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Advanced Quantum Technologies

Direct Purification of Bell and Multipartite GHZ States via Weak Measurement

Sajede Harraz, Shuang Cong

Published Dec 18, 2024• 3 citations
Publisher

Abstract

Abstract Maximally entangled states are crucial for various quantum information processing, yet their quality suffers from entanglement loss due to decoherence. In this paper, a direct purification technique that utilizes weak measurement and requires only two copies of the entangled state is proposed. The method involves transforming the damped entangled state with weak measurements and then applying a direct purification process. This approach yields a maximally entangled state with a certain probability. In the context of Bell state purification, two weak measurement‐based operations designed to convert amplitude‐damped Bell states into the desired state for direct purification method are presented. The first operation is applied prior to the noisy channel, while the second is employed after the noisy channel. Through comprehensive analysis, the performances of these two schemes are evaluated and compared and benchmarked them against a pioneering weak measurement‐based scheme. The findings demonstrate that the proposed methods exhibit superior performance in terms of both fidelity and probability. Additionally, a novel technique for purifying multipartite amplitude‐damped Greenberger–Horne–Zeilinger (GHZ) states is introduced, which existing methods cannot address. By employing only two copies of the entangled states and harnessing the unique advantages of weak measurement operations, the method achieves both resource efficiency and effectiveness.

DOI
10.1002/qute.202400450

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Abstract Maximally entangled states are crucial for various quantum information processing, yet their quality suffers from entanglement loss due to decoherence. In this paper, a direct purification technique that utilizes weak measurement and requires only two copies of the entangled state is proposed. The method involves transforming the damped entangled state with weak measurements and then applying a direct purification process. This approach yields a maximally entangled state with a certain probability. In the context of Bell state purification, two weak measurement‐based operations designed to convert amplitude‐damped Bell states into the desired state for direct purification method are presented. The first operation is applied prior to the noisy channel, while the second is employed after the noisy channel. Through comprehensive analysis, the performances of these two schemes are evaluated and compared and benchmarked them against a pioneering weak measurement‐based scheme. The findings demonstrate that the proposed methods exhibit superior performance in terms of both fidelity and probability. Additionally, a novel technique for purifying multipartite amplitude‐damped Greenberger–Horne–Zeilinger (GHZ) states is introduced, which existing methods cannot address. By employing only two copies of the entangled states and harnessing the unique advantages of weak measurement operations, the method achieves both resource efficiency and effectiveness.
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.
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.
In practical application, since quantum entanglement is inevitably destroyed by thermal noise resulted from environment, protecting entanglement from decoherence in thermal bath environment becomes a critically important issue in the field of quantum information. In this work, we propose a scheme to protect the tripartite Greenberger–Horne–Zeilinger (GHZ) entanglement state subjected to the generalized amplitude damping (GAD) channel based on the quantum technique of weak measurement (WM) and quantum measurement reversal (QMR). By optimizing the strengths of WM and QMR, the decoherence caused by the channel can be effectively suppressed. The results show that after performing the optimal pre- and post-channel weak measurement, the negativity of the GHZ state can be effectively improved with a certain probability, and the phenomenon of sudden death can be delayed to a certain degree. This scheme can be applied to the decoherence suppression of GAD channel at finite temperature.
The entangled coherent states (ECSs) have been widely used to realize quantum information processing tasks. However, the ECSs may suffer from photon loss and decoherence due to the inherent noise in quantum channel, which may degrade the fidelity of ECSs. To overcome these obstacles, we present a measurement-based entanglement purification protocol (MBEPP) for ECSs to distill some high-quality ECSs from a large number of low-quality copies. We first show the principle of this MBEPP without considering the photon loss. After that, we prove that this MBEPP is feasible to correct the error resulted from the photon loss. Additionally, this MBEPP only requires to operate the Bell state measurement without performing local two-qubit gates on the noisy pairs and the purified high-quality ECSs can be preserved for other applications. This MBEPP may have application potential in the implementation of long-distance quantum communication.
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.
Being able to implement effective entanglement distribution in noisy environments is a key step towards practical quantum communication, and long-term efforts have been made on the development of it. Recently, it has been found that the null-result weak measurement (NRWM) can be used to enhance probabilistically the entanglement of a single copy of amplitude-damped entangled state. This paper investigates remote distributions of bipartite and multipartite entangled states in the amplitudedamping environment by combining NRWMs and entanglement distillation protocols (EDPs). We show that the NRWM has no positive effect on the distribution of bipartite maximally entangled states and multipartite Greenberger-Horne-Zeilinger states, although it is able to increase the amount of entanglement of each source state (noisy entangled state) of EDPs with a certain probability. However, we find that the NRWM would contribute to remote distributions of multipartite W states. We demonstrate that the NRWM can not only reduce the fidelity thresholds for distillability of decohered W states, but also raise the distillation efficiencies of W states. Our results suggest a new idea for quantifying the ability of a local filtering operation in protecting entanglement from decoherence.