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Advanced Quantum Technologies

Entanglement Purification For Arbitrary Multipartite High‐Dimensional Greenberger‐Horne‐Zeilinger State

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

Published Jul 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1002/qute.70364

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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 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.
Multipartite entanglement plays an important role in controlled quantum teleportation, quantum secret sharing, quantum metrology and some other important quantum information branches. However, the maximally multipartite entangled state will degrade into the mixed state because of the noise. We present an efficient multipartite entanglement purification protocol (EPP) which can distill the high quality entangled states from low quality entangled states for N-photon systems in a Greenberger-Horne-Zeilinger (GHZ) state in only linear optics. After performing the protocol, the spatial-mode entanglement is used to purify the polarization entanglement and one pair of high quality polarization entangled state will be obtained. This EPP has several advantages. Firstly, with the same purification success probability, this EPP only requires one pair of multipartite GHZ state, while existing EPPs usually require two pairs of multipartite GHZ state. Secondly, if consider the practical transmission and detector efficiency, this EPP may be extremely useful for the ratio of purification efficiency is increased rapidly with both the number of photons and the transmission distance. Thirdly, this protocol requires linear optics and does not add additional measurement operations, so that it is feasible for experiment. All these advantages will make this protocol have potential application for future quantum information processing.
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.
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.
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.