Abstract

In recent years, a large quantity of work have been done to narrow the gap between theory and practice in quantum key distribution (QKD). However, most of them are focus on two-party protocols. Very recently, Yao Fu et al proposed a measurement-device-independent quantum cryptographic conferencing (MDI-QCC) protocol and proved its security in the limit of infinitely long keys. As a step towards practical application for MDI-QCC, we design a biased decoy-state measurement-device-independent quantum cryptographic conferencing protocol and analyze the performance of the protocol in both the finite-key and infinite-key regime. From numerical simulations, we show that our decoy-state analysis is tighter than Yao Fu et al. That is, we can achieve the nonzero asymptotic secret key rate in long distance with approximate to 200km and we also demonstrate that with a finite size of data (say 1011 to 1013 signals) it is possible to perform secure MDI-QCC over reasonable distances.

© 2016 Optical Society of America

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2015 (5)

C. Zhou, W.-S. Bao, H.-L Zhang, H.-W. Li, Y. Wang, Y. Li, and X. Wang, “Biased decoy-state measurement-device-independent quantum key distribution with finite resources,” Phys. Rev. A 91, 022313 (2015)
[Crossref]

F. Xu, “Measurement-device-independent quantum communication with an untrusted source,” Phys. Rev. A 92, 012333 (2015)
[Crossref]

Y. Fu, H.-L. Yin, T.-Y. Chen, and Z.-B. Chen, “Long-distance measurement-device-independent multiparty quantum communication,” Phys. Rev. Lett. 114, 090501 (2015)
[Crossref] [PubMed]

R. Valivarthi, I. Lucio-Martinez, P. Chan, A. Rubenok, C. John, D. Korchinski, C. Duffin, F. Marsili, V. Verma, M. D. Shaw, J. A. Stern, S. W. Nam, D. Oblak, Q. Zhou, J. A. Slater, and W. Tittel, “Measurement-device-independent quantum key distribution: from idea towards application,” J. Mod. Opt. 62, 1141 (2015)
[Crossref]

C. Zhu, F. Xu, and C Pei, “W-state analyzer and multi-party measurement-device-independent quantum key distribution,” Sci. Rep. 5, 17449 (2015)
[Crossref] [PubMed]

2014 (8)

B. A. Bell, D. Markham, D. A. Herrera-Mart, A. Marin, W. J. Wadsworth, J. G. Rarity, and M. S. Tame, “Experimental demonstration of graph-state quantum secret sharing,” Nat. Commun. 5, 5480 (2014)
[Crossref] [PubMed]

C.C.W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, “Concise security bounds for practical decoy-state quantum key distribution,” Phys. Rev. A 89, 022307 (2014)
[Crossref]

C. Zhou, W.-S. Bao, H.-W. Li, Y. Wang, Y. Li, Z.-Q. Yin, W. Chen, and Z.-F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89, 052328 (2014)
[Crossref]

M. Curty, F. Xu, W. Cui, C. C. W Lim, K. Tamki, and H. K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution,” Nat. Commun. 5, 3732 (2014).
[Crossref] [PubMed]

F. Xu, H. Xu, and H.-K. Lo, “Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution,” Phys. Rev. A 89, 052333 (2014)
[Crossref]

Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, and H.-K. Lo, “Experimental demonstration of polarization encoding measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 112, 190503 (2014)
[Crossref] [PubMed]

Y.-L. Tang, H.-L. Yin, S.-J. Chen, Y. Liu, W.-J. Zhang, X. Jiang, L. Zhang, J. Wang, L.-X. You, J.-Y. Guan, D.-X. Yang, Z. Wang, H. Liang, Z. Zhang, N. Zhou, X. Ma, T.-Y. Chen, Q. Zhang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over 200 km,” Phys. Rev. Lett. 113, 190501 (2014)
[Crossref] [PubMed]

M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution, ” Nat. Commun. 5, 3732 (2014)
[Crossref] [PubMed]

2013 (9)

