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arXiv:1008.4663

Security of six-state quantum key distribution protocol with threshold detectors

Go Kato, Kiyoshi Tamaki

Published Aug 27, 2010• 1 citation
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Abstract

We prove the unconditional security of the six-state protocol with threshold detectors and one-way classical communication. Unlike the four-state protocol (BB84), it has been proven that the squash operator for the six-state does not exist, i.e., the statistics of the measurements cannot be obtained via measurement on qubits. We propose a technique to determine which photon number states are important, and we consider a fictitious measurement on a qubit, which is defined through the squash operator of BB84, for the better estimation of Eve's information. As a result, we prove that the bit error rate threshold for the six-state protocol (12.611%) remains almost the same as the one of the qubit-based six-state protocol (12.619%). This clearly demonstrates the robustness of the six-state protocol against the use of the practical devices.

DOI
10.48550/arxiv.1008.4663

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Abstract. Photon number resolving detectors can enhance the performance of many practical quantum cryptographic setups. In this paper, we employ a simple method to estimate the statistics provided by such a photon number resolving detector using only a threshold detector together with a variable attenuator. This idea is similar in spirit to that of the decoy state technique, and is specially suited for those scenarios where only a few parameters of the photon number statistics of the incoming signals have to be estimated. As an illustration of the potential applicability of the method in quantum communication protocols, we use it to prove security of an entanglement based quantum key distribution scheme with an untrusted source without the need of a squash model and by solely using this extra idea. In this sense, this detector decoy method can be seen as a different conceptual approach to adapt a single photon security proof to its physical, full optical implementation. We show that in this scenario the legitimate users can now even discard the double click events from the raw key data without compromising the security of the scheme, and we present simulations on the performance of the BB84 and the 6-state quantum key distribution protocols. 1.

