## Efficient decoy-state quantum key distribution with quantified security |

Optics Express, Vol. 21, Issue 21, pp. 24550-24565 (2013)

http://dx.doi.org/10.1364/OE.21.024550

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### Abstract

We analyse the finite-size security of the efficient Bennett-Brassard 1984 protocol implemented with decoy states and apply the results to a gigahertz-clocked quantum key distribution system. Despite the enhanced security level, the obtained secure key rates are the highest reported so far at all fibre distances.

© 2013 Optical Society of America

## 1. Introduction

1. S. Wiesner, “Conjugate coding,” Sigact News **15**(1), 78–88 (1983). [CrossRef]

5. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. **81**(3), 1301–1350 (2009). [CrossRef]

6. P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Enhanced single-photon fringe visibility in a 10 km-long prototype quantum cryptography,” Electron. Lett. **29**(14), 1291 (1993). [CrossRef]

10. B. Fröhlich, J. F. Dynes, M. Lucamarini, A. W. Sharpe, Z. Yuan, and A. J. Shields, “A quantum access network,” Nature **501**(7465), 69–72 (2013). [CrossRef] [PubMed]

11. D. Mayers, D. Coppersmith, ed., in *Advances in Cryptology**:**Proceedings of CRYPTO '95*, vol. **963** of *Lecture Notes in Computer Science*, D. Coppersmith, Ed. (Springer-Verlag, 1995), pp. 124–135; in *Advances in Cryptology: Proceedings of CRYPTO '96*, vol. **1109** of *Lecture Notes in Computer Science*, N. Koblitz, Ed. (Springer-Verlag, 1996), pp. 343–357.

18. M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Renner, “Tight finite-key analysis for quantum cryptography,” Nat Commun **3**, 634 (2012). [CrossRef] [PubMed]

19. R. Renner, N. Gisin, and B. Kraus, “Information-theoretic security proof for quantum-key-distribution protocols,” Phys. Rev. A **72**(1), 012332 (2005). [CrossRef]

20. B. Kraus, N. Gisin, and R. Renner, “Lower and upper bounds on the secret-key rate for quantum key distribution protocols using one-way classical communication,” Phys. Rev. Lett. **95**(8), 080501 (2005). [CrossRef] [PubMed]

21. H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. of Crypt. **18**(2), 133–165 (2005). [CrossRef]

22. W.-Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. **91**(5), 057901 (2003). [CrossRef] [PubMed]

25. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A **72**(1), 012326 (2005). [CrossRef]

16. V. Scarani and R. Renner, “Quantum cryptography with finite resources: unconditional security bound for discrete-variable protocols with one-way postprocessing,” Phys. Rev. Lett. **100**(20), 200501 (2008). [CrossRef] [PubMed]

26. R. Y. Q. Cai and V. Scarani, “Finite-key analysis for practical implementations of quantum key distribution,” New J. Phys. **11**(4), 045024 (2009). [CrossRef]

29. Z. Wei, W. Wang, Z. Zhang, M. Gao, Z. Ma, and X. Ma, “Decoy-state quantum key distribution with biased basis choice,” Sci Rep **3**, 2453 (2013). [CrossRef] [PubMed]

19. R. Renner, N. Gisin, and B. Kraus, “Information-theoretic security proof for quantum-key-distribution protocols,” Phys. Rev. A **72**(1), 012332 (2005). [CrossRef]

20. B. Kraus, N. Gisin, and R. Renner, “Lower and upper bounds on the secret-key rate for quantum key distribution protocols using one-way classical communication,” Phys. Rev. Lett. **95**(8), 080501 (2005). [CrossRef] [PubMed]

30. N. J. Beaudry, T. Moroder, and N. Lütkenhaus, “Squashing models for optical measurements in quantum communication,” Phys. Rev. Lett. **101**(9), 093601 (2008). [CrossRef] [PubMed]

31. T. Tsurumaru and K. Tamaki, “Security proof for QKD systems with threshold detectors,” Phys. Rev. A **78**, 032302 (2008). [CrossRef]

33. M. Koashi and J. Preskill, “Secure quantum key distribution with an uncharacterized source,” Phys. Rev. Lett. **90**(5), 057902 (2003). [CrossRef] [PubMed]

26. R. Y. Q. Cai and V. Scarani, “Finite-key analysis for practical implementations of quantum key distribution,” New J. Phys. **11**(4), 045024 (2009). [CrossRef]

^{6}bits in order to provide a positive key rate. This minimum sample size becomes considerably worse, in fact more than 16 times larger, if the same simulation is run with experimental parameters similar to the ones presented in this work. Moreover, even for a reasonably large sample of 10

^{8}bits, the secure key rate is reduced to half its asymptotic value.