F. Xu, B. Qi, Z. Liao, and H.-K. Lo, “Long distance measurement-device-independent quantum key distribution with entangled photon sources,” Appl. Phys. Lett. 103, 061101 (2013)
[Crossref]

X.-B. Wang, “Three-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errors,” Phys. Rev. A 87, 012320 (2013)
[Crossref]

F. Xu, M. Curty, B. Qi, and H.-K. Lo, “Practical aspects of measurement-device-independent quantum key distribution,” New J. Phys. 15, 113007 (2013)
[Crossref]

A. Rubenok, J. A. Slater, P. Chan, I. Lucio-Martinez, and W. Tittel, “Real-world two-photon interference and proof-of-principle quantum key distribution immune to detector attacks,” Phys. Rev. Lett. 111, 130501 (2013)
[Crossref] [PubMed]

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013)
[Crossref] [PubMed]

T. Ferreira da Silva, D. Vitoreti, G. B. Xavier, G. C. do Amaral, G. P. Temporao, and J. P. von der Weid, “Proof-of-principle demonstration of measurement-device-independent quantum key distribution using polarization qubits,” Phys. Rev. A 88, 052303 (2013)
[Crossref]

C. Zhou, W.-S. Bao, W. Chen, H.-W. Li, Z.-Q. Yin, Y. Wang, and Z.-F. Han, “Phase-encoded measurement-device-independent quantum key distribution with practical spontaneous-parametric-down-conversion sources,” Phys. Rev. A 88, 052333 (2013)
[Crossref]

Y. Wang, W.-S. Bao, H.-W. Li, C. Zhou, and Y. Li, “Finite-key analysis for one-sided device-independent quantum key distribution,” Phys. Rev. A 88, 052322 (2013)
[Crossref]

M. Mafu, K. Garapo, and F. Petruccione, “Finite-size key in the Bennett 1992 quantum-key-distribution protocol for Rényi entropies,” Phys. Rev. A 88, 062306 (2013).
[Crossref]

2012 (6)

N.H.Y. Ng, M. Berta, and S. Wehner, “Min-entropy uncertainty relation for finite-size cryptography,” Phys. Rev. A 86, 042315 (2012)
[Crossref]

M. Hayashi and T. Tsurumaru, “Concise and tight security analysis of the BennettCBrassard 1984 protocol with finite key lengths,” New. J. Phys. 14, 093014 (2012)
[Crossref]

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

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108, 130503 (2012)
[Crossref] [PubMed]

X. Ma, C.-H.F. Fung, and M. Razavi, “Statistical fluctuation analysis for measurement-device-independent quantum key distribution,” Phys. Rev. A 86, 052305 (2012)
[Crossref]

X. Ma and M. Razavi, “Alternative schemes for measurement-device-independent quantum key distribution,” Phy. Rev. A 86, 062319 (2012)
[Crossref]

2011 (2)

I. Gerhardt, Q. Liu, A. Lamas-Linares, J. Skaar, C. Kurtsiefer, and V. Makarov, “Full-field implementation of a perfect eavesdropper on a quantum cryptography system,” Nat. Commun. 2, 349 (2011)
[Crossref] [PubMed]

H.-W. Li, S. Wang, J.-Z. Huang, W. Chen, Z.-Q. Yin, F.-Y. Li, Z. Zhou, D. Liu, Y. Zhang, G.-C. Guo, W.-S. Bao, and Z.-F. Han, “Attacking a practical quantum-key-distribution system with wavelength-dependent beam-splitter and multiwavelength sources,” Phys. Rev. A 84, 062308 (2011)
[Crossref]

2010 (4)

F. Xu, B. Qi, and H.-K. Lo, “Experimental demonstration of phase-remapping attack in a practical quantum key distribution system,” New J. Phys. 12, 113026 (2010)
[Crossref]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4, 686 (2010)
[Crossref]

Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 800 (2010)
[Crossref]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Avoiding the blinding attack in QKD,” Nat. Photonics 4, 801 (2010)
[Crossref]

2009 (3)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301 (2009)
[Crossref]

