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arXiv:quant-ph/0610048 · Physical Review A

Key distillation from quantum channels using two-way communication protocols

Joonwoo Bae, Antonio Acín

Published Jan 26, 2007• 76 citations
arXivHTMLPDF

Abstract

We provide a general formalism to characterize the cryptographic properties of quantum channels in the realistic scenario where the two honest parties employ prepare and measure protocols and the known two-way communication reconciliation techniques. We obtain a necessary and sufficient condition to distill a secret key using this type of schemes for Pauli qubit channels and generalized Pauli channels in higher dimension. Our results can be applied to standard protocols such as Bennett-Brassard 1984 or six-state, giving a critical error rate of 20% and 27.6%, respectively. We explore several possibilities to enlarge these bounds, without any improvement. These results suggest that there may exist weakly entangling channels useless for key distribution using prepare and measure schemes.

DOI
10.1103/physreva.75.012334

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We provide a general formalism to characterize the cryptographic properties of quantum channels in the realistic scenario where the two honest parties employ prepare and measure protocols and the known two-way communication reconciliation techniques. We obtain a necessary and sufficient condition to distill a secret key using this type of schemes for Pauli qubit channels and generalized Pauli channels in higher dimension. Our results can be applied to standard protocols such as Bennett-Brassard 1984 or six-state, giving a critical error rate of 20% and 27.6%, respectively. We explore several possibilities to enlarge these bounds, without any improvement. These results suggest that there may exist weakly entangling channels useless for key distribution using prepare and measure schemes.
Quantum cryptography is one of the most important quantum information applications. The present thesis covers several topics on quantum cryptography, such as the security analysis of quantum channels for key distribution protocols and the study of quantum cloning. First, we introduce a general formalism to characterize the cryptographic properties of quantum channels in the realistic scenario where the two honest parties employ prepare and measure protocols and the known two-way communication reconciliation techniques. We derive a necessary and sufficient condition to distill a secret key using this type of schemes for arbitrary bipartite quantum systems of finite dimension. The obtained results suggest that there may exist weakly entangling channels useless for key distribution using prepare and measure schemes. Next, we consider Gaussian states and Gaussian operations for cryptographic tasks and derive a new security condition. As it happens for quantum systems of finite dimension, our results suggest that there may also exist weakly entangled Gaussian states useless for key distribution, using Gaussian operations. Finally, we study the connection between cloning and state estimation. It was a long-standing problem to show whether state estimation becomes equivalent to quantum cloning in the asymptotic limit of an infinite number of clones. The equivalence is proven here using two known results in quantum information theory, the monogamy of quantum states and the properties of entanglement-breaking channels.
We identify those properties of a quantum channel that are relevant for cryptography. We focus on general key distribution protocols that use prepare and measure schemes and the existing classical reconciliation techniques, as these are the protocols feasible with current technology. Given a channel, we derive an easily computable necessary condition of security for such protocols. In spite of its simplicity, this condition is shown to be tight for the Bennett-Brassard 1984 and six-state protocols. We show that the condition becomes also sufficient in the event of a so-called collective attack.
We investigate the error tolerance of quantum cryptographic protocols using ddd-level systems. In particular, we focus on prepare-and-measure schemes that use two mutually unbiased bases and a key-distillation procedure with two-way classical communication. For arbitrary quantum channels, we obtain a sufficient condition for secret-key distillation which, in the case of isotropic quantum channels, yields an analytic expression for the maximally tolerable error rate of the cryptographic protocols under consideration. The difference between the tolerable error rate and its theoretical upper bound tends slowly to zero for sufficiently large dimensions of the information carriers.
Quantum key distribution (QKD) achieves information-theoretic security for a shared secret key, using distributed quantum states. To establish secret bits, two honest parties apply key distillation protocols to the resulting measurement outcomes. In this work, we establish a rigorous connection between the key rate achievable by applying two-way key distillation, such as advantage distillation, and quantum hypothesis testing, via an integral representation of the conditional entropy in terms of the error probabilities of binary state discrimination. This connection significantly improves bounds on key rates at finite blocklengths relative to existing fidelity-based bounds. Moreover, in an important special case, it closes the gap between the known sufficient and conjectured necessary conditions for key generation in the large-blocklength limit. More generally, it establishes a matching asymptotic necessary condition, leaving the corresponding finite-blocklength result open. More broadly, our work shows how results in quantum hypothesis testing can directly sharpen the security analysis of QKD.
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.