Abstract

We present a security analysis of a multiparty quantum key distribution (QKD) based on Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) type multipartite nonlocality. Our analysis provides an asymptotic secret key rate of the multiparty QKD under the restriction that successive trials are completely independent. In our analysis, we construct the relation between a secret key rate of the multiparty QKD and an expectation value of MABK operator. We show that side channel attacks, which can threaten the information theoretic (IT) security analysis of multiparty QKD, are prevented in our analysis. We compare secret key rates obtained by using the IT analysis, the existing fully device-independent analysis, and our analysis. It is shown that efficiency of the multiparty QKD can be improved with increasing number of authorized parties in our analysis as well.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

Y. Jo, K. Bae, and W. Son, “Enhanced Bell state measurement for efficient measurement-device-independent quantum key distribution using 3-dimensional quantum states,” Sci. Rep. 9(1), 687 (2019).
[Crossref]

2018 (6)

L. Dellantonio, A. S. Sørensen, and D. Bacco, “High-dimensional measurement-device-independent quantum key distribution on two-dimensional subspaces,” Phys. Rev. A 98(6), 062301 (2018).
[Crossref]

J. Ribeiro, G. Murta, and S. Wehner, “Fully device-independent conference key agreement,” Phys. Rev. A 97(2), 022307 (2018).
[Crossref]

M. Choi, Y. Lee, and S. Lee, “Quantum secret sharing and Mermin operator,” Quantum Inf. Process. 17(10), 258 (2018).
[Crossref]

F. Grasselli, H. Kampermann, and D. Bruß, “Finite-key effects in multipartite quantum key distribution protocols,” New J. Phys. 20(11), 113014 (2018).
[Crossref]

K. Kostrzewa, W. Laskowski, and T. Vértesi, “Closing the detection loophole in multipartite Bell experiments with a limited number of efficient detectors,” Phys. Rev. A 98(1), 012138 (2018).
[Crossref]

Y. Tsujimoto, M. Tanaka, N. Iwasaki, R. Ikuta, S. Miki, T. Yamashita, H. Terai, T. Yamamoto, M. Koashi, and N. Imoto, “High-fidelity entanglement swapping and generation of three-qubit GHZ state using asynchronous telecom photon pair sources,” Sci. Rep. 8(1), 1446 (2018).
[Crossref]

2017 (3)

M. Epping, H. Kampermann, C. Macchiavello, and D. Bruß, “Multi-partite entanglement can speed up quantum key distribution in networks,” New J. Phys. 19(9), 093012 (2017).
[Crossref]

W. Rosenfeld, D. Burchardt, R. Garthoff, K. Redeker, N. Ortegel, M. Rau, and H. Weinfurter, “Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes,” Phys. Rev. Lett. 119(1), 010402 (2017).
[Crossref]

C. Zhou, P. Xu, W.-S. Bao, Y. Wang, Y. Zhang, M.-S. Jiang, and H.-W. Li, “Finite-key bound for semi-device-independent quantum key distribution,” Opt. Express 25(15), 16971–16980 (2017).
[Crossref]

2016 (5)

Y. Jo and W. Son, “Key-rate enhancement using qutrit states for quantum key distribution with askew aligned sources,” Phys. Rev. A 94(5), 052316 (2016).
[Crossref]

A. Huang, S. Sajeed, P. Chaiwongkhot, M. Soucarros, M. Legre, and V. Makarov, “Testing Random-Detector-Efficiency Countermeasure in a Commercial System Reveals a Breakable Unrealistic Assumption,” IEEE J. Quantum Electron. 52(11), 1–11 (2016).
[Crossref]

D. Alsina, A. Cervera, D. Goyeneche, J. I. Latorre, and K. Życzkowski, “Operational approach to bell inequalities: Application to qutrits,” Phys. Rev. A 94(3), 032102 (2016).
[Crossref]

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, and J.-W. Pan, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117(21), 210502 (2016).
[Crossref]