H.-W. Li, Y.-B. Zhao, Z.-Q. Yin, S. Wang, Z.-F. Han, W.-S. Bao, and G.-C. Guo, “Security of decoy states QKD with finite resources against collective attacks,” Opt. Commun. 282, 4162 (2009)
[Crossref]

M. Christandl, R. König, and R. Renner, “Postselection technique for quantum channels with applications to quantum cryptography,” Phys. Rev. Lett. 102, 020504 (2009)
[Crossref] [PubMed]

2008 (1)

Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008)
[Crossref]

2007 (5)

B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, “Time-shift attack in practical quantum cryptosystems,” Quantum Inf. Comput. 7, 73 (2007)

C.-H. F. Fung, B. Qi, K. Tamaki, and H.-K. Lo, “Phase-remapping attack in practical quantum-key-distribution systems,” Phys. Rev. A 75, 032314 (2007)
[Crossref]

K. Chen and H.-K. Lo, “Multi-partite quantum cryptographic protocols with noisy GHZ states,” Quantum Inf. Comput. 7, 689 (2007)

M. Hayashi, “Upper bounds of eavesdroppers performances in finite-length code with the decoy method,” Phys. Rev. A 76, 012329 (2007)
[Crossref]

S. Gaertner, C. Kurtsiefer, M. Bourennane, and H. Weinfurter, “Experimental demonstration of four-party quantum secret sharing,” Phys. Rev. Lett. 98, 020503 (2007)
[Crossref] [PubMed]

2005 (5)

C. Schmid, P. Trojek, M. Bourennane, C. Kurtsiefer, M. Żukowski, and H. Weinfurter, “Experimental single qubit quantum secret sharing,” Phys. Rev. Lett. 95, 230505 (2005)
[Crossref] [PubMed]

Y.-A. Chen, A.-N. Zhang, Z. Zhao, X.-Q. Zhou, C.-Y. Lu, C.-Z. Peng, T. Yang, and J.-W. Pan, “Experimental quantum secret sharing and third-man quantum cryptography,” Phys. Rev. Lett. 95, 200502 (2005)
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H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005)
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X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94, 230503 (2005)
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X. F. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72, 012326 (2005)
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2003 (1)

W.-Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003)
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2002 (2)

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2001 (2)

D. Mayers, “Unconditional security in quantum cryptography,” J. ACM 48, 351 (2001)
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W. Tittel, H. Zbinden, and N. Gisin, “Experimental demonstration of quantum secret sharing,” Phys. Rev. A 63, 042301 (2001).
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2000 (1)

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1999 (4)

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1998 (2)

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1967 (1)

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1963 (1)

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H.-W. Li, S. Wang, J.-Z. Huang, W. Chen, Z.-Q. Yin, F.-Y. Li, Z. Zhou, D. Liu, Y. Zhang, G.-C. Guo, W.-S. Bao, and Z.-F. Han, “Attacking a practical quantum-key-distribution system with wavelength-dependent beam-splitter and multiwavelength sources,” Phys. Rev. A 84, 062308 (2011)
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H.-W. Li, Y.-B. Zhao, Z.-Q. Yin, S. Wang, Z.-F. Han, W.-S. Bao, and G.-C. Guo, “Security of decoy states QKD with finite resources against collective attacks,” Opt. Commun. 282, 4162 (2009)
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H.-K. Lo and H.F. Chau, “Unconditional security Of quantum key distribution over arbitrarily long distances,” Science 283, 2050 (1999)
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K. Chen and H.-K. Lo, “Multi-partite quantum cryptographic protocols with noisy GHZ states,” Quantum Inf. Comput. 7, 689 (2007)

