Abstract

We present a quantum fingerprinting protocol relying on two-photon interference which does not require a shared phase reference between the parties preparing optical signals carrying data fingerprints. We show that the scaling of the protocol, in terms of transmittable classical information, is analogous to the recently proposed and demonstrated scheme based on coherent pulses and first-order interference, offering comparable advantage over classical fingerprinting protocols without access to shared prior randomness. We analyze the protocol taking into account non-Poissonian photon statistics of optical signals and a variety of imperfections, such as transmission losses, dark counts, and residual distinguishability. The impact of these effects on the protocol performance is quantified with the help of Chernoff information.

© 2017 Optical Society of America

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References

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  1. H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
    [Crossref]
  2. J. M. Arrazola and N Lüthenhaus, “Quantum fingerprinting with coherent states and a constant mean number of photons,” Phys. Rev. A 89, 062305 (2014).
    [Crossref]
  3. N. Kumar, E. Diamanti, and I. Kerenidis, “Efficient quantum communications with coherent state fingerprints over multiple channels,” Phys. Rev. A 95, 032337 (2017).
    [Crossref]
  4. F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
    [Crossref]
  5. J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
    [Crossref] [PubMed]
  6. P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
    [Crossref] [PubMed]
  7. R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
    [Crossref] [PubMed]
  8. S. Massar, “Quantum fingerprinting protocol with a single particle,” Phys. Rev. A 71, 012310 (2005).
    [Crossref]
  9. K. Kitayama, Optical Code Division Multiple Access: A Practical Perspective (Cambridge University, 2014), Chap. 4.
    [Crossref]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Chap. 4.3.
    [Crossref]
  11. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [Crossref] [PubMed]
  12. J. C. Garcia-Escartin and P. Chamorro-Posada, “Swap test and Hong-Ou-Mandel effect are equivalent,” Phys. Rev. A 87, 052330 (2013).
    [Crossref]
  13. R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
    [Crossref] [PubMed]
  14. J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
    [Crossref]
  15. T. M. Cover and J. A. Thomas, Elements of Information Theory (2nd edition) (Wiley, 1991), Chap. 12.
    [Crossref]
  16. C. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2004), Chap. 6.
    [Crossref]
  17. R. Loudon, The Quantum Theory of Light (Oxford University, 2014), Chap. 6.8.
  18. J. H. van Lint, Introduction to Coding Theory (3rd edition) (Springer, 1987), Chap. 5.
  19. C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66, 481–537 (1994).
    [Crossref]
  20. R. G. Gallager, Stochastic Processes: Theory and Applications (Cambridge University Press, 2014), Chap. 8.
  21. C. Su and K. Wódkiewicz, “Quantum versus stochastic or hidden-variable fluctuations in two-photon interference effects,” Phys. Rev. A 44, 6097 (1991).
    [Crossref]
  22. R. Chrapkiewicz, M. Dabrowski, and W. Wasilewski, “High-Capacity Angularly Multiplexed Holographic Memory Operating at the Single-Photon Level,” Phys. Rev. Lett. 118, 063603 (2017).
    [Crossref] [PubMed]
  23. S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
    [Crossref]
  24. T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
    [Crossref]
  25. R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
    [Crossref]
  26. J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
    [Crossref] [PubMed]
  27. D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
    [Crossref]

2017 (2)

N. Kumar, E. Diamanti, and I. Kerenidis, “Efficient quantum communications with coherent state fingerprints over multiple channels,” Phys. Rev. A 95, 032337 (2017).
[Crossref]

R. Chrapkiewicz, M. Dabrowski, and W. Wasilewski, “High-Capacity Angularly Multiplexed Holographic Memory Operating at the Single-Photon Level,” Phys. Rev. Lett. 118, 063603 (2017).
[Crossref] [PubMed]

2016 (3)

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
[Crossref]

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

2015 (1)

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

2014 (2)

J. M. Arrazola and N Lüthenhaus, “Quantum fingerprinting with coherent states and a constant mean number of photons,” Phys. Rev. A 89, 062305 (2014).
[Crossref]

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

2013 (3)

J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
[Crossref] [PubMed]

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
[Crossref]

J. C. Garcia-Escartin and P. Chamorro-Posada, “Swap test and Hong-Ou-Mandel effect are equivalent,” Phys. Rev. A 87, 052330 (2013).
[Crossref]

2012 (1)

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

2006 (1)

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

2005 (2)

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

S. Massar, “Quantum fingerprinting protocol with a single particle,” Phys. Rev. A 71, 012310 (2005).
[Crossref]

2003 (1)

T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
[Crossref]

2001 (1)

H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
[Crossref]

1994 (1)

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66, 481–537 (1994).
[Crossref]

1991 (1)

C. Su and K. Wódkiewicz, “Quantum versus stochastic or hidden-variable fluctuations in two-photon interference effects,” Phys. Rev. A 44, 6097 (1991).
[Crossref]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Andersson, E.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Araneda, G.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Arrazola, J. M.