C. Zhang, C.-J. Zhang, Y.-F. Huang, Z.-B. Hou, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of genuine multipartite nonlocality under the no-signalling principle,” Sci. Rep. 6(1), 39327 (2016).
[Crossref]

2015 (3)

C. Zhang, Y.-F. Huang, Z. Wang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental greenberger-horne-zeilinger-type six-photon quantum nonlocality,” Phys. Rev. Lett. 115(26), 260402 (2015).
[Crossref]

B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526(7575), 682–686 (2015).
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A. Maitra, S. J. De, G. Paul, and A. K. Pal, “Proposal for quantum rational secret sharing,” Phys. Rev. A 92(2), 022305 (2015).
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2014 (6)

D. R. Hamel, L. K. Shalm, H. Hübel, A. J. Miller, F. Marsili, V. B. Verma, R. P. Mirin, S. W. Nam, K. J. Resch, and T. Jennewein, “Direct generation of three-photon polarization entanglement,” Nat. Photonics 8(10), 801–807 (2014).
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Y. Wang, W.-S. Bao, H.-W. Li, C. Zhou, and Y. Li, “Security of a practical semi-device-independent quantum key distribution protocol against collective attacks,” Chin. Phys. B 23(8), 080303 (2014).
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U. Vazirani and T. Vidick, “Fully Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 113(14), 140501 (2014).
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J.-Å. Larsson, M. Giustina, J. Kofler, B. Wittmann, R. Ursin, and S. Ramelow, “Bell-inequality violation with entangled photons, free of the coincidence-time loophole,” Phys. Rev. A 90(3), 032107 (2014).
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C. Erven, E. Meyer-Scott, K. Fisher, J. Lavoie, B. L. Higgins, Z. Yan, C. J. Pugh, J.-P. Bourgoin, R. Prevedel, L. K. Shalm, L. Richards, N. Gigov, R. Laflamme, G. Weihs, T. Jennewein, and K. J. Resch, “Experimental three-photon quantum nonlocality under strict locality conditions,” Nat. Photonics 8(4), 292–296 (2014).
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B. A. Bell, D. Markham, D. A. Herrera-Martí, A. Marin, W. J. Wadsworth, J. G. Rarity, and M. S. Tame, “Experimental demonstration of graph-state quantum secret sharing,” Nat. Commun. 5(1), 5480 (2014).
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2013 (5)

J. Barrett, R. Colbeck, and A. Kent, “Memory Attacks on Device-Independent Quantum Cryptography,” Phys. Rev. Lett. 110(1), 010503 (2013).
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B. W. Reichardt, F. Unger, and U. Vazirani, “Classical command of quantum systems,” Nature 496(7446), 456–460 (2013).
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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(5), 052322 (2013).
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M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. Kofler, J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. W. Nam, R. Ursin, and A. Zeilinger, “Bell violation using entangled photons without the fair-sampling assumption,” Nature 497(7448), 227–230 (2013).
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B. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calkins, T. Gerrits, A. E. Lita, A. Miller, L. K. Shalm, Y. Zhang, S. W. Nam, N. Brunner, C. C. W. Lim, N. Gisin, and P. G. Kwiat, “Detection-Loophole-Free Test of Quantum Nonlocality, and Applications,” Phys. Rev. Lett. 111(13), 130406 (2013).
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2012 (4)

H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
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C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85(1), 010301 (2012).
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J. Barrett, R. Colbeck, and A. Kent, “Unconditionally secure device-independent quantum key distribution with only two devices,” Phys. Rev. A 86(6), 062326 (2012).
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X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nat. Photonics 6(4), 225–228 (2012).
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2011 (3)

