Abstract

We propose a method to prepare different non-Poissonian signal pulses from sources of Poissonian photon number distribution, using only linear optical elements and threshold photon detectors. This method allows a simple passive preparation of decoy states for quantum key distribution. We show that the resulting key rates are comparable with the performance of active choices of intensities of Poissonian signals.

© 2009 Optical Society of America

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  1. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
    [CrossRef]
  2. W.-Y. Hwang, Phys. Rev. Lett. 91, 057901 (2003).
    [CrossRef] [PubMed]
  3. H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
    [CrossRef] [PubMed]
  4. X.-B. Wang, Phys. Rev. Lett. 94, 230503 (2005).
    [CrossRef] [PubMed]
  5. Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
    [CrossRef] [PubMed]
  6. D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
    [CrossRef] [PubMed]
  7. T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
    [CrossRef] [PubMed]
  8. W. Mauerer and C. Silberhorn, Phys. Rev. A 75, 050305(R) (2007).
    [CrossRef]
  9. Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
    [CrossRef] [PubMed]
  10. X. Ma and H.-K. Lo, New J. Phys. 10, 073018 (2008).
    [CrossRef]
  11. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).
  12. D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).
  13. H.-K. Lo, Quantum Inf. Comput. 5, 413 (2005).
  14. C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), p. 175.
  15. H.-K. Lo, H. F. C. Chau, and M. Ardehali, J. Cryptology 18, 133 (2005).
    [CrossRef]
  16. X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
    [CrossRef]
  17. C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004).
    [CrossRef]

2009

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

2008

X. Ma and H.-K. Lo, New J. Phys. 10, 073018 (2008).
[CrossRef]

2007

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

W. Mauerer and C. Silberhorn, Phys. Rev. A 75, 050305(R) (2007).
[CrossRef]

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
[CrossRef] [PubMed]

2006

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

2005

H.-K. Lo, Quantum Inf. Comput. 5, 413 (2005).

H.-K. Lo, H. F. C. Chau, and M. Ardehali, J. Cryptology 18, 133 (2005).
[CrossRef]

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
[CrossRef]

H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
[CrossRef] [PubMed]

X.-B. Wang, Phys. Rev. Lett. 94, 230503 (2005).
[CrossRef] [PubMed]

2004

C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004).
[CrossRef]

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).

2003

W.-Y. Hwang, Phys. Rev. Lett. 91, 057901 (2003).
[CrossRef] [PubMed]

Adachi, Y.

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
[CrossRef] [PubMed]

Ardehali, M.

H.-K. Lo, H. F. C. Chau, and M. Ardehali, J. Cryptology 18, 133 (2005).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

Bennett, C. H.

C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), p. 175.

Brassard, G.

C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), p. 175.

Cerf, N. J.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

Chau, H. F. C.

H.-K. Lo, H. F. C. Chau, and M. Ardehali, J. Cryptology 18, 133 (2005).
[CrossRef]

Chen, K.

H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
[CrossRef] [PubMed]

Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

Fürst, M.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Gobby, C.

C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004).
[CrossRef]

Gottesman, D.

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).

Harrington, J. W.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Hiskett, P. A.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Hughes, R. J.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Hwang, W.-Y.

W.-Y. Hwang, Phys. Rev. Lett. 91, 057901 (2003).
[CrossRef] [PubMed]

Imoto, N.

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
[CrossRef] [PubMed]

Koashi, M.

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
[CrossRef] [PubMed]

Kurtsiefer, C.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Lita, A. E.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Lo, H.-K.

X. Ma and H.-K. Lo, New J. Phys. 10, 073018 (2008).
[CrossRef]

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
[CrossRef] [PubMed]

H.-K. Lo, H. F. C. Chau, and M. Ardehali, J. Cryptology 18, 133 (2005).
[CrossRef]

H.-K. Lo, Quantum Inf. Comput. 5, 413 (2005).

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
[CrossRef]

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).

Lütkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).

Ma, X.

X. Ma and H.-K. Lo, New J. Phys. 10, 073018 (2008).
[CrossRef]

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
[CrossRef] [PubMed]

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
[CrossRef]

Mauerer, W.

W. Mauerer and C. Silberhorn, Phys. Rev. A 75, 050305(R) (2007).
[CrossRef]

Nam, S. W.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Nordholt, J. E.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

Perdigues, J.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Peterson, C. G.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Preskill, J.

