Abstract

We propose a continuous variable based quantum key distribution protocol that makes use of discretely signaled coherent light and reverse error reconciliation. We present a rigorous security proof against collective attacks with realistic lossy, noisy quantum channels, imperfect detector efficiency, and detector electronic noise. This protocol is promising for convenient, high-speed operation at link distances up to 50 km with the use of post-selection.

© 2009 Optical Society of America

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  1. C. H. Bennett and G. Grassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, Newyork, 1984), pp. 175–179.
  2. A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
    [Crossref] [PubMed]
  3. M. A. Nielsen and I. L. Chuang, ‘Quantum Computation and Quantum Information, (Cambridge University Press, UK, 2000).
  4. S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
    [Crossref]
  5. F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
    [Crossref] [PubMed]
  6. N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
    [Crossref]
  7. F. Grosshans and P. Grangier, “Reverse reconciliation protocols for quantum cryptography with continuous variables,”quant-ph/0204127 (2002).
  8. F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. 88, 057902 (2002)
    [Crossref] [PubMed]
  9. R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A 74, 032302 (2006).
    [Crossref]
  10. F. Grosshans, “CollectiveAttacks and Unconditional Security in Continuous Variable Quantum KeyDistribution,” Phys. Rev. Lett. 94, 020504 (2005).
    [Crossref] [PubMed]
  11. M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. 94, 020505 (2005).
    [Crossref] [PubMed]
  12. R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 97, 190503 (2006).
    [Crossref] [PubMed]
  13. M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
    [Crossref] [PubMed]
  14. J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
    [Crossref]
  15. M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A 76, 022313 (2007).
    [Crossref]
  16. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).
  17. Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
    [Crossref]
  18. Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).
  19. C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
    [Crossref]
  20. A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
    [Crossref] [PubMed]
  21. T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
    [Crossref]
  22. J. Singh, O. Dabeer, and U. Madhow, “Capacity of the Discrete-Time AWGN Channel Under Output Quantization,” arXiv:0801.1185v1 (2008).
  23. V. Buẑek and G. Drobný, “Quantum tomography via the MaxEnt principle,” J. Mod. Opt. 47, 14/15 pp. 2823–2839 (2000).
  24. R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory,  47, 1 pp 474–478 (2001).
    [Crossref]
  25. K. Khandekar and R. J. McEliece, “On the complexity of reliable communication on the erasure channel,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2001, p. 1.
  26. Z. Zhang and P. L. Voss, “A path towards 10 Gb/s continuous variable QKD,” LPHYS08, Trondheim, Norway. July 2008.
  27. J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
    [Crossref]
  28. R. Jozsa and J. Schlienz, “Distinguishability of States and von Neumann Entropy,” arXiv:quant-ph/9911009v1, 3 (1999).

2009 (1)

Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
[Crossref]

2007 (3)

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A 76, 022313 (2007).
[Crossref]

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

2006 (4)

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A 74, 032302 (2006).
[Crossref]

R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 97, 190503 (2006).
[Crossref] [PubMed]

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

2005 (4)

F. Grosshans, “CollectiveAttacks and Unconditional Security in Continuous Variable Quantum KeyDistribution,” Phys. Rev. Lett. 94, 020504 (2005).
[Crossref] [PubMed]

M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. 94, 020505 (2005).
[Crossref] [PubMed]

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

2003 (1)

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

2002 (2)

F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. 88, 057902 (2002)
[Crossref] [PubMed]

Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).

2001 (2)

N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory,  47, 1 pp 474–478 (2001).
[Crossref]

2000 (1)

V. Buẑek and G. Drobný, “Quantum tomography via the MaxEnt principle,” J. Mod. Opt. 47, 14/15 pp. 2823–2839 (2000).

1995 (1)

J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref] [PubMed]

Acín, A.

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. 94, 020505 (2005).
[Crossref] [PubMed]

Adam, P.

J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
[Crossref]

Ahlswede, R.

R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory,  47, 1 pp 474–478 (2001).
[Crossref]

Alton, D. J.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

Assad, S. M.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

Bennett, C. H.

C. H. Bennett and G. Grassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, Newyork, 1984), pp. 175–179.

Bloch, M.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Bowen, W. P.

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Brouri, R.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

Bu?ek, V.

V. Buẑek and G. Drobný, “Quantum tomography via the MaxEnt principle,” J. Mod. Opt. 47, 14/15 pp. 2823–2839 (2000).

Cerf, N. J.

