Reliability of Calderbank?Shor?Steane codes and security of quantum key distribution

After Mayers (1996 Advances in Cryptography: Proc. Crypto'96 pp 343?57; 2001 J. Assoc. Comput. Mach. 48 351?406) gave a proof of the security of the Bennett?Brassard (1984 Proc. IEEE Int. Conf. on Computers, Systems and Signal Processing (Bangalore, India) pp 175?9) (BB84) quantum key distribution protocol, Shor and Preskill (2000 Phys. Rev. Lett. 85 441?4) made a remarkable observation that a Calderbank?Shor?Steane (CSS) code had been implicitly used in the BB84 protocol, and suggested its security could be proved by bounding the fidelity, say Fn, of the incorporated CSS code of length n in the form for some positive number E. This work presents such a number E = E(R) as a function of the rate of codes R, and a threshold R0 such that E(R) > 0 whenever R < R0, which is larger than the achievable rate based on the Gilbert?Varshamov bound that is essentially given by Shor and Preskill. The codes in the present work are robust against fluctuations of channel parameters, which fact is needed to establish the security rigorously and was not proved for rates above the Gilbert?Varshamov rate before in the literature. As a byproduct, the security of a modified BB84 protocol against any joint (coherent) attacks is proved quantitatively.

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