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Y. Fu, H.-L. Yin, T.-Y. Chen, and Z.-B. Chen, “Long-distance measurement-device-independent multiparty quantum communication,” Phys. Rev. Lett. 114, 090501 (2015)
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C. Zhou, W.-S. Bao, H.-W. Li, Y. Wang, Y. Li, Z.-Q. Yin, W. Chen, and Z.-F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89, 052328 (2014)
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C. Zhou, W.-S. Bao, W. Chen, H.-W. Li, Z.-Q. Yin, Y. Wang, and Z.-F. Han, “Phase-encoded measurement-device-independent quantum key distribution with practical spontaneous-parametric-down-conversion sources,” Phys. Rev. A 88, 052333 (2013)
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H.-W. Li, S. Wang, J.-Z. Huang, W. Chen, Z.-Q. Yin, F.-Y. Li, Z. Zhou, D. Liu, Y. Zhang, G.-C. Guo, W.-S. Bao, and Z.-F. Han, “Attacking a practical quantum-key-distribution system with wavelength-dependent beam-splitter and multiwavelength sources,” Phys. Rev. A 84, 062308 (2011)
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Chen, Y.-A.

Y.-A. Chen, A.-N. Zhang, Z. Zhao, X.-Q. Zhou, C.-Y. Lu, C.-Z. Peng, T. Yang, and J.-W. Pan, “Experimental quantum secret sharing and third-man quantum cryptography,” Phys. Rev. Lett. 95, 200502 (2005)
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Chen, Z.-B.

Y. Fu, H.-L. Yin, T.-Y. Chen, and Z.-B. Chen, “Long-distance measurement-device-independent multiparty quantum communication,” Phys. Rev. Lett. 114, 090501 (2015)
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Chernoff, H.

H. Chernoff, “A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations,” Ann. Math. Sat. 23, 493 (1952)
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M. Christandl, R. König, and R. Renner, “Postselection technique for quantum channels with applications to quantum cryptography,” Phys. Rev. Lett. 102, 020504 (2009)
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W. Dür, J. I. Cirac, and R. Tarrach, “Separability and distillability of multiparticle quantum systems,” Phys. Rev. Lett. 83, 3562 (1999)
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Cleve, R.

R. Cleve, D. Gottesman, and H.-K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83, 648 (1999).
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Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, L. Li, X. Ma, J. S. Pelc, M. M. Fejer, C.-Z. Peng, Q. Zhang, and J.-W. Pan, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111, 130502 (2013)
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M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution, ” Nat. Commun. 5, 3732 (2014)
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M. Curty, F. Xu, W. Cui, C. C. W. Lim, K. Tamaki, and H.-K. Lo, “Finite-key analysis for measurement-device-independent quantum key distribution, ” Nat. Commun. 5, 3732 (2014)
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C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996)
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T. Ferreira da Silva, D. Vitoreti, G. B. Xavier, G. C. do Amaral, G. P. Temporao, and J. P. von der Weid, “Proof-of-principle demonstration of measurement-device-independent quantum key distribution using polarization qubits,” Phys. Rev. A 88, 052303 (2013)
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Duffin, C.

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Dür, W.

W. Dür, J. I. Cirac, and R. Tarrach, “Separability and distillability of multiparticle quantum systems,” Phys. Rev. Lett. 83, 3562 (1999)
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Dusek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301 (2009)
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Y. Fu, H.-L. Yin, T.-Y. Chen, and Z.-B. Chen, “Long-distance measurement-device-independent multiparty quantum communication,” Phys. Rev. Lett. 114, 090501 (2015)
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Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, “Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems,” Phys. Rev. A 78, 042333 (2008)
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H.-W. Li, Y.-B. Zhao, Z.-Q. Yin, S. Wang, Z.-F. Han, W.-S. Bao, and G.-C. Guo, “Security of decoy states QKD with finite resources against collective attacks,” Opt. Commun. 282, 4162 (2009)
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Han, Z.-F.