26. R. Y. Q. Cai and V. Scarani, “Finite-key analysis for practical implementations of quantum key distribution,” New J. Phys. **11**(4), 045024 (2009). [CrossRef]

^{8}bits and remains positive for sample sizes as small as 1.4 × 10

^{5}bits.

## 2. Protocol

25. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A **72**(1), 012326 (2005). [CrossRef]

_{Z}ñ, |1

_{Z}ñ (Z basis) and |0

_{X}ñ= (|0

_{Z}ñ+|1

_{Z}ñ)/√2, |1

_{X}ñ= (|0

_{Z}ñ-|1

_{Z}ñ)/√2 (X basis). The bases Z and X are selected with probabilities

*sifting*, to select the non-empty counts with matching bases;

*error correction*(EC), to determine the number of transmission errors,

*privacy amplification*(PA), to remove from a potential eavesdropper (Eve) the information which has possibly leaked to her;

*authentication*and

*verification*, to prevent man-in-the-middle attacks and guarantee that the users strings match with probability arbitrarily close to 1.

*advanced data analysis*necessary for the T12 protocol. Whenever the bases do not match, the data are discarded through the sifting procedure. From the results with matching bases, the quantities summarized in Table 1 can be drawn.

## 3. Secure key rate

16. V. Scarani and R. Renner, “Quantum cryptography with finite resources: unconditional security bound for discrete-variable protocols with one-way postprocessing,” Phys. Rev. Lett. **100**(20), 200501 (2008). [CrossRef] [PubMed]

**11**(4), 045024 (2009). [CrossRef]

16. V. Scarani and R. Renner, “Quantum cryptography with finite resources: unconditional security bound for discrete-variable protocols with one-way postprocessing,” Phys. Rev. Lett. **100**(20), 200501 (2008). [CrossRef] [PubMed]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**11**(4), 045024 (2009). [CrossRef]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

*rate per detected qubit*

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**11**(4), 045024 (2009). [CrossRef]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**11**(4), 045024 (2009). [CrossRef]

23. X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. **94**(23), 230503 (2005). [CrossRef] [PubMed]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

**100**(20), 200501 (2008). [CrossRef] [PubMed]

^{−10}in the present work. We choose and fix the numerical values of all the epsilon values so to fulfil the chain relation

**100**(20), 200501 (2008). [CrossRef] [PubMed]

## 4. Finite-size statistical analysis and parameter estimation

24. H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. **94**(23), 230504 (2005). [CrossRef] [PubMed]

23. X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. **94**(23), 230503 (2005). [CrossRef] [PubMed]

23. X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. **94**(23), 230503 (2005). [CrossRef] [PubMed]

25. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A **72**(1), 012326 (2005). [CrossRef]

**11**(4), 045024 (2009). [CrossRef]

29. Z. Wei, W. Wang, Z. Zhang, M. Gao, Z. Ma, and X. Ma, “Decoy-state quantum key distribution with biased basis choice,” Sci Rep **3**, 2453 (2013). [CrossRef] [PubMed]

40. C. Clopper and E. S. Pearson, “The use of confidence or fiducial limits illustrated in the case of the binomial,” Biometrika **26**(4), 404–413 (1934). [CrossRef]

**72**(1), 012326 (2005). [CrossRef]

41. R. Renner, “Symmetry of large physical systems implies independence of subsystems,” Nat. Phys. **3**(9), 645–649 (2007). [CrossRef]

## 5. Experimental implementation and numerical simulation

^{−5}and after pulse probability 5.25%. A software program controls all the equipment continuously, calculates the QBER for the fibre-stretcher to counteract any drift in the phase and corrects the detector gate delay and polarisation so as to maximise the count rate. The average photon number is stabilised using the feedback from a power meter connected to the main channel through a beam splitter with fixed known splitting ratio.

