Measurement-based entanglement purification for entangled coherent states

Abstract

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 highquality 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.

Graphical abstract

Keywords

measurement-based entanglement purification / entangled coherent state / photon loss / decoherence

References

[1] C. H. Bennett and S. J. Wiesner, Communication via oneand two-particle operators on Einstein–Podolsky–Rosen states, Phys. Rev. Lett. 69(20), 2881 (1992)

[2] T. Das, R. Prabhu, A. Sen(De), and U. Sen, Distributed quantum dense coding with two receivers in noisy environments, Phys. Rev. A 92(5), 052330 (2015)

[3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels, Phys. Rev. Lett. 70(13), 1895 (1993)

[4] B. G. Taketani, F. de Melo, and R. L. de Matos Filho, Optimal teleportation with a noisy source, Phys. Rev. A 85(2), 020301(R) (2012)

[5] X. M. Hu, C. Zhang, C. J. Zhang, B. H. Liu, Y. F. Huang, Y. J. Han, C. F. Li, and G. C. Guo, Experimental certification for nonclassical teleportation, Quant. Engineering 1(2), e3 (2019)

[6] Z. H. Yan, J. L. Qin, Z. Z. Qin, X. L. Su, X. J. Jia, C. D. Xie, and K. C. Peng, Generation of non-classical states of light and their application in deterministic quantum teleportation, Fundamental Res. 1(1), 43 (2021)

[7] A. K. Ekert, Quantum cryptography based on Bells theorem, Phys. Rev. Lett. 67(6), 661 (1991)

[8] H. K. Lo, M. Curty, and B. Qi, Measurement-device independent quantum key distribution, Phys. Rev. Lett.108(13), 130503 (2012)

[9] F. H. Xu, X. F. Ma, Q. Zhang, H. K. Lo, and J. W. Pan, Secure quantum key distribution with realistic devices, Rev. Mod. Phys.92(2), 025002 (2020)

[10] Z. X. Cui, W. Zhong, L. Zhou, and Y. B. Sheng, Measurement-device-independent quantum key distribution with hyper-encoding, Sci. China Phys. Mech. Astron.62(11), 110311 (2019)

[11] Y. F. Yan, L. Zhou, W. Zhong, and Y. B. Sheng, Measurement-device-independent quantum key distribution of multiple degrees of freedom of a single photon, Front. Phys.16(1), 11501 (2021)

[12] G. L. Long and X. S. Liu, Theoretically efficient high capacity quantum-key-distribution scheme, Phys. Rev. A 65(3), 032302 (2002)

[13] F. G. Deng, G. L. Long, and X. S. Liu, Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block, Phys. Rev. A 68(4), 042317 (2003)

[14] F. G. Deng and G. L. Long, Secure direct communication with a quantum one-time pad, Phys. Rev. A 69(5), 052319 (2004)

[15] W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, Quantum secure direct communication with quantum memory, Phys. Rev. Lett.118(22), 220501 (2017)

[16] F. Z. Wu, G. J. Yang, H. B. Wang, J. Xiong, F. Alzahran, A. Hobiny, and F. G. Deng, High-capacity quantum secure direct communication with two-photon six-qubit hyperentangled states, Sci. China Phys. Mech. Astron.60(12), 120313 (2017)

[17] S. S. Chen, L. Zhou, W. Zhong, and Y. B. Sheng, Three step three-party quantum secure direct communication, Sci. China Phys. Mech. Astron.61(9), 90312 (2018)

[18] L. Zhou, Y. B. Sheng, and G. L. Long, Device independent quantum secure direct communication against collective attacks, Sci. Bull.65(1), 12 (2020)

[19] Z. R. Zhou, Y. B. Sheng, P. H. Niu, L. G. Yin, G. L. Long, and L. Hanzo, Measurement-device-independent quantum secure direct communication, Sci. China Phys. Mech. Astron.63(3), 230362 (2020)

[20] T. Li, Z. K. Gao, and Z. H. Li, Measurement-device independent quantum secure direct communication: Direct quantum communication with imperfect measurement device and untrusted operator, EPL 131(6), 60001 (2020)