M. Pawłowski and N. Brunner, “Semi-device-independent security of one-way quantum key distribution,” Phys. Rev. A 84(1), 010302 (2011).
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M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD Network,” Opt. Express 19(11), 10387 (2011).
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D. Stucki, M. Legré, F. Buntschu, B. Clausen, N. Felber, N. Gisin, L. Henzen, P. Junod, G. Litzistorf, P. Monbaron, L. Monat, J.-B. Page, D. Perroud, G. Ribordy, A. Rochas, S. Robyr, J. Tavares, R. Thew, P. Trinkler, S. Ventura, R. Voirol, N. Walenta, and H. Zbinden, “Long-term performance of the swissquantum quantum key distribution network in a field environment,” New J. Phys. 13(12), 123001 (2011).
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2010 (1)

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(10), 686–689 (2010).
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2009 (5)

M. Koashi, “Simple security proof of quantum key distribution based on complementarity,” New J. Phys. 11(4), 045018 (2009).
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S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, “Device-independent quantum key distribution secure against collective attacks,” New J. Phys. 11(4), 045021 (2009).
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M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New J. Phys. 11(7), 075001 (2009).
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F. Xu, W. Chen, S. Wang, Z. Yin, Y. Zhang, Y. Liu, Z. Zhou, Y. Zhao, H. Li, D. Liu, Z. Han, and G. Guo, “Field experiment on a robust hierarchical metropolitan quantum cryptography network,” Chin. Sci. Bull. 54(17), 2991–2997 (2009).
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R. Ceccarelli, G. Vallone, F. De Martini, P. Mataloni, and A. Cabello, “Experimental entanglement and nonlocality of a two-photon six-qubit cluster state,” Phys. Rev. Lett. 103(16), 160401 (2009).
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2007 (5)

P. Walther, M. Aspelmeyer, and A. Zeilinger, “Heralded generation of multiphoton entanglement,” Phys. Rev. A 75(1), 012313 (2007).
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S. Gaertner, C. Kurtsiefer, M. Bourennane, and H. Weinfurter, “Experimental demonstration of four-party quantum secret sharing,” Phys. Rev. Lett. 98(2), 020503 (2007).
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A. Acín, J. I. Cirac, and M. Lewenstein, “Entanglement percolation in quantum networks,” Nat. Phys. 3(4), 256–259 (2007).
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R. Matsumoto, “Multiparty quantum-key-distribution protocol without use of entanglement,” Phys. Rev. A 76(6), 062316 (2007).
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A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
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2006 (5)

V. Makarov, A. Anisimov, and J. Skaar, “Effects of detector efficiency mismatch on security of quantum cryptosystems,” Phys. Rev. A 74(2), 022313 (2006).
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A. Acín, N. Gisin, and L. Masanes, “From Bell’s Theorem to Secure Quantum Key Distribution,” Phys. Rev. Lett. 97(12), 120405 (2006).
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A. Acín, S. Massar, and S. Pironio, “Efficient quantum key distribution secure against no-signalling eavesdroppers,” New J. Phys. 8(8), 126 (2006).
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V. Scarani, N. Gisin, N. Brunner, L. Masanes, S. Pino, and A. Acín, “Secrecy extraction from no-signaling correlations,” Phys. Rev. A 74(4), 042339 (2006).
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M. Koashi, “Unconditional security of quantum key distribution and the uncertainty principle,” J. Phys. Conf. Ser. 36, 98–102 (2006).
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2005 (4)

I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. R. Soc. London, Ser. A 461(2053), 207–235 (2005).
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R. Renner, N. Gisin, and B. Kraus, “Information-theoretic security proof for quantum-key-distribution protocols,” Phys. Rev. A 72(1), 012332 (2005).
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J. Barrett, L. Hardy, and A. Kent, “No signaling and quantum key distribution,” Phys. Rev. Lett. 95(1), 010503 (2005).
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Z.-j. Zhang, Y. Li, and Z.-x. Man, “Multiparty quantum secret sharing,” Phys. Rev. A 71(4), 044301 (2005).
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2003 (2)

