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New Journal of Physics

Simple security proof of quantum key distribution based on complementarity

Masato Koashi

Published Apr 30, 2009• 287 citations
Publisher

Abstract

We present an approach to the unconditional security of quantum key distribution protocols based on a complementarity argument. The approach is applicable to, but not limited to, every case that has been treated via the argument by Shor and Preskill based on entanglement distillation, with a benefit of decoupling of the error correction from the privacy amplification. It can also treat cases with uncharacterized apparatuses. We derive a secure key rate for the Bennett–Brassard-1984 protocol with an arbitrary source characterized only by a single parameter representing the basis dependence.

DOI
10.1088/1367-2630/11/4/045018

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We present an approach to the unconditional security of quantum key distribution protocols based on a complementarity argument. The approach is applicable to, but not limited to, every case that has been treated via the argument by Shor and Preskill based on entanglement distillation, with a benefit of decoupling of the error correction from the privacy amplification. It can also treat cases with uncharacterized apparatuses. We derive a secure key rate for the Bennett–Brassard-1984 protocol with an arbitrary source characterized only by a single parameter representing the basis dependence.
We present an approach to the unconditional security of quantum key distribution protocols based on the uncertainty principle. The approach applies to every case that has been treated via the argument by Shor and Preskill, and relieve them from the constraints of finding quantum error correcting codes. It can also treat the cases with uncharacterized apparatuses. We derive a secure key rate for the Bennett-Brassard-1984 protocol with an arbitrary source characterized only by a single parameter representing the basis dependence.
It is generally accepted that quantum key distribution (QKD) could supply legitimate users with unconditional security during their communication. Quite a lot of satisfactory efforts have been achieved on experimentations with quantum cryptography. However, when the eavesdropper has extra-powerful computational ability, has access to a quantum computer, for example, and can carry into execution any eavesdropping measurement that is allowed by the laws of physics, the security against such attacks has not been widely studied and rigorously proved for most QKD protocols. Quite recently, Shor and Preskill proved concisely the unconditional security of the Bennett-Brassard 1984 (BB84) protocol. Their method is highly valued for its clarity of concept and concision of form. In order to take advantage of the Shor-Preskill technique in their proof of the unconditional security of the BB84 QKD protocol, we introduced in this paper a transformation that can translate the Bennett 1992 (B92) protocol into the BB84 protocol. By proving that the transformation leaks no more information to the eavesdropper, we proved the unconditional security of the B92 protocol. We also settled the problem proposed by Lo about how to prove the unconditional security of the B92 protocol with the Shor-Preskill method.
We present a technique for proving the security of quantum-key-distribution (QKD) protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel.
Shor and Preskill (see Phys. Rev. Lett., vol.85, p.441, 2000) have provided a simple proof of security of the standard quantum key distribution scheme by Bennett and Brassard (1984) by demonstrating a connection between key distribution and entanglement purification protocols (EPPs) with one-way communications. Here, we provide proofs of security of standard quantum key distribution schemes, Bennett and Brassard and the six-state scheme, against the most general attack, by using the techniques of two-way entanglement purification. We demonstrate clearly the advantage of classical post-processing with two-way classical communications over classical post-processing with only one-way classical communications in quantum key distribution (QKD). This is done by the explicit construction of a new protocol for (the error correction/detection and privacy amplification of) Bennett and Brassard that can tolerate a bit error rate of up to 18.9%, which is higher than what any Bennett and Brassard scheme with only one-way classical communications can possibly tolerate. Moreover, we demonstrate the advantage of the six-state scheme over Bennett and Brassard by showing that the six-state scheme can strictly tolerate a higher bit error rate than Bennett and Brassard. In particular, our six-state protocol can tolerate a bit error rate of 26.4%, which is higher than the upper bound of 25% bit error rate for any secure Bennett and Brassard protocol. Consequently, our protocols may allow higher key generation rate and remain secure over longer distances than previous protocols. Our investigation suggests that two-way entanglement purification is a useful tool in the study of advantage distillation, error correction, and privacy amplification protocols.