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

J. M. Arrazola and N Lüthenhaus, “Quantum fingerprinting with coherent states and a constant mean number of photons,” Phys. Rev. A 89, 062305 (2014).
[Crossref]

Babichev, S. A.

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

Banaszek, K.

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
[Crossref]

Blatt, R.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Buhrman, H.

H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
[Crossref]

Buller, G. S.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Castelli, F.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
[Crossref]

Caves, C. M.

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66, 481–537 (1994).
[Crossref]

Chamorro-Posada, P.

J. C. Garcia-Escartin and P. Chamorro-Posada, “Swap test and Hong-Ou-Mandel effect are equivalent,” Phys. Rev. A 87, 052330 (2013).
[Crossref]

Chen, S.-J.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Chen, T.-Y.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Chrapkiewicz, R.

R. Chrapkiewicz, M. Dabrowski, and W. Wasilewski, “High-Capacity Angularly Multiplexed Holographic Memory Operating at the Single-Photon Level,” Phys. Rev. Lett. 118, 063603 (2017).
[Crossref] [PubMed]

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
[Crossref]

Cialdi, S.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
[Crossref]

Clarke, P. J.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Cleve, R.

H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
[Crossref]

Collins, R. J.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (2nd edition) (Wiley, 1991), Chap. 12.
[Crossref]

Cristani, M.

J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
[Crossref] [PubMed]

Dabrowski, M.

R. Chrapkiewicz, M. Dabrowski, and W. Wasilewski, “High-Capacity Angularly Multiplexed Holographic Memory Operating at the Single-Photon Level,” Phys. Rev. Lett. 118, 063603 (2017).
[Crossref] [PubMed]

de Riedmatten, H.

J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
[Crossref] [PubMed]

de Wolf, R.

H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
[Crossref]

Diamanti, E.

N. Kumar, E. Diamanti, and I. Kerenidis, “Efficient quantum communications with coherent state fingerprints over multiple channels,” Phys. Rev. A 95, 032337 (2017).
[Crossref]

Donaldson, R. J.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

Drummond, P. D.

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66, 481–537 (1994).
[Crossref]

Du, J.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Dunjko, V.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Ekert, A.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Fekete, J.

J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
[Crossref] [PubMed]

Feng, C.

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Filip, R.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Gallager, R. G.

R. G. Gallager, Stochastic Processes: Theory and Applications (Cambridge University Press, 2014), Chap. 8.

Garcia-Escartin, J. C.

J. C. Garcia-Escartin and P. Chamorro-Posada, “Swap test and Hong-Ou-Mandel effect are equivalent,” Phys. Rev. A 87, 052330 (2013).
[Crossref]

Gerry, C.

C. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2004), Chap. 6.
[Crossref]

Guan, J.-Y.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Hennrich, M.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Higginbottom, D. B.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Horn, R. T.

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

Jachura, M.

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
[Crossref]

Jeffers, J.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Kerenidis, I.

N. Kumar, E. Diamanti, and I. Kerenidis, “Efficient quantum communications with coherent state fingerprints over multiple channels,” Phys. Rev. A 95, 032337 (2017).
[Crossref]

Kitayama, K.

K. Kitayama, Optical Code Division Multiple Access: A Practical Perspective (Cambridge University, 2014), Chap. 4.
[Crossref]

Knight, P.

C. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2004), Chap. 6.
[Crossref]

Kuhn, A.

T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
[Crossref]

Kumar, N.

N. Kumar, E. Diamanti, and I. Kerenidis, “Efficient quantum communications with coherent state fingerprints over multiple channels,” Phys. Rev. A 95, 032337 (2017).
[Crossref]

Kwek, L.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Lachman, L.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Legero, T.

T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
[Crossref]

Li, L.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Li, Y.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Lo, H-K

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 2014), Chap. 6.8.

Lüthenhaus, N

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

J. M. Arrazola and N Lüthenhaus, “Quantum fingerprinting with coherent states and a constant mean number of photons,” Phys. Rev. A 89, 062305 (2014).
[Crossref]

Lvovsky, A. I.

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Chap. 4.3.
[Crossref]

Marzlin, K. P.

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

Massar, S.