R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 97, 190503 (2006).
[Crossref] [PubMed]

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

Cerf, N.J.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, ‘Quantum Computation and Quantum Information, (Cambridge University Press, UK, 2000).

Dabeer, O.

J. Singh, O. Dabeer, and U. Madhow, “Capacity of the Discrete-Time AWGN Channel Under Output Quantization,” arXiv:0801.1185v1 (2008).

Debuisschert, T.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Diamanti, E.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Domokos, P.

J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
[Crossref]

Drobný, G.

V. Buẑek and G. Drobný, “Quantum tomography via the MaxEnt principle,” J. Mod. Opt. 47, 14/15 pp. 2823–2839 (2000).

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

Ekert, A. K.

A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref] [PubMed]

Fossier, S.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Garcia-Patron, R.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

García-Patrón, R.

R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 97, 190503 (2006).
[Crossref] [PubMed]

Grangier, P.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. 88, 057902 (2002)
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Reverse reconciliation protocols for quantum cryptography with continuous variables,”quant-ph/0204127 (2002).

Grassard, G.

C. H. Bennett and G. Grassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, Newyork, 1984), pp. 175–179.

Grosshans, F.

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

F. Grosshans, “CollectiveAttacks and Unconditional Security in Continuous Variable Quantum KeyDistribution,” Phys. Rev. Lett. 94, 020504 (2005).
[Crossref] [PubMed]

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. 88, 057902 (2002)
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Reverse reconciliation protocols for quantum cryptography with continuous variables,”quant-ph/0204127 (2002).

Heid, M.

Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
[Crossref]

M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A 76, 022313 (2007).
[Crossref]

Hirano, T.

R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A 74, 032302 (2006).
[Crossref]

Janszky, J.

J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
[Crossref]

Jozsa, R.

R. Jozsa and J. Schlienz, “Distinguishability of States and von Neumann Entropy,” arXiv:quant-ph/9911009v1, 3 (1999).

Karpov, E.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Khandekar, K.

K. Khandekar and R. J. McEliece, “On the complexity of reliable communication on the erasure channel,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2001, p. 1.

Lam, P. K.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Lance, A. M.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Leuchs, G.

Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).

Lévy, M.

N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

Lober, P.

R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory,  47, 1 pp 474–478 (2001).
[Crossref]

Lodewyck, J.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Lütkenhaus, N.

Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
[Crossref]

M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A 76, 022313 (2007).
[Crossref]

Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

Madhow, U.

J. Singh, O. Dabeer, and U. Madhow, “Capacity of the Discrete-Time AWGN Channel Under Output Quantization,” arXiv:0801.1185v1 (2008).

McEliece, R. J.

K. Khandekar and R. J. McEliece, “On the complexity of reliable communication on the erasure channel,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2001, p. 1.

McLaughlin, S. W.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Namiki, R.

R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A 74, 032302 (2006).
[Crossref]

Navascués, M.

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. 94, 020505 (2005).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, ‘Quantum Computation and Quantum Information, (Cambridge University Press, UK, 2000).

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

Ralph, T. C.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
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C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Ralph, T.C.

Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).

Rigas, J.

Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
[Crossref]

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

Schlienz, J.

R. Jozsa and J. Schlienz, “Distinguishability of States and von Neumann Entropy,” arXiv:quant-ph/9911009v1, 3 (1999).

Sharma, V.

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Silberhorn, Ch.

Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).

Singh, J.

J. Singh, O. Dabeer, and U. Madhow, “Capacity of the Discrete-Time AWGN Channel Under Output Quantization,” arXiv:0801.1185v1 (2008).

Symul, T.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Szabó, S.

J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
[Crossref]

Tualle-Brouri, R.

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

Van Assche, G.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

Van Loock, P.

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Voss, P. L.

Z. Zhang and P. L. Voss, “A path towards 10 Gb/s continuous variable QKD,” LPHYS08, Trondheim, Norway. July 2008.

Weedbrook, C.

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Wenger, J.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

Zhang, Z.

Z. Zhang and P. L. Voss, “A path towards 10 Gb/s continuous variable QKD,” LPHYS08, Trondheim, Norway. July 2008.

Zhao, Y.

Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
[Crossref]

IEEE Trans. Inf. Theory (1)

R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory,  47, 1 pp 474–478 (2001).
[Crossref]

J. Mod. Opt. (1)

V. Buẑek and G. Drobný, “Quantum tomography via the MaxEnt principle,” J. Mod. Opt. 47, 14/15 pp. 2823–2839 (2000).

Nature (1)

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

Phys. Rev. A (9)

N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76, 042305 (2007).
[Crossref]

M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A 76, 022313 (2007).
[Crossref]

Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79, 012307 (2009).
[Crossref]

Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A 89, 167901 (2002).