[21] T. Li and G. L. Long, Quantum secure direct communication based on single-photon Bell-state measurement, New J. Phys.22(6), 063017 (2020)

[22] D. Pan, Z. S. Lin, J. W. Wu, H. R. Zhang, Z. Sun, D. Ruan, L. G. Yin, and G. L. Long, Experimental free-space quantum secure direct communication and its security analysis, Photon. Res.8(9), 1522 (2020)

[23] C. Wang, Quantum secure direct communication: Intersection of communication and cryptography, Fundamental Res.1(1), 91 (2021)

[24] Z. D. Ye, D. Pan, Z. Sun, C. G. Du, L. G. Yin, and G. L. Long, Generic security analysis framework for quantum secure direct communication, Front. Phys.16(2), 21503 (2021)

[25] J. Y. Quan, Q. Li, C. D. Liu, J. J. Shi, and Y. Peng, A simplified verifiable blind quantum computing protocol with quantum input verification, Quant. Engineering 3(1), e58 (2021)

[26] D. X. Li, C. Yang, and X. Q. Shao, Dissipative engineering of a tripartite Greenberger–Horne–Zeilinger state for neutral atoms, Quant. Engineering 3(2), e66 (2021)

[27] H. Jeong and M. S. Kim, Efficient quantum computation using coherent states, Phys. Rev. A 65(4), 042305 (2002)

[28] M. Paternostro, M. S. Kim, and P. L. Knight, Vibrational coherent quantum computation, Phys. Rev. A 71(2), 022311 (2005)

[29] A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Fault tolerant linear optical quantum computing with small amplitude coherent states, Phys. Rev. Lett.100(3), 030503 (2008)

[30] L. M. Zhang, T. Gao, and F. L. Yan, Transformations of multilevel coherent states under coherence-preserving operations, Sci. China Phys. Mech. Astron.64(6), 260312 (2021)

[31] A. Mecozzi and P. Tombesi, Distinguishable quantum states generated via nonlinear birefringence, Phys. Rev. Lett.58(11), 1055 (1987)

[32] B. C. Sanders, Entangled coherent states, Phys. Rev. A 45(9), 6811 (1992)

[33] H. Jeong, M. S. Kim, and J. Lee, Quantum-information processing for a coherent superposition state via a mixed entangled coherent channel, Phys. Rev. A 64(5), 052308 (2001)

[34] X. G. Wang, Quantum teleportation of entangled coherent states, Phys. Rev. A 64(2), 022302 (2001)

[35] S. J. van Enk and O. Hirota, Entangled coherent states: Teleportation and decoherence, Phys. Rev. A 64(2), 022313 (2001)

[36] J. Joo and E. Ginossar, Efficient scheme for hybrid teleportation via entangled coherent states in circuit quantum electrodynamics, Sci. Rep.6(1), 26338 (2016)

[37] K. Park and H. Jeong, Entangled coherent states versus entangled photon pairs for practical quantum information processing, Phys. Rev. A 82(6), 062325 (2010)

[38] D. S. Simon, G. Jaeger, and A. V. Sergienko, Entangled coherent-state quantum key distribution with entanglement witnessing, Phys. Rev. A 89(1), 012315 (2014)

[39] S. L. Zhang, Improving long-distance distribution of entangled coherent state with the method of twin-field quantum key distribution, Opt. Express 27(25), 37087 (2019)

[40] X. F. Ma, P. Zeng, and H. Y. Zhou, Phase-matching quantum key distribution, Phys. Rev. X 8(3), 031043 (2018)

[41] X. D. Wu, Y. J. Wang, H. Zhong, Q. Liao, and Y. Guo, Plug-and-play dual-phase-modulated continuous variable quantum key distribution with photon subtraction, Front. Phys.14(4), 41501 (2019)

[42] J. J. Ma, Y. Zhou, X. Yuan, and X. F. Ma, Operational interpretation of coherence in quantum key distribution, Phys. Rev. A 99(6), 062325 (2019)