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).

Qi, B.

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
[CrossRef]

Qian, L.

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

Rarity, J. G.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Rice, P. R.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Rosenberg, D.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

Scheidl, T.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Schmitt-Manderbach, T.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Shields, A. J.

C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004).
[CrossRef]

Silberhorn, C.

W. Mauerer and C. Silberhorn, Phys. Rev. A 75, 050305(R) (2007).
[CrossRef]

Sodnik, Z.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Tiefenbacher, F.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Ursin, R.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Wang, X.-B.

X.-B. Wang, Phys. Rev. Lett. 94, 230503 (2005).
[CrossRef] [PubMed]

Weier, H.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Weinfurter, H.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Yamamoto, T.

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
[CrossRef] [PubMed]

Yuan, Z. L.

C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004).
[CrossRef]

Zeilinger, A.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Zhao, Y.

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
[CrossRef]

Appl. Phys. Lett.

C. Gobby, Z. L. Yuan, and A. J. Shields, Appl. Phys. Lett. 84, 3762 (2004).
[CrossRef]

J. Cryptology

H.-K. Lo, H. F. C. Chau, and M. Ardehali, J. Cryptology 18, 133 (2005).
[CrossRef]

New J. Phys.

X. Ma and H.-K. Lo, New J. Phys. 10, 073018 (2008).
[CrossRef]

Phys. Rev. A

W. Mauerer and C. Silberhorn, Phys. Rev. A 75, 050305(R) (2007).
[CrossRef]

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Phys. Rev. A 72, 012326 (2005).
[CrossRef]

Phys. Rev. Lett.

Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. Lett. 99, 180503 (2007).
[CrossRef] [PubMed]

W.-Y. Hwang, Phys. Rev. Lett. 91, 057901 (2003).
[CrossRef] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, Phys. Rev. Lett. 94, 230504 (2005).
[CrossRef] [PubMed]

X.-B. Wang, Phys. Rev. Lett. 94, 230503 (2005).
[CrossRef] [PubMed]

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Phys. Rev. Lett. 98, 010503 (2007).
[CrossRef] [PubMed]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Phys. Rev. Lett. 98, 010504 (2007).
[CrossRef] [PubMed]

Quantum Inf. Comput.

D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).

H.-K. Lo, Quantum Inf. Comput. 5, 413 (2005).

Rev. Mod. Phys.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
[CrossRef]

Other

C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), p. 175.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Cited By

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Figures (3)

Fig. 1
Fig. 1

Interference of two phase randomized WCPs, ρ μ 1 and ρ μ 2 , at a BS of transmittance t; a and b represent the two output modes.

Fig. 2
Fig. 2

(A) Conditional photon number distribution in mode a (see Fig. 1): r n c (black) and r n c ¯ (gray) for the case μ 1 = μ 2 = 1 and t = 1 2 . (B) r n c (black) versus a Poissonian distribution of the same mean photon number (gray).

Fig. 3
Fig. 3

Lower bound on the secret key rate R given by Eq. (4) on a logarithmic scale for a passive decoy state setup with two intensity settings (dashed curve). The transmittance of the BS is t = 1 2 . The solid curve represents a lower bound on R for an active asymptotic decoy state system [3].

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

p n , m = υ n + m e υ n ! m ! 1 2 π 0 2 π γ n ( 1 γ ) m d θ .
p n t = m = 0 p n , m = υ n n ! 1 2 π 0 2 π γ n e υ γ d θ ,
p n c ¯ = p n , 0 = υ n e υ n ! 1 2 π 0 2 π γ n d θ .
R max { R c , 0 } + max { R c ¯ , 0 } ,
Y 1 Y 1 l max { 0 , p 2 c ¯ Q t p 2 t Q c ¯ ( p 2 c ¯ p 0 t p 2 t p 0 c ¯ ) Y 0 u p 2 c ¯ p 1 t p 2 t p 1 c ¯ } ,
e 1 e 1 u min { E c ¯ Q c ¯ p 0 c ¯ Y 0 l e 0 p 1 c ¯ Y 1 l , E c Q c p 0 c Y 0 l e 0 p 1 c Y 1 l , p 0 c ¯ E t Q t p 0 t E c ¯ Q c ¯ ( p 0 c ¯ p 1 t p 0 t p 1 c ¯ ) Y 1 l } ,

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