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The impact of constellation cardinality on discrete unidimensional CVQKD protocols

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Abstract

We introduce a new unidimensional CV-QKD protocol based on classical capacity achieving constellations and inspect how its cardinality translates into the secret key rate. The protocol requires simple modulation and detection procedures with intensity modulation on a single quadrature and homodyne detection. The secret key rate is computed for several constellation shapes, sizes and detection quantum efficiency for a pure loss quantum channel. We show that the capacity achieving constellations with increasing cardinality in a classical context preserve the same pattern in a QKD context by approximating the performance of Gaussian modulation. We also verify that the non-unit quantum efficiency detection effect is more severe on discrete modulation, diminishing the protocol performance and limiting its link range to no more than 100 km.

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Notes

  1. We call attention the difference between \({\tilde{V}}_m\) and \(V_m\): the first refers to the modulation variance (or average energy), while the second corresponds to the quadrature operator \(\hat{q}\) variance.

  2. Here, we assume the channel to be symmetric on its symplectic invariants, that is, the transmittance and excess noise are equal for both quadratures. The original paper presented a general case where the channel symmetry is initially discarded, forcing sporadic measurements on the non-modulated quadrature.

  3. Here, we consider the operators relations in shot noise units, that is, \(\hat{q}= \hat{a}^{\dagger }+\hat{a}\) and \(\hat{p}= i(\hat{a}^{\dagger }- \hat{a})\), which is slightly different from the development found in [7] but causes no harm as its it will mainly differ on the normalization factor for the output current expected value. Check [12] for a good comparison between the canonical operators relations in different unit systems.

  4. Note that it is a valid bosonic annihilation operator as it preserves the canonical commutation relation \([\hat{\varLambda },\hat{\varLambda }^\dagger ] = 1.\)

References

  1. Assche, G.V., Cardinal, J., Cerf, N.J.: Reconciliation of a quantum-distributed Gaussian key. IEEE TIT 50(2), 394–400 (2004). https://doi.org/10.1109/tit.2003.822618

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, D., Huang, P., Zhu, Y., Ma, H., Xiao, T., Wang, T., Zeng, G.: Unidimensional continuous-variable measurement-device-independent quantum key distribution. Quantum Inf. Process. 19(2), 53 (2020). https://doi.org/10.1007/s11128-019-2546-5

    Article  ADS  Google Scholar 

  3. Bai, Z., Yang, S., Li, Y.: High-efficiency reconciliation for continuous variable quantum key distribution. Jpn. J. Appl. Phys. 56(4), 44401 (2017). https://doi.org/10.7567/jjap.56.044401

    Article  Google Scholar 

  4. Bloch, M., Thangaraj, A., McLaughlin, S., Merolla, J.M.: LDPC-based Gaussian key reconciliation. In: 2006 IEEE Information Theory Workshop, pp. 116–120. IEEE, Punta del Este, Uruguay (2006). https://doi.org/10.1109/ITW.2006.1633793

  5. Brádler, K., Weedbrook, C.: Security proof of continuous-variable quantum key distribution using three coherent states. Phys. Rev. A 97(2), 1–16 (2018). https://doi.org/10.1103/PhysRevA.97.022310

    Article  Google Scholar 

  6. Brassard, G., Salvail, L.: Secret-key reconciliation by public discussion. In: EUROCRYPT, pp. 410–423. Springer, Berlin (1994). https://doi.org/10.1007/3-540-48285-735

  7. Cerf, N.J., Leuchs, G., Polzik, E.S.: Quantum information with continuous variables of atoms and light. Icp (2007). https://doi.org/10.1142/P489

    Article  MATH  Google Scholar 

  8. Sun, F.-W., van Tilborg, H.: Approaching capacity by equiprobable signaling on the Gaussian channel. IEEE Tran. Inf. Theory 39(5), 1714–1716 (1993). https://doi.org/10.1109/18.259663

    Article  MATH  Google Scholar 

  9. Ghorai, S., Grangier, P., Diamanti, E., Leverrier, A.: Asymptotic security of continuous-variable quantum key distribution with a discrete modulation. Phys. Rev. X 9(2), 021059 (2019). https://doi.org/10.1103/PhysRevX.9.021059

    Article  Google Scholar 

  10. Grosshans, F., Grangier, P.: Continuous variable quantum cryptography using coherent states. Phys. Rev. Lett. 88(5), 57902 (2002). https://doi.org/10.1103/PhysRevLett.88.057902

    Article  ADS  Google Scholar 

  11. Jouguet, P., Elkouss, D., Kunz-Jacques, S.: High-bit-rate continuous-variable quantum key distribution. Phys. Rev. A 90(4), 42329 (2014)

    Article  ADS  Google Scholar 

  12. 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). https://doi.org/10.1002/qute.201800011

    Article  Google Scholar 

  13. Leonhardt, U.: Measuring the Quantum State of Light No. 22 in Cambridge Studies in Modern Optics. Cambridge University Press, New York (1997)

  14. Leverrier, A., Grangier, P.: Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation. Phys. Rev. Lett. (2009). https://doi.org/10.1103/PhysRevLett.102.180504

