Continuous Variable Entanglement in a Thermal Environment

Abstract

Continuous-variable quantum teleportation (CVQT) plays a vital role in practical quantum communications. However, the degradation of quantum signals, usually caused by the absorption and scattering of light in actual scenarios, has an influence on the transmission performance and hence hinders its implementations. In this paper, we propose a non-Gaussian approach to improve the performance of CVQT via photon catalysis through a lossy optical fiber channel. The photon catalysis-enabled scheme can be performed on both sides of Einstein–Podolsky–Rosen state prepared by the receiver, where it gives birth to the enhancement of entanglement of the system. Numerical simulations show that the performance of the photon catalysis-enabled CVQT has been improved in terms of both fidelity and maximal transmission distance. Moreover, the larger the squeezing parameter results in the more enhancement of the CVQT system. The results may provide a useful insight for the practical implementation of CVQT.

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References

1. Bennett, C.H., DiVincenzo, D.P.: Quantum information and computation. Nature 404(6775), 247–255 (2000)

   Article  ADS  Google Scholar

2. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)

   Article  ADS  MathSciNet  Google Scholar

3. Ekert, A.K.: Quantum Cryptography Based on Bell's Theorem. Phys. Rev. Lett. 67(6), 661 (1991)

   Article  ADS  MathSciNet  Google Scholar

4. Furusawa, A., Van Loock, P.: Quantum teleportation and entanglement: a hybrid approach to optical quantum information processing. John Wiley and Sons (2011)

5. Bennett, C. H., Brassard, G., Cr peau, C., Jozsa, R., Peres, A., Wootters, W. K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev .Lett 70(13), 1895 (1993)

   Article  ADS  MathSciNet  Google Scholar

6. Bouwmeester, D., Pan, J.W., Mattle, K., Eibl, M., Weinfurter, H., Zeilinger, A.: Experimental quantum teleportation. Nature 390(6660), 575–579 (1997)

   Article  ADS  Google Scholar

7. Braunstein, S.L., Kimble, H.J.: Teleportation of continuous quantum variables. Phys. Rev. Lett. 80(4), 869 (1998)

   Article  ADS  Google Scholar

8. Ralph, T.C., Lam, P.K.: Teleportation with bright squeezed light. Phys. Rev. Lett. 81(25), 5668 (1998)

   Article  ADS  Google Scholar

9. Xu, J.Y., Xu, X.W., Zuo, Z.Y., Guo, Y.: Continuous variable quantum teleportation through turbulent channels. Phys. Scr. 97(4), 045103 (2022)

10. Zuo, Z., Wang, Y., Liao, Q., Guo, Y.: Overcoming the uplink limit of satellite-based quantum communication with deterministic quantum teleportation. Phys. Rev. A 104(2), 022615 (2021)

    Article  ADS  MathSciNet  Google Scholar

11. Andersen, U.L., Ralph, T.C.: High-fidelity teleportation of continuous-variable quantum states using delocalized single photons. Phys. Rev. Lett. 111(5), 050504 (2013)

    Article  ADS  Google Scholar

12. Hofmann, H.F., Ide, T., Kobayashi, T., Furusawa, A.: Information losses in continuous-variable quantum teleportation. Phys. Rev. A 64(4), 040301 (2001)

    Article  ADS  Google Scholar

13. Chizhov, A.V., Knöll, L., Welsch, D.G.: Continuous-variable quantum teleportation through lossy channels. Phys. Rev. A 65(2), 022310 (2002)

    Article  ADS  Google Scholar

14. Furusawa, A., Srensen, J.L., Braunstein, S.L., Fuchs, C.A., Kimble, H.J., Polzik, E.S.: Unconditional quantum teleportation. Science 282(5389), 706–709 (1998)

    Article  ADS  Google Scholar

15. Takei, N., Aoki, T., Koike, S., Yoshino, K.I., Wakui, K., Yonezawa, H., Hiraoka, T., Mizuno, J., Takeoka, M., Ban, M., Furusawa, A.: Experimental demonstration of quantum teleportation of a squeezed state. Phys. Rev. A 72(4), 042304 (2005)

    Article  ADS  Google Scholar

16. He, G., Zhang, J., Zhu, J., Zeng, G.: Continuous-variable quantum teleportation in bosonic structured environments. Phys. Rev. A 84(3), 034305 (2011)

    Article  ADS  Google Scholar

17. Laudenbach, F., Pacher, C., Fung, C.H.F., Poppe, A., Peev, M., Schrenk, B., Hentschel, M., Walther, P., Hübel, H.: Continuous-variable quantum key distribution with Gaussian modulation-the theory of practical implementations. Adv. Quantum Technol. 1(1), 1800011 (2018)

