency long-distance quantum communication.
High-dimensional multipartite entanglement plays a crucial role in quantum information science. However, existing schemes for generating such entanglement become complex and costly as the dimension of quantum units increases. In this Letter, we overcome the limitation by proposing a significantly enhanced linear optical heralded scheme that generates the d-level N-partite Greenberger-Horne-Zeilinger (GHZ) state with single-photon sources and linear operations. Our scheme requires dN photons, which is the minimal required photon number, with substantially improved success probability from previous schemes. It employs linear optical logic gates compatible with any qudit encoding system and can generate generalized GHZ states with installments of beam splitters. With efficient generations of high-dimensional resource states, our work opens avenues for further exploration in high-dimensional quantum information processing.
The generation of a high-dimensional quantum entangled state provides a crucial resource for achieving more complex and diverse quantum information processing tasks. In this paper, we present two deterministic protocols designed for the generation of a four-dimensional two-qudit Bell state and three-qudit Greenberger-Horne-Zeilinger (GHZ) state assisted by cross-Kerr nonlinearity, where the four-dimensional qudit is encoded in the hybrid polarization-spatial state of the single photon to conserve quantum resources and alleviate environmental noise. The proposed protocols for generating a four-dimensional two-qudit Bell state and three-qudit GHZ state are circuit concise, high success, robust fidelity, noise resistant, and resource saving. Moreover, the four-dimensional NNN-qudit (N>3N>3N>3) GHZ states can be generated with the help of the $N\ensuremath{-}1$ generating devices of the four-dimensional Bell state. Consequently, its efficiency and fidelity decay exponentially by a factor of $N\ensuremath{-}1$. Further, the arbitrary high-dimensional qudit-encoded entangled states can be achieved with linear-optics four-dimensional single-qudit XXX gates and ZZZ gates. These protocols offer a reference for further advancements in high-dimensional state generation technology and providing insights for the practical implementations of metrological enhancement, quantum simulation, high-dimensional quantum computation, and quantum error correction.
We present an efficient and general multipartite entanglement purification protocol (MEPP) for N-photon systems in Greenberger-Horne-Zeilinger (GHZ) states with non-identical input states. As a branch of entanglement purification, besides the cases of successful purification, the recurrence MEPP actually has the reusable discarded items which are usually regarded as a failure. Our protocol contains two parts for bit-flip error correction. The first one is the conventional MEPP, corresponding successful cases. The second one includes two efficient approaches, recycling purification with entanglement link and direct residual entanglement purification, that can utilize discarded items. We also make a comparison between two approaches. Which method to use depends on initial input states, and in most cases the approach of direct residual purification is optimal for it not only may obtain a higher fidelity entangled state but also it does not require additional sophisticated links. In addition, for phase-flip errors, the discarded items still have available residual entanglement in the case of different input states. With these approaches, this MEPP has a higher efficiency than all previous MEPPs and it may have potential applications in the future long-distance quantum communications and networks.