S. Gaertner, M. Bourennane, M. Eibl, C. Kurtsiefer, and H. Weinfurter, “High-fidelity source of four-photon entanglement,” Appl. Phys. B 77(8), 803–807 (2003).
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Z. Zhao, T. Yang, Y.-A. Chen, A.-N. Zhang, M. Żukowski, and J.-W. Pan, “Experimental violation of local realism by four-photon greenberger-horne-zeilinger entanglement,” Phys. Rev. Lett. 91(18), 180401 (2003).
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2002 (1)

M. Żukowski and Č. Brukner, “Bell’s Theorem for General N -Qubit States,” Phys. Rev. Lett. 88(21), 210401 (2002).
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2001 (5)

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86(20), 4435–4438 (2001).
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H. Weinfurter and M. Żukowski, “Four-photon entanglement from down-conversion,” Phys. Rev. A 64(1), 010102 (2001).
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V. Scarani and N. Gisin, “Quantum communication between $\mathit {N}$N partners and bell’s inequalities,” Phys. Rev. Lett. 87(11), 117901 (2001).
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V. Scarani and N. Gisin, “Quantum key distribution between N partners: Optimal eavesdropping and Bell’s inequalities,” Phys. Rev. A 65(1), 012311 (2001).
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D. Mayers, “Unconditional security in quantum cryptography,” J. Assoc. Comput. Mach. 48(3), 351–406 (2001).
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2000 (4)

P. W. Shor and J. Preskill, “Simple Proof of Security of the BB84 Quantum Key Distribution Protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000).
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G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. 85(6), 1330–1333 (2000).
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R. F. Werner and M. M. Wolf, “Bell’s inequalities for states with positive partial transpose,” Phys. Rev. A 61(6), 062102 (2000).
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W. Dür and J. I. Cirac, “Classification of multiqubit mixed states: Separability and distillability properties,” Phys. Rev. A 61(4), 042314 (2000).
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1999 (2)

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
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A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59(1), 162–168 (1999).
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1996 (1)

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels,” Phys. Rev. Lett. 77(13), 2818–2821 (1996).
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1995 (1)

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1993 (1)

A. V. Belinskiĭ and D. N. Klyshko, “Interference of light and Bell’s theorem,” Phys. Usp. 36(8), 653–693 (1993).
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1992 (1)

M. Ardehali, “Bell inequalities with a magnitude of violation that grows exponentially with the number of particles,” Phys. Rev. A 46(9), 5375–5378 (1992).
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1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
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1990 (2)

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1982 (1)

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1973 (1)

A. S. Holevo, “Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel,” Probl. Inf. Transm. 9, 177–183 (1973).

1969 (1)

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B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526(7575), 682–686 (2015).
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Acín, A.

S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, “Device-independent quantum key distribution secure against collective attacks,” New J. Phys. 11(4), 045021 (2009).
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A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent Security of Quantum Cryptography against Collective Attacks,” Phys. Rev. Lett. 98(23), 230501 (2007).
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A. Acín, J. I. Cirac, and M. Lewenstein, “Entanglement percolation in quantum networks,” Nat. Phys. 3(4), 256–259 (2007).
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V. Scarani, N. Gisin, N. Brunner, L. Masanes, S. Pino, and A. Acín, “Secrecy extraction from no-signaling correlations,” Phys. Rev. A 74(4), 042339 (2006).
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A. Acín, N. Gisin, and L. Masanes, “From Bell’s Theorem to Secure Quantum Key Distribution,” Phys. Rev. Lett. 97(12), 120405 (2006).
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V. Makarov, A. Anisimov, and J. Skaar, “Effects of detector efficiency mismatch on security of quantum cryptosystems,” Phys. Rev. A 74(2), 022313 (2006).
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Figures (5)