    Article  Google Scholar

18. Wu, X.D., Wang, Y.J., Zhong, H., Ye, W., Huang, D., Guo, Y.: Performance improvement of plug-and-play dual-phase-modulated continuous-variable quantum key distribution with quantum catalysis. Quant. Inf. Process. 19(8), 1–15 (2020)

    Article  ADS  MathSciNet  Google Scholar

19. Shao, Y., Wang, H., Pi, Y.D., Huang, W., Li, Y., Liu, J.L., Zhang, Y.C., Xu, B.J.: Phase noise model for continuous-variable quantum key distribution using a local local oscillator. Phys. Rev. A 104(3), 032608 (2021)

    Article  ADS  Google Scholar

20. Lund, A.P., Ralph, T.C.: Continuous-variable entanglement distillation over a general lossy channel. Phys. Rev. A 80(3), 032309 (2009)

    Article  ADS  Google Scholar

21. Opatrn, T., Kurizki, G., Welsch, D.G.: Improvement on teleportation of continuous variables by photon subtraction via conditional measurement. Phys. Rev. A 61(3), 032302 (2000)

    Article  ADS  Google Scholar

22. Braunstein, S.L., Fuchs, C.A., Kimble, H.J.: Criteria for continuous-variable quantum teleportation. J. Modern Opt. 47(2–3), 267–278 (2000)

    Article  ADS  MathSciNet  Google Scholar

23. Ide, T., Hofmann, H.F., Kobayashi, T., Furusawa, A.: Continuous-variable teleportation of single-photon states. Phys. Rev. A 65(1), 012313 (2001)

    Article  ADS  Google Scholar

24. Bowen, W.P., Treps, N., Buchler, B.C., Schnabel, R., Ralph, T.C., Symul, T., Lam, P.K.: Unity gain and nonunity gain quantum teleportation. IEEE J. Sel. Topics Quantum Electron. 9(6), 1519–1532 (2003)

    Article  ADS  Google Scholar

25. Huo, M., Qin, J., Cheng, J., Yan, Z., Qin, Z., Su, X., Jia, X., Xie, C., Peng, K.: Deterministic quantum teleportation through fiber channels. Sci. Adv. 4(10), eaas9401 (2018)

    Article  ADS  Google Scholar

26. Jozsa, R.: Fidelity for mixed quantum states. J. Modern Opt. 41(12), 2315–2323 (1994)

    Article  ADS  MathSciNet  Google Scholar

27. Fiurasek, J.: Improving the fidelity of continuous-variable teleportation via local operations. Phys. Rev. A 66(1), 012304 (2002)

    Article  ADS  MathSciNet  Google Scholar

28. Lvovsky, A.I., Mlynek, J.: Quantum-optical catalysis: generating nonclassical states of light by means of linear optics. Phys. Rev. Lett. 88(25), 250401 (2002)

    Article  ADS  Google Scholar

29. Hu, L.Y., Liao, Z., Zubairy, M.S.: Continuous-variable entanglement via multiphoton catalysis. Phys. Rev. A 95(1), 012310 (2017)

    Article  ADS  Google Scholar

30. Guo, Y., Ye, W., Zhong, H., Liao, Q.: Continuous-variable quantum key distribution with non-Gaussian quantum catalysis. Phys. Rev. A 99(3), 032327 (2019)

    Article  ADS  Google Scholar

31. Ye, W., Zhong, H., Liao, Q., Huang, D., Hu, L., Guo, Y.: Improvement of self-referenced continuous-variable quantum key distribution with quantum photon catalysis. Opt. Exp. 27(12), 17186–17198 (2019)

    Article  Google Scholar

32. Ye, W., Zhong, H., Wu, X., Hu, L., Guo, Y.: Continuous-variable measurement-device-independent quantum key distribution via quantum catalysis. Quant. Inf. Process. 19(10), 1–22 (2020)

    Article  ADS  MathSciNet  Google Scholar

33. Hu, L., Al-amri, M., Liao, Z., Zubairy, M.S.: Continuous-variable quantum key distribution with non-Gaussian operations. Phys. Rev. A 102(1), 012608 (2020)

    Article  ADS  MathSciNet  Google Scholar

34. Ulanov, A.E., Fedorov, I.A., Pushkina, A.A., Kurochkin, Y.V., Ralph, T.C., Lvovsky, A.I.: Undoing the effect of loss on quantum entanglement. Nat. Photon. 9(11), 764–768 (2015)

    Article  ADS  Google Scholar

35. Hu, L.Y., Wu, J.N., Liao, Z., Zubairy, M.S.: Multiphoton catalysis with coherent state input: nonclassicality and decoherence. J. Phys. B At. Mol. Phys. 49(17), 175504 (2016)

    Article  ADS  Google Scholar

36. Zhou, W., Ye, W., Liu, C., Hu, L., Liu, S.: Entanglement improvement of entangled coherent state via multiphoton catalysis. Laser Phys. Lett. 15(6), 065203 (2018)

    Article  ADS  Google Scholar

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61871407) and Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing (Grant No. GXKL06200208).

