Abstract
Quantum secret sharing is well known as a method for players to share a classical secret for secret sharing in quantum mechanical ways. Most of the results associated with quantum secret sharing are based on pure multipartite entangled states. In reality, however, it is difficult for players to share a pure entangled state, although they can share a state close to the state. Thus, it is necessary to study how to perform the quantum secret sharing based on a general multipartite state. We here present a quantum secret sharing protocol on an N-qubit state close to a pure N-qubit Greenberger–Horne–Zeilinger state. In our protocol, N players use an inequality derived from the Mermin inequality to check secure correlation of classical key bits for secret sharing. We show that if our inequality holds then every legitimate player can have key bits with positive key rate. Therefore, for sufficiently many copies of the state, the players can securely share a classical secret with high probability by means of our protocol.
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References
Blakley, G.R.: Safeguarding cryptographic keys. Proc. Natl. Comput. Conf. 48, 313 (1979)
Shamir, A.: How to share a secret. Commun. ACM 22, 612 (1979)
Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829 (1999)
Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going Beyond Bell’s Theorem. In: Bell’s Theorem, Quantum Theory, and Conceptions of the Universe ed M Kafatos (Dordrecht: Kluwer) p. 69 (1989)
Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A 59, 162 (1999)
Gottesman, D.: Theory of quantum secret sharing. Phys. Rev. A 61, 042311 (2000)
Xiao, L., Long, G.L., Deng, F.G., Pan, J.W.: Efficient multiparty quantum-secret-sharing schemes. Phys. Rev. A 69, 052307 (2004)
jun Zhang, Z., xiao Man, Z.: Multiparty quantum secret sharing of classical messages based on entanglement swapping. Phys. Rev. A 72, 022303 (2005)
Qin, S.J., Gao, F., Wen, Q.Y., Zhu, F.C.: Cryptanalysis of the Hillery–Bužek–Berthiaume quantum secret-sharing protocol. Phys. Rev. A 76, 062324 (2007)
Schauer, S., Huber, M., Hiesmayr, B.C.: Experimentally feasible security check for n-qubit quantum secret sharing. Phys. Rev. A 82, 062311 (2010)
Marin, A., Markham, D.: Equivalence between sharing quantum and classical secrets and error correction. Phys. Rev. A 88, 042332 (2013)
Kogias, I., Xiang, Y., He, Q., Adesso, G.: Unconditional security of entanglement-based continuous-variable quantum secret sharing. Phys. Rev. A 95, 012315 (2017)
Wei, C.Y., Cai, X.Q., Liu, B., Wang, T.Y., Gao, F.: A generic construction of quantum-oblivious-key-transfer-based private query with ideal database security and zero failure. IEEE Trans. Comput. 67, 2 (2018)
Tittel, W., Zbinden, H., Gisin, N.: Experimental demonstration of quantum secret sharing. Phys. Rev. A 63, 042301 (2001)
Lance, A.M., Symul, T., Bowen, W.P., Sanders, B.C., Lam, P.K.: Tripartite quantum state sharing. Phys. Rev. Lett. 92, 177903 (2004)
Chen, Y.A., Zhang, A.N., Zhao, Z., Zhou, X.Q., Lu, C.Y., Peng, C.Z., Yang, T., Pan, J.W.: Experimental quantum secret sharing and third-man quantum cryptography. Phys. Rev. Lett. 95, 200502 (2005)
Gaertner, S., Kurtsiefer, C., Bourennane, M., Weinfurter, H.: Experimental demonstration of four-party quantum secret sharing. Phys. Rev. Lett. 98, 020503 (2007)
Bell, B.A., Markham, D., Herrera-Mart, D.A., Marin, A., Wadsworth, W.J., Rarity, J.G., Tame, M.S.: Experimental demonstration of graph-state quantum secret sharing. Nat. Commun. 5, 5480 (2014)
Chen, K., Lo, H.K.: Multi-partite quantum cryptographic protocols with noisy GHZ states. Q. Inf. Comput. 7, 689 (2007)
Lee, S., Park, J.: Three methods to distill multipartite entanglement over bipartite noisy channels. Phys. Lett. A 372, 3157 (2008)
Dür, W., Cirac, J.I., Tarrach, R.: Separability and distillability of multiparticle quantum systems, separability and distillability of multiparticle quantum systems. Phys. Rev. Lett. 83, 3562 (1999)
Mermin, N.D.: Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1838 (1990)
Belinski, A.V., Klyshko, D.N.: Interference of light and Bell’s theorem. Phys. Usp. 36, 653 (1993)
Werner, R.F.: Werner state. Phys. Rev. A 40, 4277 (1989)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, Quantum Computation and Quantum Information. Cambridge University, Cambridge (2000)
Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A 461, 207 (2005)
Arnon-Friedman, R., Renner, R., Vidick, T.: arXiv:1607.01797
Dupuis, F., Fawzi, O., Renner, R.: arXiv:1607.01796
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2016R1A2B4014928).
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Choi, M., Lee, Y. & Lee, S. Quantum secret sharing and Mermin operator. Quantum Inf Process 17, 258 (2018). https://doi.org/10.1007/s11128-018-2035-2
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DOI: https://doi.org/10.1007/s11128-018-2035-2