S. Massar, “Quantum fingerprinting protocol with a single particle,” Phys. Rev. A 71, 012310 (2005).
[Crossref]

Oh, C.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Oi, D. K.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Olivares, S.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
[Crossref]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Palacios-Avila, P.

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Pan, J.-W.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Paris, M. G. A.

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
[Crossref]

Peng, X.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Rempe, G.

T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
[Crossref]

Rielander, D.

J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
[Crossref] [PubMed]

Sajeed, S.

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Sanders, B. C.

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

Slodicka, L.

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Su, C.

C. Su and K. Wódkiewicz, “Quantum versus stochastic or hidden-variable fluctuations in two-photon interference effects,” Phys. Rev. A 44, 6097 (1991).
[Crossref]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (2nd edition) (Wiley, 1991), Chap. 12.
[Crossref]

van Lint, J. H.

J. H. van Lint, Introduction to Coding Theory (3rd edition) (Springer, 1987), Chap. 5.

Wallden, P.

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

Wang, W.

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Wang, Z.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Wasilewski, W.

R. Chrapkiewicz, M. Dabrowski, and W. Wasilewski, “High-Capacity Angularly Multiplexed Holographic Memory Operating at the Single-Photon Level,” Phys. Rev. Lett. 118, 063603 (2017).
[Crossref] [PubMed]

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
[Crossref]

Watrous, J.

H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
[Crossref]

Wei, K.

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Wilk, T.

T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
[Crossref]

Wódkiewicz, K.

C. Su and K. Wódkiewicz, “Quantum versus stochastic or hidden-variable fluctuations in two-photon interference effects,” Phys. Rev. A 44, 6097 (1991).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Chap. 4.3.
[Crossref]

Xu, F.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

Yang, X.-Y.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Yin, H.-L.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

You, L.-X.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Zhang, Q.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Zhang, W.-J.

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

Zou, P.

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Appl. Phys. B (1)

T. Legero, T. Wilk, A. Kuhn, and G. Rempe, “Time-Resolved Two-Photon Quantum Interference,” Appl. Phys. B 77, 797–802 (2003).
[Crossref]

Nat. Commun. (2)

F. Xu, J. M. Arrazola, K. Wei, W. Wang, P. Palacios-Avila, C. Feng, S. Sajeed, N Lüthenhaus, and H-K Lo, “Experimental quantum fingerprinting with weak coherent pulses,” Nat. Commun. 6, 8735 (2015).
[Crossref]

P. J. Clarke, R. J. Collins, V. Dunjko, E. Andersson, J. Jeffers, and G. S. Buller, “Experimental demonstration of quantum digital signatures using phase-encoded coherent states of light,” Nat. Commun. 3, 1174 (2012).
[Crossref] [PubMed]

Nat. Photon. (1)

R. Chrapkiewicz, M. Jachura, K. Banaszek, and W. Wasilewski, “Hologram of a single photon,” Nat. Photon. 10, 576–579 (2016).
[Crossref]

New J. Phys. (1)

D. B. Higginbottom, L. Slodička, G. Araneda, L. Lachman, R. Filip, M. Hennrich, and R. Blatt, “Pure single photons from a trapped atom source,” New J. Phys. 18, 093038 (2016).
[Crossref]

Phys. Rev. A (7)

S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, “Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion,” Phys. Rev. A 87, 0503032013).
[Crossref]

C. Su and K. Wódkiewicz, “Quantum versus stochastic or hidden-variable fluctuations in two-photon interference effects,” Phys. Rev. A 44, 6097 (1991).
[Crossref]

S. Massar, “Quantum fingerprinting protocol with a single particle,” Phys. Rev. A 71, 012310 (2005).
[Crossref]

J. M. Arrazola and N Lüthenhaus, “Quantum fingerprinting with coherent states and a constant mean number of photons,” Phys. Rev. A 89, 062305 (2014).
[Crossref]

N. Kumar, E. Diamanti, and I. Kerenidis, “Efficient quantum communications with coherent state fingerprints over multiple channels,” Phys. Rev. A 95, 032337 (2017).
[Crossref]

J. C. Garcia-Escartin and P. Chamorro-Posada, “Swap test and Hong-Ou-Mandel effect are equivalent,” Phys. Rev. A 87, 052330 (2013).
[Crossref]

J. Du, P. Zou, X. Peng, D. K. Oi, L. Kwek, C. Oh, and A. Ekert, “Experimental quantum multimeter and one-qubit Fingerprinting,” Phys. Rev. A 74042319 (2006).
[Crossref]

Phys. Rev. Lett. (7)

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett.,  95, 150502 (2005).
[Crossref] [PubMed]

H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum Fingerinting,” Phys. Rev. Lett. 87, 167902 (2001).
[Crossref]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