[43] X. D. Wu, Y. J. Wang, D. Huang, and Y. Guo, Simultaneous measurement-device-independent continuous variable quantum key distribution with realistic detector compensation, Front. Phys.15(3), 31601 (2020)

[44] S. Y. Lee, Y. S. Ihn, and Z. Kim, Optimal entangled coherent states in lossy quantum-enhanced metrology, Phys. Rev. A 101(1), 012332 (2020)

[45] N. Sangouard, C. Simon, N. Gisin, J. Laurat, R. Tualle-Brouri, and P. Grangier, Quantum repeaters with entangled coherent states, J. Opt. Soc. Am. B 27(6), A137 (2010)

[46] L. M. Kuang, Z. B. Chen, and J. W. Pan, Generation of entangled coherent states for distant Bose–Einstein condensates via electromagnetically induced transparency, Phys. Rev. A 76(5), 052324 (2007)

[47] A. P. Lund, T. C. Ralph, and H. Jeong, Generation of distributed entangled coherent states over a lossy environment with inefficient detectors, Phys. Rev. A 88(5), 052335 (2013)

[48] Z. R. Zhong, X. J. Huang, Z. B. Yang, L. T. Shen, and S. B. Zheng, Generation and stabilization of entangled coherent states for the vibrational modes of a trapped ion, Phys. Rev. A 98(3), 032311 (2018)

[49] B. Xiong, X. Li, S. L. Chao, Z. Yang, W. Z. Zhang, and L. Zhou, Generation of entangled Schrödinger cat state of two macroscopic mirrors, Opt. Express 27(9), 13547 (2019)

[50] L. Tian, S. P. Shi, Y. H. Tian, Y. J. Wang, Y. H. Zheng, and K. C. Peng, Resource reduction for simultaneous generation of two types of continuous variable nonclassical states, Front. Phys.16(2), 21502 (2021)

[51] Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs, Phys. Rev. A 85(1), 012307 (2012)

[52] Y. B. Sheng, L. Zhou, and S. M. Zhao, Efficient two-step entanglement concentration for arbitrary Wstates, Phys. Rev. A 85(4), 042302 (2012)

[53] B. C. Ren and G. L. Long, General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities, Opt. Express 22(6), 6547 (2014)

[54] H. Wang, B. C. Ren, A. H. Wang, A. Alsaedi, T. Hayat, and F. G. Deng, General hyperentanglement concentration for polarization-spatial-time-bin multi-photon systems with linear optics, Front. Phys.13(5), 130315 (2018)

[55] H. Wang, B. C. Ren, F. Alzahrani, A. Hobiny, and F. G. Deng, Hyper entanglement concentration for polarizationspatial-time-bin hyperentangled photon systems with linear optics, Quantum Inform. Process.16(10), 237 (2017)

[56] M. Sisodia, C. Shukla, and G. L. Long, Linear optics based entanglement concentration protocols for cluster type entangled coherent state, Quantum Inform. Process.18(8), 253 (2019)

[57] J. Liu, L. Zhou, W. Zhong, and Y. B. Sheng, Logic Bell state concentration with parity check measurement, Front. Phys.14(2), 21601 (2019)

[58] L. Zhou, J. Liu, Z. K. Liu, W. Zhong, and Y. B. Sheng, Logic W-state concentration with parity check, Quant. Engineering 3(2), e63 (2021)

[59] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of noise entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett.76(5), 722 (1996)

[60] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Quantum privacy amplification and the security of quantum cryptography over noisy channels, Phys. Rev. Lett.77(13), 2818 (1996)

[61] M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, Multiparticle entanglement purification protocols, Phys. Rev. A 57(6), R4075 (1998)

[62] J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, Entanglement purification for quantum communication, Nature 410(6832), 1067 (2001)

[63] C. Simon and J. W. Pan, Polarization entanglement purification using spatial entanglement, Phys. Rev. Lett.89(25), 257901 (2002)

