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Journal of the Optical Society of America B

Single-copy entanglement purification for Greenberger–Horne–Zeilinger states

Peishun Yan, Lan Zhou, Yu‐Bo Sheng

Published Jun 20, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.1364/josab.491914

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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.
Abstract Multipartite entanglement has been widely employed in quantum information processing. However, the unavoidable noise in quantum channels causes the maximally multipartite entangled state to degrade to the mixed state. In this paper, we propose a highly efficient multipartite entanglement purification protocol (MEPP) for the polarization Greenberger–Horne–Zeilinger state. Unlike conventional MEPPs, where two pairs of polluted entanglement are usually required, our protocol only requires one noisy copy of polarization-time-bin hyperentanglement and exploits the time-bin entanglement to purify the polarization entanglement. This feature increases the efficiency of our MEPP. Our MEPP may have potential applications in long-distance quantum communication.
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.
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 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.
Entanglement purification is a powerful method to distill the high-quality entanglement from low-quality entanglement. In the paper, we propose an efficient two-step entanglement purification protocol (EPP) for the polarization entanglement by using only one copy of two-photon hyperentangled state in polarization, spatial-mode, and time-bin DOFs. We suppose that the entanglement in all DOFs suffer from channel noise. In two purification steps, the parties can reduce the bit-flip error and phase-flip error in polarization DOF by consuming the imperfect entanglement in the spatial-mode and time-bin DOFs, respectively. This EPP effectively reduces the consumption of entanglement pairs and the experimental difficulty. Moreover, if consider the practical photon transmission and detector efficiencies, our EPP has much higher purification efficiency than previous recurrence EPPs. Meanwhile, when one or two purification steps fail, the distilled mixed state may have residual entanglement. Taking use of the residual entanglement, the parties may still distill higher-quality polarization entanglement. Even if not, they can still reuse the residual entanglement in the next purification round. The existence of residual entanglement benefits for increasing the yield of the EPP. All the above advantages make our EPP have potential application in future quantum information processing.
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.