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A 73,022316 (2006).
[Crossref]

R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A 74, 032302 (2006).
[Crossref]

T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A 76, 030303(R) (2007).
[Crossref]

J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A 51, 5 (1995).
[Crossref]

Phys. Rev. Lett. (7)

F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. 88, 057902 (2002)
[Crossref] [PubMed]

F. Grosshans, “CollectiveAttacks and Unconditional Security in Continuous Variable Quantum KeyDistribution,” Phys. Rev. Lett. 94, 020504 (2005).
[Crossref] [PubMed]

M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. 94, 020505 (2005).
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R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. 97, 190503 (2006).
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M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
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A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
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A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. 95, 180503 (2005).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Other (8)

C. H. Bennett and G. Grassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, Newyork, 1984), pp. 175–179.

M. A. Nielsen and I. L. Chuang, ‘Quantum Computation and Quantum Information, (Cambridge University Press, UK, 2000).

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008).

F. Grosshans and P. Grangier, “Reverse reconciliation protocols for quantum cryptography with continuous variables,”quant-ph/0204127 (2002).

J. Singh, O. Dabeer, and U. Madhow, “Capacity of the Discrete-Time AWGN Channel Under Output Quantization,” arXiv:0801.1185v1 (2008).

R. Jozsa and J. Schlienz, “Distinguishability of States and von Neumann Entropy,” arXiv:quant-ph/9911009v1, 3 (1999).

K. Khandekar and R. J. McEliece, “On the complexity of reliable communication on the erasure channel,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2001, p. 1.

Z. Zhang and P. L. Voss, “A path towards 10 Gb/s continuous variable QKD,” LPHYS08, Trondheim, Norway. July 2008.

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Figures (6)

Fig. 1.
Fig. 1.

Alice’s encoding scheme in which she only sends four different coherent states.

Fig. 2.
Fig. 2.

Model of relevant quantum channels. ρ̂εn and ρ̂εr , density matrices produced by Eve’s EPR source; ρ̂a, density matrix of signal sent by Alice; ρ̂b and ρ̂ b, density matrix before Bob’s detector inefficiencies and after detector inefficiencies; ρ̂εn , density matrix post-beamsplitter, measured by Eve; ρ^ hom, density matrix of equivalent mode consisting of light lost to detector inefficiencies. τ is the squeezing parameter of EPR source, η is the channel efficiency, and η m is Bob’s detector efficiency.

Fig. 3.
Fig. 3.

Eve’s attack operator M̂ can be decomposed into three sub-operators Ô,P̂ and Q̂, which give the same output quantum states.

Fig. 4.
Fig. 4.

Bob’s decision rule under post-selection. Here σS =√VS and σel =√Vel .

Fig. 5.
Fig. 5.

The secrecy capacity and required reconciliation efficiency for the system without post-selection.

Fig. 6.
Fig. 6.

The secrecy capacity, required reconciliation efficiency and the error rate on the BSC channel of the system with post-selection

Tables (1)

Tables Icon

Table 1. Differences of the secrecy capacity with ΔIref . Here 25km denotes the case of 25 km QIQO CVQKD without post-selection. 25km-ps denotes the case of 25 km QIQO CVQKD with post-selection. 50 km denotes the case of 50 km QIQO CVQKD without post-selection. 50 km-ps denotes the case of 50km QIQO CVQKD with post-selection.

Equations (59)

