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Generalized isometric tensor based quantum key distribution protocols of squeezed multiphoton entangled states

Lai Hong

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Generalized isometric tensor based quantum key distribution protocols of squeezed multiphoton entangled states

Lai Hong
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  • Isometric tensor offers a novel and powerful tool that can compress an entangled state into its tensor network state (TNS). The resulting quantum compression provides a new opportunity for enhancing quantum key distribution (QKD) protocols. The main idea explored in this work is to use the quantum compression to improve the efficiency of QKD. In a nut-shell, a collection of any multi-photon entangled states that carry encoded classical bits is compressed into a single-photon state before the corresponding photon is sent to the receiver that measures the qubit and decompresses it. In this paper, we first show how to obtain the generalized isometric tensors for compressing any entangled states and their inverse isometric tensors for decompression. In our proposed QKD protocol, the input state consists of any multi-photon entangled states, which are first compressed into a single-photon state $ \left| 0 \right\rangle $ or $ \left| 1 \right\rangle $ or Bell states by the sender Alice. A sequence of single-photon states $ \left| 0 \right\rangle $ and $ \left| 1 \right\rangle $ and one photon from the Bell state mixed with decoy qubits is sent to the receiver Bob via a quantum channel. Bob obtains the final sifted compressed states $ \left| 0 \right\rangle $ and $ \left| 1 \right\rangle $ and conjugate transpose of the isometric tensors. Using our protocols, Bob can decompress the received states $ \left| 0 \right\rangle $ and $ \left| 1 \right\rangle $ into original entangled states. Since quantum processors that are used to send quantum information between nodes are relatively primitive and low in power and the preparation of many-photon entanglement is relatively difficult at present, finding suitable protocols for the compression of transmitted quantum data brings important practical benefits. More generally, the quantum information theory primarily investigates quantum data manipulation under locality constraints, so our protocols connect naturally to these investigations. Our protocols increase the encoding capacity of QKD protocols. Not only our proposed processes of compression and decompression are very simple, but also entanglement compression using isometric tensors can be implemented by using quantum circuits and current technology. Because many ideas for designing of quantum information processing equipment envision that a network composed of relatively small quantum processors sending quantum information between nodes, it is greatly significant to find appropriate protocols for compressing the transmitted quantum data .
      Corresponding author: Lai Hong, hlai@swu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61702427), the Natural Science Foundation of Chongqing, China (Grant No. CSTB2022NSCQ-MSX0749), and the Southwest University’s 2022School-level Teaching Reform Project, China (Grant No. 2022JY086).
    [1]

    Bennett C H, Brassard G 1984 Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing Bangalore, India, December 10–12, 1984 pp175–179

    [2]

    Wang B, Zhang B F, Zou F C, et al. 2021 Optik 235 166628Google Scholar

    [3]

    Ekert A K 1991 Phys. Rev. Lett. 67 661Google Scholar

    [4]

    Bennett C H, Brassard G, Mermin N D 1992 Phys. Rev. Lett. 68 557Google Scholar

    [5]

    Greenberger D M, Horne M A, Zeilinger A 1989 Bell’s Theorem, Quantum Theory and Conceptions of the Universe (Dordrecht: Springer) pp69–72

    [6]

    Bouwmeester D, Pan J W, Daniell M, Weinfurter H, Zeilinger A 1999 Phys. Rev. Lett. 82 1345Google Scholar

    [7]

    Guo Y, Shi R, Zeng G 2010 Phys. Scr. 81 045006Google Scholar

    [8]

    Xu G B, Wen Q Y, Gao F, Qin S J 2014 Quantum Inf. Process. 13 2587Google Scholar

    [9]

    Castañeda Valle D, Quezada L F, Dong S H 2021 Ann. Phys. Berlin 533 2100116Google Scholar

    [10]

    Zhao N, Guo X, Wu T 2021 Phys. Rev. A 104 062616Google Scholar

    [11]

    Upadhyaya T, van Himbeeck T, Lin J, et al. 2021 PRX Quantum. 2 020325Google Scholar

    [12]

    Jiang C, Yu Z W, Hu X L, Wang X B 2021 Phys. Rev. A 103 012402Google Scholar

    [13]

    Lim C C W, Xu F, Pan J W, Ekert A 2021 Phys. Rev. Lett. 126 100501Google Scholar

    [14]

    Long G L and Liu X S 2002 Phys. Rev. A 65 032302Google Scholar

    [15]