[64] Y. B. Sheng, L. Zhou, and G. L. Long, Hybrid entanglement purification for quantum repeaters, Phys. Rev. A 88(2), 022302 (2013)

[65] M. Zwerger, H. J. Briegel, and W. Dür, Universal and optimal error thresholds for measurement-based entanglement purification, Phys. Rev. Lett.110(26), 260503 (2013)

[66] M. Zwerger, H. J. Briegel, and W. Dür, Robustness of hashing protocols for entanglement purification, Phys. Rev. A 90(1), 012314 (2014)

[67] G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, Faithful entanglement purification for high capacity quantum communication with two-photon four qubit systems, Phys. Rev. Appl.10(5), 054058 (2018)

[68] L. Zhou, W. Zhong, and Y. B. Sheng, Purification of the residual entanglement, Opt. Express 28(2), 2291 (2020)

[69] M. Y. Wang, F. L. Yan, and T. Gao, Entanglement purification of two-photon systems in multiple degrees of freedom, Quantum Inform. Process.19(7), 206 (2020)

[70] D. Y. Chen, Z. Lin, M. Yang, Q. Yang, X. P. Zang, and Z. L. Cao, Distillation of lossy hyperentangled states, Phys.Rev. A 102(2), 022425 (2020)

[71] X. M. Hu, C. X. Huang, Y. B. Sheng, L. Zhou, B. H. Liu, Y. Guo, C. Zhang, W. B. Xing, Y. F. Huang, C. F. Li, and G. C. Guo, Long-distance entanglement purification for quantum communication, Phys. Rev. Lett.126(1), 010503 (2021)

[72] P. S. Yan, L. Zhou, W. Zhong, and Y. B. Sheng, Feasible time-bin entanglement purification based on sum frequency generation, Opt. Express 29(2), 571 (2021)

[73] P. S. Yan, L. Zhou, W. Zhong, and Y. B. Sheng, Feasible measurement-based entanglement purification in linear optics, Opt. Express 29(6), 9363 (2021)

[74] H. Jeong and M. S. Kim, Purification of entangled coherent states, Quantum Inf. Comput.2(3), 208 (2002)

[75] J. Clausen, L. Knöll, and D. G. Welsch, Lossy purification and detection of entangled coherent states, Phys. Rev. A 66(6), 062303 (2002)

[76] U. L. Andersen, R. Filip, J. Fiurášek, V. Josse, and G. Leuchs, Experimental purification of coherent states, Phys. Rev. A 72(6), 060301(R) (2005)

[77] M. Zwerger, W. Dür, and H. J. Briegel, Measurement based quantum repeaters, Phys. Rev. A 85(6), 062326 (2012)

[78] M. Zwerger, H. J. Briegel, and W. Dür, Measurement based quantum communication, Appl. Phys. B 122(3), 50 (2016)

[79] S. W. Lee and H. Jeong, Near-deterministic quantum teleportation and resource-efficient quantum computation using linear optics and hybrid qubits, Phys. Rev. A 87(2), 022326 (2013)

[80] A. E. Lita, A. J. Miller, and S. W. Nam, Counting near infrared single-photons with 95% efficiency, Opt. Express 16(5), 3032 (2008)

[81] M. ö, M. Swillo, S. Gyger, V. Zwiller, and G. Björk, Temporal array with superconducting nanowire single photon detectors for photon-number-resolution, Phys. Rev. A 102(5), 052616 (2020)

[82] R. Guo, L. Zhou, S. P. Gu, X. F. Wang, and Y. B. Sheng, Generation of concatenated Greenberger–Horne– Zeilinger-type entangled coherent state based on linear optics, Quantum Inform. Process. 16(3), 68 (2017)

[83] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Light speed reduction to 17 metres per second in an ultracold atomic gas, Nature 397(6720), 594 (1999)

[84] K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. B. Plenio, A. Serafini, M. M. Wolf, and E. S. Polzik, Quantum memory for entangled continuous-variable states, Nat. Phys. 7(1), 13 (2011)

RIGHTS & PERMISSIONS

2022 Higher Education Press