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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    [CrossRef] [PubMed]
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    [CrossRef]
  10. F. Grosshans, "Collective Attacks and Unconditional Security in Continuous Variable Quantum Key Distribution," Phys. Rev. Lett. 94, 020504 (2005).
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  12. R. Garcıa-Patron 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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    [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. Lutkenhaus, "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. Dusek, N. Lutkenhaus, and M. Peev, "Title: A Framework for Practical Quantum Cryptography"arXiv:0802.4155(2008).
  17. Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, "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. Lutkenhaus, 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, P. K. Lam, "No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light," Phys. Rev. Lett. 95, 180503 (2005).
    [CrossRef] [PubMed]
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2009 (1)

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

2007 (2)

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. Lutkenhaus, "Security of coherent-state quantum cryptography in the presence of Gaussian noise" Phys. Rev. A 76, 022313 (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. Garcıa-Patron and N. J. Cerf, "Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution," Phys. Rev. Lett. 97, 190503 (2006).
[CrossRef] [PubMed]

M. Navascues, F. Grosshans, and A. Acın, "Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography," Phys. Rev. Lett. 97, 190502 (2006).
[CrossRef] [PubMed]

2005 (4)

F. Grosshans, "Collective Attacks and Unconditional Security in Continuous Variable Quantum Key Distribution," Phys. Rev. Lett. 94, 020504 (2005).
[CrossRef] [PubMed]

M. Navascues and A. Acın, "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, 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. Lutkenhaus, and G. Leuchs, "Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit" Phys. Rev. A 89, 167901 (2002).

2001 (1)

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

1995 (1)

J. Janszky, P. Domokos, S. Szabo, 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]

Acin, A.

M. Navascues, F. Grosshans, and A. Acın, "Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography," Phys. Rev. Lett. 97, 190502 (2006).
[CrossRef] [PubMed]

M. Navascues and A. Acın, "SecurityBounds for Continuous Variables Quantum Key Distribution," Phys. Rev. Lett. 94, 020505 (2005).
[CrossRef] [PubMed]

Adam, P.

J. Janszky, P. Domokos, S. Szabo, and P. Adam, "Quantum-state engineering via discrete coherent-state superpositions," Phys. Rev. A 51, 5 (1995).
[CrossRef]

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]

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]

R. Garcıa-Patron 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. Levy, and G. Van Assche, "Quantum distribution of Gaussian keys using squeezed states," Phys. Rev. A 63, 052311 (2001).
[CrossRef]

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. Szabo, and P. Adam, "Quantum-state engineering via discrete coherent-state superpositions," Phys. Rev. A 51, 5 (1995).
[CrossRef]

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]

R. Garcıa-Patron 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]

Grosshans, F.

M. Navascues, F. Grosshans, and A. Acın, "Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography," Phys. Rev. Lett. 97, 190502 (2006).
[CrossRef] [PubMed]

F. Grosshans, "Collective Attacks and Unconditional Security in Continuous Variable Quantum Key Distribution," 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]

Heid, M.

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

M. Heid and N. Lutkenhaus, "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. Szabo, and P. Adam, "Quantum-state engineering via discrete coherent-state superpositions," Phys. Rev. A 51, 5 (1995).
[CrossRef]

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]

Lam, P. K.

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, P. K. Lam, "No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light," Phys. Rev. Lett. 95, 180503 (2005).
[CrossRef] [PubMed]

Lance, A. M.

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, 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. Lutkenhaus, and G. Leuchs, "Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit" Phys. Rev. A 89, 167901 (2002).

Levy, M.

N. J. Cerf, M. Levy, and G. Van Assche, "Quantum distribution of Gaussian keys using squeezed states," Phys. Rev. A 63, 052311 (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]

Lutkenhaus, N.

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

M. Heid and N. Lutkenhaus, "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. Lutkenhaus, and G. Leuchs, "Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit" Phys. Rev. A 89, 167901 (2002).

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]

Navascues, M.

M. Navascues, F. Grosshans, and A. Acın, "Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography," Phys. Rev. Lett. 97, 190502 (2006).
[CrossRef] [PubMed]

M. Navascues and A. Acın, "SecurityBounds for Continuous Variables Quantum Key Distribution," Phys. Rev. Lett. 94, 020505 (2005).
[CrossRef] [PubMed]

Ralph, T. C.

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, P. K. Lam, "No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light," Phys. Rev. Lett. 95, 180503 (2005).
[CrossRef] [PubMed]

Ch. Silberhorn, T. C. Ralph, N. Lutkenhaus, 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. Lutkenhaus, "Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks," Phys. Rev. A 79, 012307 (2009).
[CrossRef]

Sharma, V.

A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, 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. Lutkenhaus, and G. Leuchs, "Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit" Phys. Rev. A 89, 167901 (2002).

Symul, T.

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, P. K. Lam, "No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light," Phys. Rev. Lett. 95, 180503 (2005).
[CrossRef] [PubMed]

Szabo, S.

J. Janszky, P. Domokos, S. Szabo, 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. Levy, 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]

Weedbrook, C.

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, 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]

Zhao, Y.

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

Nature (1)

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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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