•Jan 1, 2009•15 citations
We prove the unconditional security of the six-state protocol with threshold detectors and one-way classical communication. Unlike the four-state protocol (BB84), it has been proven that the squash operator for the six-state does not exist, i.e., the statistics of the measurements cannot be obtained via measurement on qubits. We propose a technique to determine which photon number states are important, and we consider a fictitious measurement on a qubit, which is defined through the squash operator of BB84, for the better estimation of Eve's information. As a result, we prove that the bit error rate threshold for the six-state protocol (12.611%) remains almost the same as the one of the qubit-based six-state protocol (12.619%). This clearly demonstrates the robustness of the six-state protocol against the use of the practical devices.
The security of quantum key distribution (QKD) is established by a security proof, and the security proof puts some assumptions on the devices consisting of a QKD system. Among such assumptions, security proofs of the six-state protocol assume the use of photon number resolving (PNR) detector, and as a result the bit error rate threshold for secure key generation for the six-state protocol is higher than that for the BB84 protocol. Unfortunately, however, this type of detector is demanding in terms of technological level compared to the standard threshold detector, and removing the necessity of such a detector enhances the feasibility of the implementation of the six-state protocol. Here, we develop the security proof for the six-state protocol and show that we can use the threshold detector for the six-state protocol. Importantly, the bit error rate threshold for the key generation for the six-state protocol (12.611%) remains almost the same as the one (12.619%) that is derived from the existing security proofs assuming the use of PNR detectors. This clearly demonstrates feasibility of the six-state protocol with practical devices.
We prove the unconditional security of the standard six-state scheme for quantum key distribution (QKD). We demonstrate its unconditional security up to a bit error rate of 12.7 percents, by allowing only one-way classical communications in the error correction/privacy amplification procedure between Alice and Bob. This shows a clear advantage of the six-state scheme over another standard scheme---BB84, which has been proven to be secure up to only about 11 percents, if only one-way classical communications are allowed. Our proof technique is a generalization of that of Shor-Preskill's proof of security of BB84. We show that a advantage of the six-state scheme lies in the Alice and Bob's ability to establish rigorously from their test sample the non-trivial mutual information between the bit-flip and phase error patterns. A modified version of the degenerate quantum codes studied by DiVincenzo, Shor and Smolin is employed in our proof.
Standard security proofs of quantum-key-distribution (QKD) protocols often rely on symmetry arguments. In this paper, we prove the security of a three-state protocol that does not possess rotational symmetry. The three-state QKD protocol we consider involves three qubit states, where the first two states $\ensuremath{\mid}{0}_{z}⟩$ and $\ensuremath{\mid}{1}_{z}⟩$ can contribute to key generation, and the third state $\ensuremath{\mid}+⟩=(\ensuremath{\mid}{0}_{z}⟩+\ensuremath{\mid}{1}_{z}⟩)∕\sqrt{2}$ is for channel estimation. This protocol has been proposed and implemented experimentally in some frequency-based QKD systems where the three states can be prepared easily. Thus, by founding on the security of this three-state protocol, we prove that these QKD schemes are, in fact, unconditionally secure against any attacks allowed by quantum mechanics. The main task in our proof is to upper bound the phase error rate of the qubits given the bit error rates observed. Unconditional security can then be proved not only for the ideal case of a single-photon source and perfect detectors, but also for the realistic case of a phase-randomized weak coherent light source and imperfect threshold detectors. Our result in the phase error rate upper bound is independent of the loss in the channel. Also, we compare the three-state protocol with the Bennett-Brassard 1984 (BB84) protocol. For the single-photon source case, our result proves that the BB84 protocol strictly tolerates a higher quantum bit error rate than the three-state protocol, while for the coherent-source case, the BB84 protocol achieves a higher key generation rate and secure distance than the three-state protocol when a decoy-state method is used.
In this paper, we rigorously prove the intuition that in security proofs for the Bennett-Brassard 1984 (BB84) protocol, one may regard an incoming signal to Bob as a qubit state. From this result, it follows that all security proofs for BB84 protocol based on a virtual qubit entanglement distillation protocol, which was originally proposed by Lo and Chau [Science 283, 2050 (1999)] and by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)], are all valid even if Bob's actual apparatus cannot distill a qubit state explicitly. As a consequence, especially, the well-known result that a higher bit error rate of 20% can be tolerated for BB84 protocol by using two-way classical communications is still valid even when Bob uses threshold detectors. Using the same technique, we also prove the security of Bennett-Brassard-Mermin 1992 (BBM92) protocol where Alice and Bob both use threshold detectors.
We investigate the resilience to noise of single-qubit quantum key distribution (QKD) protocols in the scenario of security against independent eavesdropping attacks and key distillation based on one-way classical communication. To this end, we introduce a noise-based metric that quantifies the efficiency of QKD protocols. Within this framework, we analyze the maximal noise levels that allow Alice and Bob to asymptotically establish a secure secret key. Using this assumption, we compare the noise tolerance of general single-qubit QKD protocols, in particular the BB84, B92, E91, and six-state protocols. Our main result determines the noise level threshold for QKD allowing one to distill an asymptotically secure secret key. Additionally, we demonstrate that the six-state protocol achieves the greatest resistance to noise while simultaneously yielding a higher post-selection efficiency than the other analyzed single-qubit protocols, confirming its robustness within the considered security model. Finally, we perform an analysis of the proposed noise-based metric and the conventional quantum bit error rate (QBER) metric.
Abstract. Photon number resolving detectors can enhance the performance of many practical quantum cryptographic setups. In this paper, we employ a simple method to estimate the statistics provided by such a photon number resolving detector using only a threshold detector together with a variable attenuator. This idea is similar in spirit to that of the decoy state technique, and is specially suited for those scenarios where only a few parameters of the photon number statistics of the incoming signals have to be estimated. As an illustration of the potential applicability of the method in quantum communication protocols, we use it to prove security of an entanglement based quantum key distribution scheme with an untrusted source without the need of a squash model and by solely using this extra idea. In this sense, this detector decoy method can be seen as a different conceptual approach to adapt a single photon security proof to its physical, full optical implementation. We show that in this scenario the legitimate users can now even discard the double click events from the raw key data without compromising the security of the scheme, and we present simulations on the performance of the BB84 and the 6-state quantum key distribution protocols. 1.