    Chang C H, Yang C W, Hwang T 2016 Int. J. Theor. Phys. 55 3993Google Scholar

    [16]

    Pivoluska M, Huber M, Malik M 2018 Phys. Rev. A 97 032312Google Scholar

    [17]

    Zhu K N, Zhou N R, Wang Y Q, et al. 2018 Int. J. Theor. Phys. 57 3621Google Scholar

    [18]

    Zhou H, Lv K, Huang L, et al. 2022 IEEE/ACM T. Network. 30 1328Google Scholar

    [19]

    Wang X L, Chen L K, Li W, Huang H L, Liu C, Chen C, Luo Y H, Su Z E, Wu D, Li Z D, Lu H, Hu Y, Jiang X, Peng C Z, Li L, Liu N L, Chen Y A, Lu C Y, Pan J W 2016 Phys. Rev. Lett. 117 210502Google Scholar

    [20]

    Xia Y, Lu P M, Zeng Y Z 2012 Quantum Inf. Process. 11 605Google Scholar

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    Huang Y F, Liu B H, Peng L, et al. 2011 Nat. Commun. 2 1

    [22]

    Xia Y, Song J, Ning Y, et al. 2010 JETP Lett. 90 735Google Scholar

    [23]

    Svozil K 2022 Found. Phys. 52 4Google Scholar

    [24]

    Yin H L, Fu Y, Li C L, et al. 2023 Nati. Sci. Rev. 10 228Google Scholar

    [25]

    Xie Y M, Lu Y S, Weng C X, et al. 2022 PRX Quantum. 3 020315Google Scholar

    [26]

    Gu J, Cao X Y, Fu Y, et al. 2022 Sci. Bull. 67 2167Google Scholar

    [27]

    Fan C R, Lu B, Feng X T, et al. 2021 Quant. Engineer. 3 e67

    [28]

    Bostroem K, Felbinger T 2002 Phys. Rev. A 65 032313Google Scholar

    [29]

    Datta N, Renes J M, Renner R, et al. 2013 IEEE Inform. Theory 59 8057Google Scholar

    [30]

    Plesch M, Bužek V 2010 Phys. Rev. A 81 032317Google Scholar

    [31]

    Rozema L A, Mahler D H, Hayat A, Turner P S, Steinberg A M 2014 Phys. Rev. Lett. 113 160504Google Scholar

    [32]

    Yang Y, Chiribella G, Ebler D 2016 Phys. Rev. Lett. 116 080501Google Scholar

    [33]

    Yang Y, Chiribella G, Hayashi M 2016 Phys. Rev. Lett. 117 090502Google Scholar

    [34]

    Romero J, Olson J P, Aspuru-Guzik A 2017 Quantum Sci. Technol. 2 045001Google Scholar

    [35]

    Pepper A, Tischler N, Pryde G J 2019 Phys. Rev. Lett. 122 060501Google Scholar

    [36]

    Van Acoleyen K, Hallam A, Bal M, Hauru M, Haegeman J, Verstraete F 2020 Phys. Rev. B 102 165131Google Scholar

    [37]

    Lai H, Pieprzyk J, Pan L 2023 Sci. China Inf. Sc. 66 180510Google Scholar

    [38]

    Evenbly G 2022 Front. Phys. 10 1146

    [39]

    Bennett C H, DiVincenzo D P, Smolin J A, Wootters W K 1996 Phys. Rev. A 54 3824Google Scholar

  • 图 1  利用wN1, wN2, N光子GHZ 纠缠态$ ({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right) $ 被随机压缩为数态$| 0 \rangle $$| 1\rangle $, 利用${\boldsymbol{w}}_{N1}^{\dagger},{\boldsymbol{w}}_{N2}^{\dagger}$,$| 0 \rangle $$| 1 \rangle $被解压缩为N光子GHZ 纠缠态$({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right)$ 示意图

    Figure 1.  The schematic compression of $({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right)$ into $\left| 0 \right\rangle $ or $\left| 1 \right\rangle $ using ${{\boldsymbol{w}}_{N1}}, {{\boldsymbol{w}}_{N2}}$, and decompression of $\left| 0 \right\rangle $ or $\left| 1 \right\rangle $ into $({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right)$ using ${\boldsymbol{w}}_{N1}^\dagger ,{\boldsymbol{w}}_{N2}^\dagger$.