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b̂=ηâ+1ηε̂n,
Q̂b=ηQ̂a+1ηQ̂εn.
Pb(q)=Pa(ηq)*Pεn(1ηq),
ΔI=I(A;B)χ(B;E),
ΔI=β I (A;B)χ(B;E).
β0I(A;B)χ(B;E)=0 .
eAB=112πSNRex22dx.
I(A;B)=1h(eAB),
χ(B;E)=S(ρ̂E)ΣipiS(ρ̂E|q=i),
χ(B;E)=S(ρ̂E)12S(ρ̂E|q=1)12S(ρ̂E|q=1)=S(ρ̂E)S(ρ̂E|q=1)
ρ̂E=Σi=14141ηαi1ηαi.
ρ̂E|q=1=Σi=14pi|q=1η1αiη1αi.
VB=VS+Vel.
SNR=ui2VB,
SNR=2{ηηmαi}Vs+Vel .
|Φi=M̂(ΨEαi) ,
ρbi=T rE (ΦiΦi) .
ρ̂εn=Trr[Ô(ΨEΨ)ô],
P̂=[η1η1ηη].
Q̂=M̂ÔP̂.
ρ̂εn=(1τ2)Σn=0τ2nnn.
ρ̂εn=Trεr(Ψεn,εrΨ) ,
Ψεn,εr=1τ2n=0τnnεnϕ(n)εr,
ΨA=Σi=1412αiaia,
|Φ=B̂b,hom(ηm)B̂a,εn(η)|ψA|Ψεn,εr|0hom,
ΞX=bXΦΦXbXΦ,
ρ̂EX=Tra,hom(ΞXΞX).
p(ρ̂EX)=ΦXbXΦ.
ρ̂E= p (ρ̂EX) ρ̂EX dx
ρ̂E|q=1= p (ρ̂EX|q=1)ρ̂EXdx.
p(ρ̂EX|q=1)=p(ρ̂EX)p(q=1|ρ̂EX)p(q=1).
p(q=1|ρ̂EX)=12πVelXexp(x22Vel)dx.
nth=(1τ2)Σn=0nτ2n=τ21τ2.
VB=VS+12(1η)ηmnth+Vel.
SNR=2{ηηmαi}Vs+12(1η)ηmτ21τ2+Vel.
p(ρ̂EX|q0)=p(q0|ρ̂EX)p(ρ̂EX)p(q0)
p(q0|ρ̂EX)=12πVel[(TX)exp(x22Vel)+(T+X)exp(x22Vel)].
p(q0)=12πVB[(Tui)exp(x22VB)+(T+ui)exp(x22VB)].
ρ̂EX= p (ρ̂EX|q0) ρ̂EX dx .
p(q=1|ρ̂EX)=12πVel(TX)exp(x22Vel)dx.
eAB=p(q=1Aliceencodesα1)p(q0)=(T+ui)exp(x22VB)dx(Tui)exp(x22VB)dx+(T+ui)exp(x22VB)dx.
|Ψεn,εr=1τ2Σn=0EMAXτnnεnϕ(n)εr.
|n,r=c (r)n!er22(n+1)rnΣk=0n e2πin+1k|re2πin+1k,
|n,r0=n.
1cn2=n!(2n+1)!r2(n+1)+o(r4(n+1)),
Φ=c(r)1τ2Σj=1412jaΣn=0EMAXτnnεrn!e(rn)22(n+1)(rn)nΣk=0ne2πikn+1ηm(ηαj+1ηrne2πikn+1)b
1ηm(ηαj+1ηrne2πikn+1)hom1ηαjηrne2πikn+1εn.
ρ̂b,εn,εr,hom=Tra(ΦΦ)=Σi=1414ΩiΩi,
ρ̂εn,εr,hom=Trb(ρ̂b,εn,εr,hom)=Σi=1414Σk=1XMAXp(Xi,k)Xi,kΩiΩiXi,kΩiXi,kXi,kΩi=Σi=1414Σk=1XMAXp (Xi,kψi,kψi,k) ,
Trhom(ρ̂εn,εr,homi,k)=Σj=0HMAXhomj|ψi,kψi,k|jhom.
ρ̂E = Σi=14 14 Σk=1XMAX p (Xi,k) Σj=0HMAX jψi,k ψi,kj = Σi=14 14 Σk=1XMAX p (Xi,k)Σj=0HMAXp(jXi,k)εi,j,kεi,j,k,
ρ̂E=Σi,j,kp(|εi,j,k) εi,j,kεi,j,k,
Gijk,ijk=p(|εi,j,k)p (|εi,j,k) εi,j,k|εi,j,k .
S(ρ̂E)=Σi=1nλilog2(λi).
p(εi,j,kq=1)=p(q=1||εi,j,k)p(|εi,j,k)p(q=1).
p(q=1||εi,j,k)=12πVSlbi,krbi,kexp[(xui)22VS]12πVel0exp[(yx)22Vel]dydx12πVSlbi,krbi,kexp[(xui)22VS]dx .
p(εi,j,kq0)=p(q0εi,j,k)p(|εi,j,k)p(q0) .
p(q0||εi,j,k)=12πVSlbi,krbi,kexp[(xui)22VS]12πVel(Texp[(yx)22Vel]+Texp[(yx)22Vel])dydx12πVSlbi,krbi,kexp[(xui)22VS]dx .
p(q=1||εi,j,k)=12πVSlbi,krbi,kexp[(xui)22VS]12πVelTexp[(yx)22Vel]dydx12πVSlbi,krbi,kexp[(xui)22VS]dx .

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