J.-Y. Guan, F. Xu, H.-L. Yin, Y. Li, W.-J. Zhang, S.-J. Chen, X.-Y. Yang, L. Li, L.-X. You, T.-Y. Chen, Z. Wang, Q. Zhang, and J.-W. Pan, “Observation of Quantum Fingerprinting Beating the Classical Limit,” Phys. Rev. Lett. 116, 240502 (2016).
[Crossref] [PubMed]

R. J. Collins, R. J. Donaldson, V. Dunjko, P. Wallden, P. J. Clarke, E. Andersson, J. Jeffers, and G. S. Buller, “Realization of Quantum Digital Signatures without the Requirement of Quantum Memory,” Phys. Rev. Lett. 113, 040502 (2014).
[Crossref] [PubMed]

R. Chrapkiewicz, M. Dabrowski, and W. Wasilewski, “High-Capacity Angularly Multiplexed Holographic Memory Operating at the Single-Photon Level,” Phys. Rev. Lett. 118, 063603 (2017).
[Crossref] [PubMed]

J. Fekete, D. Rielander, M. Cristani, and H. de Riedmatten, “Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks,” Phys. Rev. Lett. 110, 220502 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

C. M. Caves and P. D. Drummond, “Quantum limits on bosonic communication rates,” Rev. Mod. Phys. 66, 481–537 (1994).
[Crossref]

Other (7)

R. G. Gallager, Stochastic Processes: Theory and Applications (Cambridge University Press, 2014), Chap. 8.

K. Kitayama, Optical Code Division Multiple Access: A Practical Perspective (Cambridge University, 2014), Chap. 4.
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Chap. 4.3.
[Crossref]

T. M. Cover and J. A. Thomas, Elements of Information Theory (2nd edition) (Wiley, 1991), Chap. 12.
[Crossref]

C. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2004), Chap. 6.
[Crossref]

R. Loudon, The Quantum Theory of Light (Oxford University, 2014), Chap. 6.8.

J. H. van Lint, Introduction to Coding Theory (3rd edition) (Springer, 1987), Chap. 5.

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Figures (6)

Fig. 1
Fig. 1 Quantum fingerprinting protocols using (a) coherent states and (b) photon pairs. Alice and Bob map error correcting codewords onto +/− phase modulation patterns for optical signals sent to the Referee, who combines them on a 50/50 beam splitter. (a) In the case of first order interference, a click on the detector monitoring the output port b of the beam splitter, where field amplitudes are subtracted, implies that the codewords were different. (b) The same conclusion can be drawn in the two-photon protocol when a coincidence between the two detectors is registered.
Fig. 2
Fig. 2 Modification of the error correcting code required for the two-photon quantum fingerprinting protocol. (a) A standard error correcting code of length m guarantees that the relative Hamming distance δ between any two different codewords used by Alice A and Bob B is between δmin and 1. Two extreme cases are shown. (b) Extension of the standard code by a fixed number of min ensures that the relative Hamming distance Δ for any two codewords in the modified code is bounded between δmin/(1 + δmin) and 1/(1 + δmin).
Fig. 3
Fig. 3 (a) Maximum code rates rGV and RGV given by the Gilbert-Varshamov bound respectively for the coherent-state and the two-photon fingerprinting protocol. The minimum Hamming distances δ min coh and Δmin for the two scenarios are chosen such that the misidentification probability has the same scaling with the number of transmitted photons and their dependence is shown in the inset. (b) The ratio of maximum code rates rGV / RGV which specifies the overhead in the codeword length for the two-photon protocol.
Fig. 4
Fig. 4 Comparison of the performance of classical and quantum fingerprinting protocols as a function of the input bit string length n. The gray curve represents the best known lower bound on the number of bits required in the classical protocol Iclass. The red and black curves depict the amount of classical information IS and Icoh that could be transmitted in physical systems employed respectively in the two-photon and the coherent state quantum fingerprinting protocol. The error probability is Perr = 10−6 and the minimum relative Hamming distance equals δ min coh = 0.2 for the coherent state protocol with the corresponding value for the two-photon protocol given by Eq. (9). The inset depicts the ratio IS/Icoh.
Fig. 5
Fig. 5 (a) The decision between hypotheses of different D and equal E inputs is made based on the observed number Nc of coincidence events. The chosen hypothesis corresponds to the higher probability pD(Nc) or pE(Nc). (b) The error probability Perr assuming equiprobable hypotheses for the minimum Hamming distance Δmin = 0.1, dark count contribution p d / ( η n ¯ ) = 0.01, and indistinguishability W = 98%. Results for single photons (S, blue lines) corresponding to g(2) = 0 are compared to signals with Poissonian photon number statistics (P, red lines). Solid lines are numerical results and dotted lines are asymptotic expressions based on Chernoff information. The plots are parameterized with the effective number ( η n ¯ ) 2 N of pairs composed of photons received from both Alice and Bob. Note that the curves depicting numerical results lie below the lines corresponding to the asymptotic exponential expression. This is because of additional sublinear terms in N in the exponent of the exact error probability.
Fig. 6
Fig. 6 Rescaled Chernoff information for the fingerprinting protocol based on second-order interference using (a) single photons ζS and (b) optical signals with Poissonian photon number statistics ζP as a function of the dark count contribution p d / ( η n ¯ ) and the minimum Hamming distance Δmin. Note different colour codings of numerical values in the panels. (c) The ratio ζS/ζP which quantifies the benefit of using single photons in the weak signal regime.

