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Physical Review A

Measurement-based logical qubit entanglement purification

Peishun Yan, Lan Zhou, Wei Zhong, Yu‐Bo Sheng

Published Jun 10, 2022• 43 citations
Publisher

Abstract

Entanglement purification is the distilling of high-quality entanglement from low-quality entanglement and is a key element in the quantum repeater. As a new branch of entanglement purification, the measurement-based entanglement purification protocol (MBEPP) only requires one to perform the Bell state measurement to couple resource states with noisy pairs, and it tolerates more local noise than the conventional purification protocols. Existing MBEPPs usually focus on physical qubit entanglement. In this paper, we propose a measurement-based logical qubit entanglement purification protocol (MBLEPP) with quantum nondemolition detection (QND), where the qubit is encoded in the quantum parity code. The results show that this MBLEPP can also work with photon loss under the conditions that each block of the logical Bell state measurement contains at least one physical qubit, that at least one of the blocks is intact, and that the entanglement exists between all blocks. Moreover, we also consider the MBLEPP with imperfect QND. We show that below a certain QND error threshold, this MBLEPP can still work. In this way, this MBLEPP not only obtains high-fidelity entanglement but also tolerates photon loss and the error from imperfect QND. This MBLEPP combines the benefits of the MBEPP and quantum error correction code and may have potential application in long-distance quantum communication.

DOI
10.1103/physreva.105.062418

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Entanglement purification is the distilling of high-quality entanglement from low-quality entanglement and is a key element in the quantum repeater. As a new branch of entanglement purification, the measurement-based entanglement purification protocol (MBEPP) only requires one to perform the Bell state measurement to couple resource states with noisy pairs, and it tolerates more local noise than the conventional purification protocols. Existing MBEPPs usually focus on physical qubit entanglement. In this paper, we propose a measurement-based logical qubit entanglement purification protocol (MBLEPP) with quantum nondemolition detection (QND), where the qubit is encoded in the quantum parity code. The results show that this MBLEPP can also work with photon loss under the conditions that each block of the logical Bell state measurement contains at least one physical qubit, that at least one of the blocks is intact, and that the entanglement exists between all blocks. Moreover, we also consider the MBLEPP with imperfect QND. We show that below a certain QND error threshold, this MBLEPP can still work. In this way, this MBLEPP not only obtains high-fidelity entanglement but also tolerates photon loss and the error from imperfect QND. This MBLEPP combines the benefits of the MBEPP and quantum error correction code and may have potential application in long-distance quantum communication.
Measurement-based entanglement purification protocol (MBEPP) depending on the Bell state measurement (BSM) is first proposed in [Phys. Rev. Lett.110260503 (2013)10.1103/PhysRevLett.110.260503], which is shown to tolerate much more local noise than the conventional gate-based EPPs. However, the existing MBEPPs for discrete variable (MBEPP-DV) and MBEPP for continuous variable (MBEPP-CV) fail to take full advantage of their strengths. In this paper, we study the MBEPP for logical hybrid entanglement (MBEPP-LHE) in linear optics. We first demonstrate the principle of the MBEPP-LHE in the case that all logical BSMs (LBSMs) are perfect. Additionally, we consider that one of the components of the LBSM, i.e., the BSM for DVs or the BSM for CVs, fails on specific modes. The results indicate that the MBEPP-LHE can still work provided that at least one of the components of the LBSM is successful, which is the advantage of this purification compared to the previous MBEPPs. Moreover, the MBEPP-LHE can correct bit-flip and phase-flip errors not only on logical qubits but also on physical qubits. Furthermore, we show that the success probability of MBEPP-DV is the lower bound of that of MBEPP-LHE, while the success probability of the MBEPP-LHE has the same upper bound as MBEPP-CV. Finally, we also carry out the analysis of purifying the logical hybrid entanglement polluted by the dissipation error on coherent-state mode, and this error can be converted to bit-flip error on single-photon mode.
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.
Entanglement purification is to distill high-quality entanglement from low-quality entanglement. Entanglement purification determines the efficiency and communication rates and plays a key role in quantum repeaters and long-distance quantum communications. Existing entanglement purification protocols and proof of principle experiments usually choose two identical input pairs. They are limited to focus on the successful purification case to obtain high-quality entangled states and discard the remained items if the purification fails. Recent entanglement purification theory [Opt. Express 28, 2291 (2020)OPEXFF1094-408710.1364/OE.383499] for arbitrary nonidentical input mixed states indicated that the discarded items may have residual entanglement and can be reused to increase the yield of entanglement purification, while the successful purification may not obtain a high-quality entangled state. In this Letter, using hyperentanglement, we design and experimentally demonstrate an entanglement purification protocol with nonidentical input mixed states. For the first time, we experimentally confirm the existence of residual entanglement in different error models. Furthermore, we illustrate that successful purification may decrease the fidelity of purified entangled states. We also demonstrate the approach of optimal entanglement purification. Our proof of principle results have potential applications in future high-efficiency and long-distance quantum communication.
It has been found that logic-qubit entanglement has great potential for applications in quantum communication and quantum networks in recent years. However, along with the effects of noise and decoherence, the fidelity of the communication transmission can be greatly reduced. In this paper, we investigate the entanglement purification of logic bit-flip error and phase-flip error in polarization logic-qubit entanglement based on the parity-check measurement (PCM) gate, which is constructed by the cross-Kerr nonlinearity and used to distinguish the parity information of two-photon polarization states. The probability of entanglement purification is higher than the scheme using the linear optical method. Moreover, the quality of logic-qubit entangled states can be improved by a cyclic purification process. This entanglement purification protocol will be useful in the future when faced with long-distance communication with logic-qubit entanglement states.
Logic-qubit entanglement is a promising resource in quantum information processing, especially in future large-scale quantum networks. In the paper, we put forward an efficient entanglement purification protocol (EPP) for nonlocal mixed logic entangled states with the bit-flip error in the logic qubits of the logic Bell state, resorting to the photon-atom interaction in low-quality (Q) cavity and atomic state measurement. Different from existing EPPs, this protocol can also purify the logic phase-flip error, and the bit-flip error and the phase-flip error in physic qubit. During the protocol, we only require to measure the atom states, and it is useful for improving the entanglement of photon systems in future large-scale quantum networks.
Entanglement swapping (ES) plays a key role in long-distance quantum communication. However, photon loss is inevitable due to noise in quantum channels, which makes existing ES focusing on the physical qubit fail to work, thereby degrading the efficiency of quantum communication. To address this issue, we resort to quantum parity codes (QPCs) and quantum nondemolition detection (QND) to realize a loss-tolerant ES (LTES), where each qubit is encoded into a logical qubit consisting of three levels: logical level, block level, and physical level. The research findings illustrate that the LTES can be successfully performed as long as the logical Bell state completely contains at least one block and each block contains at least one pair of physical Bell states. In addition, we evaluate the influence of the imperfect QND on the LTES, indicating that if the error probability of QND is lower than 0.47, the entanglement achieves a fidelity of 0.99 after performing the LTES. As a consequence, this LTES has the potential to pave the way for long-distance quantum communication.