Author information

Authors and Affiliations

1. School of Physics and Electronics, Central South University, Changsha, 410083, China

   Zhennan Wen, Yougen Yi & Xiaowen Xu

2. School of Computer Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

   Ying Guo

Corresponding author

Correspondence to Xiaowen Xu.

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A. Derivation of the covariance matrix \(\gamma _{B_1A_1}\) and the successful probability \(P_c\) of photon catalysis

A. Derivation of the covariance matrix \(\gamma _{B_1A_1}\) and the successful probability \(P_c\) of photon catalysis

Taking into account the TMSV state with squeezing parameter r, it is usually described by the two-mode squeezed operator \(S_2(r)=\mathrm{exp}\{r(a^\dag b^\dag -ab)\}\) and a two-mode squeezed vacuum state \(|0,0\rangle _{\text{ AB }}\) [30],

$$\begin{aligned} |\mathrm{TMSV}\rangle _{\text{ AB }}= & {} S_2(r)|0,0\rangle _{\text{ AB }}\nonumber \\= & {} \sqrt{1-\lambda ^2}\sum _{n=0}^\infty \lambda ^n|n\rangle |n\rangle , \end{aligned}$$

(A.1)

with \(\lambda =\mathrm{\tanh }r\in [0,1]\). After n-photon and m-photon catalysis, operations are performed on modes A and B of the TMSV state, it turns into the state \(|\psi \rangle _{A_1B_1}\). Before analyzing the target state, we need to describe the process of photon catalysis in advance. Generally, the n-photon catalysis process can be expressed as an equivalent operator \({\hat{O}}_n\) given by

$$\begin{aligned} {\hat{O}}_n \equiv \langle n|B(\eta _1)|n\rangle , \end{aligned}$$

(A.2)

where \(B(\eta _1)\) is the BS operator with transmittance \(\eta _1\), which is represented by using IWOP technology as follows

$$\begin{aligned} B(\eta _1) = :\mathrm{exp}[(\sqrt{\eta _1}\!-\!1)(a^\dag a\!+\!c^\dag c)\!+\!(c^\dag a\!-\!ca^\dag )\sqrt{1\!-\!\eta _1}]:, \end{aligned}$$

(A.3)

where the notation :\(\cdot \): represents the normal ordering of operator [29]. By using the Laguerre polynomials [35, 36], \({\hat{O}}_n \) can be rewritten as

$$\begin{aligned} {\hat{O}}_n = G_{\eta _1}(a^\dag a)(\sqrt{\eta _1})^{a^\dag a+n}, \end{aligned}$$

(A.4)

where

$$\begin{aligned} \!\!\!\!G_{\eta _1}(a^\dag a) = \frac{\partial ^{n}}{n! \partial t_1^{n}}\left\{ \dfrac{1}{1-t_1}\left( \dfrac{1-t_1/\eta _1}{1-t_1}\right) ^{a^\dag a} \right\} _{t_1=0}. \end{aligned}$$

(A.5)

Similarly, the process of m-photon catalysis operations performed on mode B can be characterized by

$$\begin{aligned} {\hat{O}}_m = G_{\eta _2}(b^\dag b)(\sqrt{\eta _2})^{b^\dag b+m}, \end{aligned}$$

(A.6)

and

$$\begin{aligned} \!\!\!\!G_{\eta _2}(b^\dag b) = \frac{\partial ^{m}}{m! \partial t_2^{m}}\left\{ \dfrac{1}{1-t_2}\left( \dfrac{1-t_2/\eta _2}{1-t_2}\right) ^{b^\dag b} \right\} _{t_2=0}. \end{aligned}$$

(A.7)

The target state \(|\psi \rangle _{A_1B_1}\) is obtained as

$$\begin{aligned} |\psi \rangle _{A_1B_1}= & {} \dfrac{{\hat{O}}_n{\hat{O}}_m}{\sqrt{P_c}}|\mathrm{TMSV}\rangle \nonumber \\= & {} \sum _{n=0}^\infty \ \dfrac{W_0}{\sqrt{P_c}} \frac{\partial ^{n}}{\partial t_1^{n}}\frac{\partial ^{m}}{\partial t_2^{m}}\dfrac{W^n}{(1-t_1)(1-t_2)}|n\rangle |n\rangle , \end{aligned}$$