Equations (22)

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

a ^ x = 1 m i = 0 m 1 ( 1 ) E i ( x ) a ^ i , b ^ y = 1 m i = 0 m 1 ( 1 ) E i ( y ) b ^ i .
| α x A = exp ( α a ^ x α * a ^ x ) | 0 A , | β y B = exp ( β b ^ y β * b ^ y ) | 0 B ,
a ^ i ( a ^ i + b ^ i ) / 2 , b ^ i ( a ^ i + b ^ i ) / 2 .
V = 1 m i = 0 m 1 ( 1 ) E i ( x ) + E i ( y )
| 1 x A = a ^ x | 0 A = 1 m i = 0 m 1 ( 1 ) E i ( x ) a ^ i | 0 A | 1 y B = b ^ y | 0 B = 1 m i = 0 m 1 ( 1 ) E i ( y ) b ^ i | 0 B ,
P c = 1 2 ( 1 | V | 2 ) ,
V = 1 m [ m 2 d ( E ( x ) , E ( y ) ) ] = 1 2 δ .
n M = R R GV ( Δ min ) = ( 1 Δ min ) r GV ( Δ min 1 Δ min )
Δ min = 1 2 [ 1 2 exp ( 2 δ min coh ) 1 ]
P 2 = ( η n ¯ ) 2 [ 1 + g ( 2 ) + 4 p d η n ¯ + 2 ( p d η n ¯ ) 2 ] ,
Q = P c P 2 = 1 2 ( 1 V eff ) , V eff = W | V | 2 1 + g ( 2 ) + 4 p d / ( η n ¯ ) + 2 p d 2 / ( η n ¯ ) 2 .
p ν ( N c ) = ( N 2 N c ) Q ν N c ( 1 Q ν ) N 2 N c , ν = D , E
P err = 1 2 N c = 0 N 2 min { p D ( N c ) , p E ( N c ) } .
C = log [ min 0 α 1 ( k [ p D ( k ) ] α [ p E ( k ) ] 1 α ) ] ,
C = log ( 1 P 2 + P 2 min 0 α 1 [ Q D α Q E 1 α + ( 1 Q D ) α ( 1 Q E ) 1 α ] ) .
ζ = [ 1 + g ( 2 ) + 4 p d η n ¯ + 2 ( p d η n ¯ ) 2 ] ( 1 min 0 α 1 [ Q D α Q E 1 α + ( 1 Q D ) α ( 1 Q E ) 1 α ] ) .
P err exp [ ( η n ¯ ) 2 N ζ ]
N ^ b = 1 2 i = 0 m 1 ( a ^ i b ^ i ) ( a ^ i b ^ i ) .
[ exp ( α a ^ x α a ^ x ) ] a ^ i exp ( α a ^ x α a ^ x ) = a ^ i + ( 1 ) E i ( x ) α m , [ exp ( β b ^ y β b ^ y ) ] b ^ i exp ( β b ^ y β b ^ y ) = b ^ i + ( 1 ) E i ( y ) β m .
A α x |   B β y | : exp ( N ^ b ) : | α x A | β y B = exp ( 1 2 m i = 0 m 1 | ( 1 ) E i ( x ) α ( 1 ) E i ( y ) β | 2 ) = exp [ n ¯ ( 1 Re V ) ] ,
N ^ a = 1 2 j = 0 m 1 ( a ^ j + b ^ j ) ( a ^ j + b ^ j ) .
  A 1 x | B 1 y | : N ^ a N ^ b : | 1 x A | 1 x B = 1 4 m 2 i , j = 0 m 1 | ( 1 ) E i ( x ) + E j ( y ) ( 1 ) E i ( x ) + E j ( y ) | 2 = 1 2 ( 1 | V | 2 ) .

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