C. Zhou, W.-S. Bao, H.-W. Li, Y. Wang, Y. Li, Z.-Q. Yin, W. Chen, and Z.-F. Han, “Tight finite-key analysis for passive decoy-state quantum key distribution under general attacks,” Phys. Rev. A 89, 052328 (2014)
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C. Zhou, W.-S. Bao, W. Chen, H.-W. Li, Z.-Q. Yin, Y. Wang, and Z.-F. Han, “Phase-encoded measurement-device-independent quantum key distribution with practical spontaneous-parametric-down-conversion sources,” Phys. Rev. A 88, 052333 (2013)
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H.-W. Li, S. Wang, J.-Z. Huang, W. Chen, Z.-Q. Yin, F.-Y. Li, Z. Zhou, D. Liu, Y. Zhang, G.-C. Guo, W.-S. Bao, and Z.-F. Han, “Attacking a practical quantum-key-distribution system with wavelength-dependent beam-splitter and multiwavelength sources,” Phys. Rev. A 84, 062308 (2011)
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H.-W. Li, Y.-B. Zhao, Z.-Q. Yin, S. Wang, Z.-F. Han, W.-S. Bao, and G.-C. Guo, “Security of decoy states QKD with finite resources against collective attacks,” Opt. Commun. 282, 4162 (2009)
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M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829 (1999).
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W. Hoeffding, “Probability inequalities for sums of bounded random variables,” J. Am. Stat. Assoc. 58, 13 (1963)
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M. Żukowski, A. Zeilinger, M. A. Horne, and H. Weinfurter, “Quest for GHZ states,” Acta Phys. Pol. A 93, 187 (1998)
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R. Valivarthi, I. Lucio-Martinez, P. Chan, A. Rubenok, C. John, D. Korchinski, C. Duffin, F. Marsili, V. Verma, M. D. Shaw, J. A. Stern, S. W. Nam, D. Oblak, Q. Zhou, J. A. Slater, and W. Tittel, “Measurement-device-independent quantum key distribution: from idea towards application,” J. Mod. Opt. 62, 1141 (2015)
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S. Bose, V. Vedral, and P. L. Knight, “Multiparticle generalization of entanglement swapping,” Phys. Rev. A 57, 822 (1998)
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M. Christandl, R. König, and R. Renner, “Postselection technique for quantum channels with applications to quantum cryptography,” Phys. Rev. Lett. 102, 020504 (2009)
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R. Valivarthi, I. Lucio-Martinez, P. Chan, A. Rubenok, C. John, D. Korchinski, C. Duffin, F. Marsili, V. Verma, M. D. Shaw, J. A. Stern, S. W. Nam, D. Oblak, Q. Zhou, J. A. Slater, and W. Tittel, “Measurement-device-independent quantum key distribution: from idea towards application,” J. Mod. Opt. 62, 1141 (2015)
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Figures (4)

Fig. 1
Fig. 1 MDI-QCC scheme. Alice, Bob and Charlie encode their bits in the polarization degrees of freedom of phase-randomised WCPs and David uses the linear optics quantum relay which is assumed to identify two of the eight GHZ states.
Fig. 2
Fig. 2 Secret key rate vs fiber length with different probabilities of choosing X basis. The curves from left to right are numerically optimized for a different px =0.9, 0.7, 0.5, 0.3, 0.1. A realistic finite size of data N is fixed to be 1011. The intensity of one decoy state is 0.005 and the other decoy state is a vacuum state, while the signal state is optimized for different distances. The efficiency of Davids detectors is 90%.
Fig. 3
Fig. 3 Secret key rate vs transmission distance. The curves from left to right are numerically optimized for a fixed number of total pulses N=10 j with j = 11, 11.5, 12,...,13. The intensity of one decoy state are 0.005 and the other decoy state is a vacuum state, while the signal state is optimized for different distances.The efficiency of Davids detectors is 90%.
Fig. 4
Fig. 4 Secret key rate vs transmission distance. The red dashed curve denotes the asymptotic secret key rate calculated with the decoy analysis in [23]. The blue dashed curve denotes the asymptotic secret key rate calculated with the decoy analysis presented in Sec. III. The intensity of the signal state and one decoy state are 0.4 and 0.005 respectively, while the other decoy state is a vacuum state. The efficiency of Davids detectors is 40%.