^{8}counts, the protocol performs at about 85% of its asymptotic value. Moreover, the key rate remains positive up to a sample size of 1.4×10

^{5}counts. With the same experimental parameters, the security proof by Cai and Scarani [26

**11**(4), 045024 (2009). [CrossRef]

^{8}counts, and 1.6×10

^{7}counts as the minimum size tolerated by the protocol. Therefore our model is significantly more resilient to finite size effects. In Fig. 2, we also show two pie-charts with the breakdown of the counts collected in the experiment, one for a large data sample (10

^{11}counts) and the other for the minimum data sample tolerated by the protocol (1.4×10

^{5}counts). The total detected sample is reduced by basis sifting, EC and PA applied to multi-photon events and phase-errors before reaching the final secure key fraction. The detrimental contribution of the finite-size effects becomes apparent for smaller sample sizes.

## 6. Experimental results

*advanced*data analysis of the experimental sample, which takes into account not only the basis information [21

21. H.-K. Lo, H. F. Chau, and M. Ardehali, “Efficient quantum key distribution scheme and proof of its unconditional security,” J. of Crypt. **18**(2), 133–165 (2005). [CrossRef]

44. C. Erven, X. Ma, R. Laﬂamme, and G. Weihs, “Entangled quantum key distribution with a biased basis choice,” New J. Phys. **11**(4), 045025 (2009). [CrossRef]

44. C. Erven, X. Ma, R. Laﬂamme, and G. Weihs, “Entangled quantum key distribution with a biased basis choice,” New J. Phys. **11**(4), 045025 (2009). [CrossRef]

46. D. Rosenberg, C. G. Peterson, J. W. Harrington, P. R. Rice, N. Dallmann, K. T. Tyagi, K. P. McCabe, S. Nam, B. Baek, R. H. Hadfield, R. J. Hughes, and J. E. Nordholt, “Practical long-distance quantum key distribution system using decoy levels,” New J. Phys. **11**(4), 045009 (2009). [CrossRef]

46. D. Rosenberg, C. G. Peterson, J. W. Harrington, P. R. Rice, N. Dallmann, K. T. Tyagi, K. P. McCabe, S. Nam, B. Baek, R. H. Hadfield, R. J. Hughes, and J. E. Nordholt, “Practical long-distance quantum key distribution system using decoy levels,” New J. Phys. **11**(4), 045009 (2009). [CrossRef]

44. C. Erven, X. Ma, R. Laﬂamme, and G. Weihs, “Entangled quantum key distribution with a biased basis choice,” New J. Phys. **11**(4), 045025 (2009). [CrossRef]

## 7. Conclusion

**100**(20), 200501 (2008). [CrossRef] [PubMed]

^{8}bits, finite-size effects reduce the asymptotic key rate by about 15%. For higher sample sizes, the reduction becomes negligible.

^{9}counts in a typical session of 20 minutes thus providing a key rate that is not appreciably affected by the finiteness of the sample. Despite its large security level, represented by a failure probability ε = 10

^{−10}, the system provides the highest secure key rates reported to date over tens of kilometres in optical fibre.

## Appendix

## Acknowledgments

## References and links

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**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.4510) Fiber optics and optical communications : Optical communications

(270.5568) Quantum optics : Quantum cryptography

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: August 2, 2013

Revised Manuscript: September 20, 2013

Manuscript Accepted: September 24, 2013

Published: October 7, 2013

**Citation**

M. Lucamarini, K. A. Patel, J. F. Dynes, B. Fröhlich, A. W. Sharpe, A. R. Dixon, Z. L. Yuan, R. V. Penty, and A. J. Shields, "Efficient decoy-state quantum key distribution with quantified security," Opt. Express **21**, 24550-24565 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24550

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### References

- S. Wiesner, “Conjugate coding,” Sigact News15(1), 78–88 (1983). [CrossRef]
- C. H. Bennett and G. Brassard, “Quantum cryptography: public-key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1984), pp. 175–179.
- A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett.67(6), 661–663 (1991). [CrossRef] [PubMed]
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys.74(1), 145–195 (2002). [CrossRef]
- V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys.81(3), 1301–1350 (2009). [CrossRef]
- P. D. Townsend, J. G. Rarity, and P. R. Tapster, “Enhanced single-photon fringe visibility in a 10 km-long prototype quantum cryptography,” Electron. Lett.29(14), 1291 (1993). [CrossRef]
- M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys.11(7), 075001 (2009). [CrossRef]
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- M. Hayashi, “Upper bounds of eavesdropper’s performances in finite-length code with the decoy method,” Phys. Rev. A76(1), 012329 (2007). [CrossRef]
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