Fig. 1.
Fig. 1. A schematic diagram of $N$-QKD. Alice generates a $N$-qubit GHZ state. Subsequently, Alice keeps one qubit and distributes the others to Bobs.
Fig. 2.
Fig. 2. A schematic diagram for generating the 3-photon GHZ state based on Fig. 1(a) in [67]. In this setup, 3-photon polarization GHZ state is generated by using cascade entangled photon sources (EPSs). $| {H}\rangle$ and $| {V}\rangle$ are corresponding to the eigenstates of $\hat {Z}$, $| {0} \rangle$ and $| {1}\rangle$, in the maintext. Normalization factors are ignored in the diagram. QWP : quarter wave plate.
Fig. 3.
Fig. 3. A schematic diagram for implementing photonic polarization $X$-, $Y$-, and $Z$-measurement used in [67]. Measurement basis can be chosen by controlling the motorized wave plate (MWP). Subsequently, the incoming photon goes to the polarization beam splitter (PBS). Polarization of the incoming photon is measured from a click of a single photon detector (SPD).
Fig. 4.
Fig. 4. Comparison among the secret key rates of $N$-QKD obtained by using the proposed SDI analysis (solid lines), the IT analysis (dashed lines) and the secret key rate of DI-CKA (dotted lines) under the depolarizing noise. The black lines represent the secret key rates when the number of authorized parties is $3$, and the red lines does when the number of parties is $11$.
Fig. 5.
Fig. 5. (a) The secret key rates of $N$-QKD under the depolarizing channel noise with the various number of parties $N$. The secret key rates are evaluated by using the proposed SDI analysis. It is shown that the secret key rate becomes more efficient for increasing $N$. (b) The threshold of multiparty QBER $Q_{b}$ when the secret key rate becomes zero against the number of parties $N$ in the depolarizing channel noise. The difference between the thresholds QBER of $N$-QKD and $(N+1)$-QKD becomes smaller for increasing $N$.

Equations (28)

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M^1=X^1M^N=12[M^N1(X^N+Y^N)+M¯^N1(X^NY^N)]
M^NLR1.
SN=M^NQ2N12.
|GHZN=12(|0N+eiθ(N)|1N)
|Φ+=12(|0,0+|1,1).
|Φ+=12(|0,0+|1,1)=12(|+,++|,)
|GHZ3=12(|0,0,0+i|1,1,1)=12(|i,i+,i++|i+,i,i++|i+,i+,i+|i,i,i)
|ψj±=12(|0|j±eiθ(N)|1|j¯)
D={X^N}{Z^AZ^Bi|1jN1}{R^k|1kN1}
R^k|ψjσ={|ψjσifj(k)=0,i|ψjσifj(k)=1
ρ^N=λ0+|ψ0+ψ0+|+λ0|ψ0ψ0|+j=12N11λj(|ψj+ψj+|+|ψjψj|).
rmin=min1kN1[I(A,Bk)]χ(A;E)
min1kN1[I(A,Bk)]=min1kN1[1h(QBk)]=1max1kN1[h(QBk)]
χ(A;E)=S(ρ^E)z=±1p(z)S(ρ^E|z)
|ΨABsE=j=02N11σ=±λjσ|ψj|ej.
S(ρE)=j=02N11σ=±λjσlog2λjσ.
Z^=cosφZ^+sinφX^.
|z=1+zcosφ2|0+z1zcosφ2|1
12minφ[0,π][S(ρ^E|z=1)+S(ρ^E|z=+1)]=j=02N11(λj++λj)log2(λj++λj)
χ(A;E)λ0+log2λ0+λ0log2λ0+(λ0++λ0)log2(λ0++λ0)+1λ0+λ0.
F(λ0+,λ0)h(12+12[λ0+λ0])=G(λ0+,λ0)
SN=Tr(ρ^NM^N)=2N12(λ0+λ0).
χ(A;E)G(λ0+,λ0)h(12+12max[1,(SN)2]12N11)
rN=1maxk[h(QBk)]h(12+12max[1,(SN)2]12N11).
W(S)=h(12+12max[1,(SN)2]12N11)
ipiχi(A;E)ipiW(Si)W(ipiSi)
ρ^N,dp=λ0+|ψ0+ψ0+|+1λ0+2N1(I|ψ0+ψ0+|).
λ0+=1(2N1)Qb2N2λ0=Qb2N2.

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