(A.8)

where \(P_c\) is the successful probability of photon catalysis. In order to facilitate the calculation of \(P_c\) and the covariance matrix \(\gamma _{B_1A_1}\), we rewrite the state \(|\psi \rangle _{A_1B_1}\) in the form of density operator, i.e.,

$$\begin{aligned} \rho _{A_1B_1}= & {} \frac{1}{P_c}{\hat{O}}_n{\hat{O}}_m|\mathrm{TMSV}\rangle \langle \mathrm{TMSV}|{\hat{O}}_m^\dag {\hat{O}}_n^\dag \nonumber \\= & {} \dfrac{W_0^2}{P_c}D_{m,n} \prod \mathrm{exp}\left( a^\dag b^\dag W\right) |00\rangle \langle 00|\mathrm{exp}\left( ab W_1\right) , \end{aligned}$$

(A.9)

where \(W, W_0, W_1, \prod \) and \(D_{m,n}\) is given by

$$\begin{aligned}&W = \dfrac{\lambda (\eta _1-t_1)(\eta _2-t_2)}{\sqrt{\eta _1\eta _2}(1-t_1)(1-t_2)},\nonumber \\&\quad W_0 = \dfrac{\sqrt{\eta _1^n\eta _2^m(1-\lambda ^2)}}{n!m!},\nonumber \\&\quad W_1 = \dfrac{\lambda (\eta _1-t_3)(\eta _2-t_4)}{\sqrt{\eta _1\eta _2}(1-t_3)(1-t_4)},\nonumber \\&\quad \prod = \dfrac{1}{1-t_1}\dfrac{1}{1-t_2}\dfrac{1}{1-t_3}\dfrac{1}{1-t_4},\nonumber \\&\quad D_{m,n} = \frac{\partial ^{n}}{\partial t_1^{n}}\frac{\partial ^{m}}{\partial t_2^{m}}\frac{\partial ^{n}}{\partial t_3^{n}}\frac{\partial ^{m}}{\partial t_4^{m}}\left\{ \cdot \right\} |_{t_1=t_2=t_3=t_4=0}. \end{aligned}$$

(A.10)

Based on completeness of the coherent state representation \(\int \ |z\rangle \langle z|\, d^2z\) and the target state \(\mathrm{Tr}\big (\rho _{A_1B_1}^N\big )=1\), the successful probability \(P_c\) is derived as

$$\begin{aligned} P_c= & {} W_0^2D_{m,n}\prod \langle 00|\mathrm{exp}\left( ab W_1\right) \mathrm{exp}\left( a^\dag b^\dag W\right) |00\rangle ,\nonumber \\= & {} W_0^2D_{m,n}\left\{ \dfrac{\prod }{1-W_1W}\right\} . \end{aligned}$$

(A.11)

Combining the IWOP technique with Eqs.(A.9)–(A.11), we can derive the element X,Y and Z of the covariance matrix \(\gamma _{B_1A_1}\) as follows

$$\begin{aligned} X= & {} \mathrm{Tr}\left[ 2a^\dag a+1\right] =2\langle a^\dag a\rangle +1,\nonumber \\= & {} 2\mathrm{Tr}\left[ \rho _{A_1B_1}^N(aa^\dag -1)\right] -1,\nonumber \\= & {} \dfrac{2W_0^2}{P_c}D_{m,n}\left\{ \dfrac{\prod }{(1-W_1W)^2}\right\} -n1, \end{aligned}$$

(A.12)

$$\begin{aligned} Y= & {} \mathrm{Tr}\left[ 2b^\dag b+1\right] =2\langle b^\dag b\rangle +1,\nonumber \\= & {} 2\mathrm{Tr}\left[ \rho _{A_1B_1}^N(bb^\dag -1)\right] -1,\nonumber \\= & {} \dfrac{2W_0^2}{P_c}D_{m,n}\left\{ \dfrac{\prod }{(1-W_1W)^2}\right\} -1, \end{aligned}$$

(A.13)

and

$$\begin{aligned} Z= & {} \mathrm{Tr}\left[ ab+a^\dag b^\dag \right] =2\langle ab\rangle ,\nonumber \\= & {} 2\mathrm{Tr}\left[ \rho _{A_1B_1}^N ab\right] ,\nonumber \\= & {} \dfrac{W_0^2}{P_c}D_{m,n}\left\{ \dfrac{\prod W}{(1-W_1W)^2}\right\} . \end{aligned}$$

(A.14)

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Wen, Z., Yi, Y., Xu, X. et al. Continuous-variable quantum teleportation with entanglement enabled via photon catalysis. Quantum Inf Process 21, 325 (2022). https://doi.org/10.1007/s11128-022-03666-8

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• Received: 29 November 2021

• Accepted: 23 August 2022

• Published: 17 September 2022

• DOI : https://doi.org/10.1007/s11128-022-03666-8

Keywords

• Quantum teleportation
• Continuous variable
• Photon catalysis
• Lossy channel

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