Tables (1)

Tables Icon

Table 1 List of experimental parameters for simulations: β is the loss coefficient of the fiber; ηd is the efficiency of Davids detectors; pd is the background count rate; f is the error-correction efficiency; ed represents the overall misalignment-error probability of the system.

Equations (30)

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

N 000 : = { i : ( u i a = 0 ) ( u i b = 0 ) ( u i c = 0 ) ( k i { 0 , 1 } ) }
X α β γ : = { i : ( u i a = α ) ( u i b = β ) ( u i c = γ ) ( α i a α i b α i c = X X X ) ( k i ) } ( α β γ μ μ μ , α β 000 )
Z α β γ : = { i : ( u i a = α ) ( u i b = β ) ( u i c = γ ) ( α i a α i b α i c = Z Z Z ) ( k i ) } ( α β γ 000 )
ρ υ A = k a k | k k | , ρ μ A = k a k | k k | ρ υ B = k b k | k k | , ρ μ B = k b k | k k | ρ υ C = k c k | k k | , ρ μ C = k c k | k k |
a k a k a 2 a 2 a 1 a 1 , b k b k b 2 b 2 b 1 b 1 , c k c k c 2 c 2 c 1 c 1 , k 2
n α β γ Z = n , m , k = 0 p α β γ | n m k Z s n m k Z , α , β , γ U
p α β γ | n m k Z = p α β γ , z a n b m c k τ n m k Z
τ n m k Z = α , β , γ U p α β γ , z a n b m c k
n μ a υ b υ c Z * = p μ p υ p υ p Z ( a 1 + b 1 c g 111 + a 1 b 2 c 1 g 121 + G μ a υ b υ c )
n μ a μ b μ c Z * = p μ p μ p μ p Z ( a 1 b 1 c 1 g 111 + a 1 b 2 c 1 g 121 + G μ a μ b μ c )
n μ a υ b υ c Z * p μ p υ 2 = n μ a υ b υ c Z p μ p υ 2 a 0 n 0 υ b υ c Z p 0 p υ 2 b 0 n μ a 0 υ c Z p μ p 0 p υ c 0 n μ a υ b 0 Z p μ p υ p 0 + a 0 b 0 n 00 υ c Z p 0 2 p υ + a 0 c 0 n 0 υ b 0 Z p 0 2 p υ + b 0 c 0 n μ a 00 Z p μ p 0 2 + 2 a 0 b 0 c 0 n 000 Z p 0 3
n μ a μ b μ c Z * p μ 3 = n μ a μ b μ c Z p μ 3 a 0 n 0 μ b μ c Z p 0 p μ 2 b 0 n μ a 0 μ c Z p 0 p μ 2 c 0 n μ a μ b 0 Z p 0 p μ 2 + a 0 b 0 n 00 μ c Z p 0 2 p μ + a 0 c 0 n 0 μ b 0 Z p 0 2 p μ + b 0 c 0 n μ a 00 Z p 0 2 p μ + 2 a 0 b 0 c 0 n 000 Z p 0 3
G μ a υ b υ c = n , m , k G 0 a n b m c k g n m k
G μ a μ b μ c = n , m , k G 0 a n b m c k g n m k
g m n k = s n m k Z τ n m k Z ( n 1 , m 1 , k 1 )
G 0 = { ( n , m , k ) | n 1 , m 1 , k 1 , n + m + k 4 , ( n , m , k ) { ( 2 , 1 , 1 ) , ( 1 , 1 , 2 ) }
g 111 Z = g 111 Z low + ( m , n , k G 0 ) f n m k g n m k Z