    图 2  利用${\boldsymbol{w}}'_{N1}, {\boldsymbol{w}}'_{N2}$, N 光子 GHZ 纠缠态$({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right)$被随机压缩为Bell态$({{\sqrt 2 }}/{2})\left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right)$$({{\sqrt 2 }}/{2}) ( | 01 \rangle + $$ |10\rangle)$, 利用${\boldsymbol{w}}^{\prime\dagger}_{N1}, {\boldsymbol{w}}^{\prime\dagger}_{N2}$, Bell态 $({{\sqrt 2 }}/{2})\left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right)$$({{\sqrt 2 }}/{2})\left( {\left| {01} \right\rangle + \left| {10} \right\rangle } \right)$被解压缩为N 光子 GHZ 纠缠态$({{\sqrt 2 }}/{2}) (| 00 \cdots 0\rangle + $$ | 11 \cdots 1\rangle)$示意图

    Figure 2.  The schematic compression of $({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right)$ into $({{\sqrt 2 }}/{2})\left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right)$ or $({{\sqrt 2 }}/{2})\left( {\left| {01} \right\rangle + \left| {10} \right\rangle } \right)$ using ${{\boldsymbol{w}}'_{N1}}, $$ {{\boldsymbol{w}}'_{N2}}$, and decompression of $({{\sqrt 2 }}/{2})\left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right)$ or $({{\sqrt 2 }}/{2})\left( {\left| {01} \right\rangle + \left| {10} \right\rangle } \right)$ into $({{\sqrt 2 }}/{2})\left( {\left| {00 \cdots 0} \right\rangle + \left| {11 \cdots 1} \right\rangle } \right)$ using ${\boldsymbol{w}}_{N1}^{\prime\dagger} , {\boldsymbol{w}}_{N2}^{\prime\dagger }$.

    表 1  压缩纠缠态用于在 Alice 和 Bob 之间生成密钥的例子

    Table 1.  An example of transmitting the compressed entangled state for generating a secret key between Alice and Bob.

    Alice随机制备纠缠态$ ({{\sqrt 2 }}/{2})\left( {\left| {000} \right\rangle + \left| {111} \right\rangle } \right) $$ ({{\sqrt 2 }}/{2})\left( {\left| {0000} \right\rangle + \left| {1111} \right\rangle } \right) $$ N(N \geqslant 5) $光子GHZ态
    Alice随机选择等距张量$ {{\boldsymbol{w}}_{31}} $$ {{\boldsymbol{w}}_{{4}1}} $$ {{\boldsymbol{w}}_{N2}} $
    Alice压缩纠缠态$ \left| 0 \right\rangle $$ \left| 0 \right\rangle $$ \left| 0 \right\rangle $
    Alice随机选择测量基$ {B_x} $$ {B_{\text{z}}} $${B_z}$
    Bob随机选择测量基${B_z}$$ {B_{\text{z}}} $$ {B_x} $
    对基
    筛选出的共享压缩态$ \left| 0 \right\rangle $
    解压缩共享压缩态$ ({{\sqrt 2 }}/{2})\left( {\left| {0000} \right\rangle + \left| {1111} \right\rangle } \right) $
    DownLoad: CSV

    表 2  本文的协议与文献[4, 16, 20]中的协议进行了比较

    Table 2.  Our protocol compares with the protocols in Refs.[4, 16, 20].

    BBM92[4]Pivoluska等[16]的协议Xia等[20]
    的协议
    本文的
    协议
    量子源贝尔态贝尔态和GHZ态N光子的GHZ态任何压缩纠缠态
    压缩
    抵抗测量
    攻击
    抵抗拦截和重放攻击
    抵抗纠缠测量的攻击
    抗退相干
    DownLoad: CSV
  • [1]

    Bennett C H, Brassard G 1984 Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing Bangalore, India, December 10–12, 1984 pp175–179

    [2]

    Wang B, Zhang B F, Zou F C, et al. 2021 Optik 235 166628Google Scholar

    [3]

    Ekert A K 1991 Phys. Rev. Lett. 67 661Google Scholar

    [4]

    Bennett C H, Brassard G, Mermin N D 1992 Phys. Rev. Lett. 68 557Google Scholar

    [5]

    Greenberger D M, Horne M A, Zeilinger A 1989 Bell’s Theorem, Quantum Theory and Conceptions of the Universe (Dordrecht: Springer) pp69–72

    [6]

    Bouwmeester D, Pan J W, Daniell M, Weinfurter H, Zeilinger A 1999 Phys. Rev. Lett. 82 1345Google Scholar

    [7]

    Guo Y, Shi R, Zeng G 2010 Phys. Scr. 81 045006Google Scholar

    [8]