    Article  MATH  Google Scholar 

  15. Liao, Q., Guo, Y., Xie, C., Huang, D., Huang, P., Zeng, G.: Composable security of unidimensional continuous-variable quantum key distribution. Quantum Inf. Process. 17(5), 113 (2018). https://doi.org/10.1007/s11128-018-1881-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Marie, A., Alléaume, R.: Self-coherent phase reference sharing for continuous-variable quantum key distribution. Phys. Rev. A 95(1), 12316 (2017). https://doi.org/10.1103/PhysRevA.95.012316

    Article  ADS  Google Scholar 

  17. Nguyen, K.C., Van Assche, G., Cerf, N.J.: Side-information coding with turbo codes and its application to quantum key distribution. In: International Symposium on Information Theory and Its Applications. Parma, Italy (2004)

  18. Papanastasiou, P., Lupo, C., Weedbrook, C., Pirandola, S.: Quantum key distribution with phase-encoded coherent states: asymptotic security analysis in thermal-loss channels. Phys. Rev. A 98(1), 012340 (2018). https://doi.org/10.1103/PhysRevA.98.012340

    Article  ADS  Google Scholar 

  19. Pilori, D.: Advanced Digital Signal Processing Techniques for High-Speed Optical Communications Links. Ph.D. Thesis, Politecnico di Torino (2019)

  20. Qu, Z., Djordjevic, I.B., Anderson, J.: Two-dimensional constellation shaping in fiber-optic communications. Appl. Sci. 9(9), 1889 (2019). https://doi.org/10.3390/app9091889

    Article  Google Scholar 

  21. Ralph, T.C.: Continuous variable quantum cryptography. Phys. Rev. A (1999). https://doi.org/10.1103/physreva.61.010303

    Article  Google Scholar 

  22. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. (1948). https://doi.org/10.1002/j.1538-7305.1948.tb00917.x

    Article  MathSciNet  MATH  Google Scholar 

  23. Ungerboeck, G.: Channel coding with multilevel/phase signals. IEEE Trans. Inf. Theory 28(1), 55–67 (1982). https://doi.org/10.1109/TIT.1982.1056454

    Article  MathSciNet  MATH  Google Scholar 

  24. Usenko, V.C., Grosshans, F.: Unidimensional continuous-variable quantum key distribution. Phys. Rev. A 92(6), 062337 (2015). https://doi.org/10.1103/PhysRevA.92.062337

    Article  ADS  MathSciNet  Google Scholar 

  25. Wang, P., Wang, X., Li, J., Li, Y.: Finite-size analysis of unidimensional continuous-variable quantum key distribution under realistic conditions. Opt. Express 25(23), 27995 (2017). https://doi.org/10.1364/OE.25.027995

    Article  ADS  Google Scholar 

  26. Wolf, M.M., Giedke, G., Cirac, J.I.: Extremality of Gaussian quantum states. Phys. Rev. Lett. 96(8), 080502 (2006). https://doi.org/10.1103/PhysRevLett.96.080502

    Article  ADS  MathSciNet  Google Scholar 

  27. Wu, Y., Verdú, S.: The impact of constellation cardinality on Gaussian channel capacity. In: 2010 48th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2010, pp. 620–628 (2010). https://doi.org/10.1109/ALLERTON.2010.5706965

  28. Yin, H.L., Zhu, W., Fu, Y.: Phase self-aligned continuous-variable measurement-device-independent quantum key distribution. Sci. Rep. 9(1), 49 (2019). https://doi.org/10.1038/s41598-018-36366-4

    Article  ADS  Google Scholar 

  29. Zhao, W., Guo, Y., Zhang, L., Huang, D.: Phase noise estimation using Bayesian inference for continuous-variable quantum key distribution. Opt. Express 27(3), 1838 (2019). https://doi.org/10.1364/OE.27.001838

    Article  ADS  Google Scholar 

  30. Zhao, W., Shi, R., Feng, Y., Huang, D.: Unidimensional continuous-variable quantum key distribution with discrete modulation. Phys. Lett. A 384(2), 126061 (2020). https://doi.org/10.1016/j.physleta.2019.126061

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao, Y.B., Heid, M., Rigas, J., Lütkenhaus, N.: Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks. Phys. Rev. A 79(1), 1–14 (2009). https://doi.org/10.1103/PhysRevA.79.012307

    Article  Google Scholar 

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Acknowledgements

This work was performed in the Institute for Studies in Quantum Information and Computation (iQuanta) at the Federal University of Campina Grande (UFCG), Paraíba, Brazil, and supported in part by the National Council for Scientific and Technological Development (CNPq) under research Grant No. 305918/2019-2 and the Coordination of Superior Level Staff Improvement (CAPES/PROEX).

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  1. Department of Electrical Engineering, Federal University of Campina Grande, Campina Grande, Brazil

    Micael A. Dias & Francisco M. de Assis

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Dias, M.A., de Assis, F.M. The impact of constellation cardinality on discrete unidimensional CVQKD protocols. Quantum Inf Process 20, 284 (2021). https://doi.org/10.1007/s11128-021-03222-w

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