g 111 Z low = ( b 2 c 1 p μ 2 n μ a υ b υ c Z * b 2 c 1 p υ 2 n μ a μ b μ c Z * ) a 1 c 1 c 1 ( b 1 b 2 b 1 b 2 ) p μ 3 p υ 2 p Z
f n m k = a n ( b 2 b m c 1 c k b 2 b m c 1 c k ) a 1 c 1 c 1 ( b 1 b 2 b 1 b 2 )
s 111 Z low = ( b 2 c 1 p μ 2 n ˜ μ a υ b υ c Z * b 2 c 1 p υ 2 n ˜ μ a μ b μ c Z * ) τ 111 Z low a 1 c 1 c 1 ( b 1 b 2 b 1 b 2 ) p μ 3 p υ 2 p Z
n ˜ μ a υ b υ c Z * = n ˜ μ a υ b υ c Z Δ μ υ υ ( a 0 p μ ( n ˜ 0 υ b υ c Z + Δ ^ 0 υ υ ) + b 0 p υ ( n ˜ μ a 0 υ c Z + Δ ^ μ 0 υ ) + c 0 p υ ( n ˜ μ a υ b 0 Z + Δ ^ μ υ 0 ) / p 0 + ( a 0 b 0 p μ p υ ( n ˜ 00 υ c Z Δ 00 υ ) + a 0 c 0 p μ p υ ( n ˜ 0 υ b 0 Z Δ 0 υ 0 ) + b 0 c 0 p υ 2 ( n ˜ μ a 00 Z Δ μ 00 ) ) / p 0 2 + 2 a 0 b 0 c 0 p μ p μ 2 ( n ˜ 000 Z Δ 000 ) / p 0 3
n ˜ μ a μ b μ c Z * = n ˜ μ a μ b μ c Z Δ μ μ μ p μ ( a 0 ( n ˜ 0 μ b μ c Z + Δ ^ 0 μ μ ) + b 0 ( n ˜ μ a 0 μ c Z + Δ ^ μ 0 μ ) + c 0 ( n ˜ μ a μ b 0 Z + Δ ^ μ μ 0 ) ) / p 0 + p μ 2 ( a 0 b 0 ( n ˜ 00 μ c Z Δ 00 μ ) + a 0 c 0 ( n ˜ 0 μ b 0 Z Δ 0 μ 0 ) + b 0 c 0 ( n ˜ μ a 00 Z Δ μ 00 ) ) / p 0 2 + p μ 3 ( 2 a 0 b 0 c 0 ( n ˜ 000 Z Δ 000 ) ) / p 0 3
w v v v X * = p v p v p v p X ( a 1 b 1 c 1 r 111 + a 1 b 1 c 2 r 112 + a 2 b 1 c 1 r 211 + a 1 b 2 c 1 r 121 + R v v v )
w v v v X * p υ 3 = w v v v X p υ 3 1 p 0 p υ 2 ( a 0 w 0 v v X + b 0 w v 0 v X + c 0 w v v 0 X ) + 1 p 0 2 p υ ( a 0 b 0 w 00 v X + a 0 c 0 w 0 v 0 X + b 0 c 0 w v 00 X ) + 2 a 0 b 0 c 0 w 000 X p 0 3
R v v v = ( n , m , k ) G 0 a n b m c k r n m k
r n m k = v n m k X τ n m k X ( n 1 , m 1 , k 1 )
v 111 X τ 111 X w v v v X * a 1 b 1 c 1 p v 3 p X
v 111 X τ 111 X w ˜ v v v X * a 1 b 1 c 1 p v 3 p X
w v v v X * = ( w ˜ v v v X + Λ ^ v v v ) p υ p 0 ( a 0 ( w ˜ 0 v v X Λ 0 v v ) + b 0 ( w ˜ v 0 v X Λ v 0 v ) + c 0 ( w ˜ v v 0 X Λ v v 0 ) ) + p υ 2 p 0 2 ( a 0 b 0 ( w ˜ 00 v X + Λ ^ 00 v ) + a 0 c 0 ( w ˜ 0 v 0 X + Λ ^ 0 v 0 ) + b 0 c 0 ( w ˜ v 00 X + Λ ˜ v 00 ) ) + 2 a 0 b 0 c 0 p υ 3 ( w ˜ 000 X + Λ ^ 00 ) p 0 3
R QCC = ( n v Z + s 111 Z [ 1 H ( v 111 X / N q ) ] H ( E μ ν ω Z * ) f n μ ν ω Z ) / N

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