    Xu G B, Wen Q Y, Gao F, Qin S J 2014 Quantum Inf. Process. 13 2587Google Scholar

    [9]

    Castañeda Valle D, Quezada L F, Dong S H 2021 Ann. Phys. Berlin 533 2100116Google Scholar

    [10]

    Zhao N, Guo X, Wu T 2021 Phys. Rev. A 104 062616Google Scholar

    [11]

    Upadhyaya T, van Himbeeck T, Lin J, et al. 2021 PRX Quantum. 2 020325Google Scholar

    [12]

    Jiang C, Yu Z W, Hu X L, Wang X B 2021 Phys. Rev. A 103 012402Google Scholar

    [13]

    Lim C C W, Xu F, Pan J W, Ekert A 2021 Phys. Rev. Lett. 126 100501Google Scholar

    [14]

    Long G L and Liu X S 2002 Phys. Rev. A 65 032302Google Scholar

    [15]

    Chang C H, Yang C W, Hwang T 2016 Int. J. Theor. Phys. 55 3993Google Scholar

    [16]

    Pivoluska M, Huber M, Malik M 2018 Phys. Rev. A 97 032312Google Scholar

    [17]

    Zhu K N, Zhou N R, Wang Y Q, et al. 2018 Int. J. Theor. Phys. 57 3621Google Scholar

    [18]

    Zhou H, Lv K, Huang L, et al. 2022 IEEE/ACM T. Network. 30 1328Google Scholar

    [19]

    Wang X L, Chen L K, Li W, Huang H L, Liu C, Chen C, Luo Y H, Su Z E, Wu D, Li Z D, Lu H, Hu Y, Jiang X, Peng C Z, Li L, Liu N L, Chen Y A, Lu C Y, Pan J W 2016 Phys. Rev. Lett. 117 210502Google Scholar

    [20]

    Xia Y, Lu P M, Zeng Y Z 2012 Quantum Inf. Process. 11 605Google Scholar

    [21]

    Huang Y F, Liu B H, Peng L, et al. 2011 Nat. Commun. 2 1

    [22]

    Xia Y, Song J, Ning Y, et al. 2010 JETP Lett. 90 735Google Scholar

    [23]

    Svozil K 2022 Found. Phys. 52 4Google Scholar

    [24]

    Yin H L, Fu Y, Li C L, et al. 2023 Nati. Sci. Rev. 10 228Google Scholar

    [25]

    Xie Y M, Lu Y S, Weng C X, et al. 2022 PRX Quantum. 3 020315Google Scholar

    [26]

    Gu J, Cao X Y, Fu Y, et al. 2022 Sci. Bull. 67 2167Google Scholar

    [27]

    Fan C R, Lu B, Feng X T, et al. 2021 Quant. Engineer. 3 e67

    [28]

    Bostroem K, Felbinger T 2002 Phys. Rev. A 65 032313Google Scholar

    [29]

    Datta N, Renes J M, Renner R, et al. 2013 IEEE Inform. Theory 59 8057Google Scholar

    [30]

    Plesch M, Bužek V 2010 Phys. Rev. A 81 032317Google Scholar

    [31]

    Rozema L A, Mahler D H, Hayat A, Turner P S, Steinberg A M 2014 Phys. Rev. Lett. 113 160504Google Scholar

    [32]

    Yang Y, Chiribella G, Ebler D 2016 Phys. Rev. Lett. 116 080501Google Scholar

    [33]

    Yang Y, Chiribella G, Hayashi M 2016 Phys. Rev. Lett. 117 090502Google Scholar

    [34]

    Romero J, Olson J P, Aspuru-Guzik A 2017 Quantum Sci. Technol. 2 045001Google Scholar

    [35]

    Pepper A, Tischler N, Pryde G J 2019 Phys. Rev. Lett. 122 060501Google Scholar

    [36]

    Van Acoleyen K, Hallam A, Bal M, Hauru M, Haegeman J, Verstraete F 2020 Phys. Rev. B 102 165131Google Scholar

    [37]

    Lai H, Pieprzyk J, Pan L 2023 Sci. China Inf. Sc. 66 180510Google Scholar

    [38]

    Evenbly G 2022 Front. Phys. 10 1146

    [39]

    Bennett C H, DiVincenzo D P, Smolin J A, Wootters W K 1996 Phys. Rev. A 54 3824Google Scholar

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  • Received Date:  13 April 2023
  • Accepted Date:  02 June 2023
  • Available Online:  18 July 2023
  • Published Online:  05 September 2023

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