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arXiv:quant-ph/0412075

Efficient and Robust Quantum Key Distribution With Minimal State Tomography

Berthold‐Georg Englert, Dagomir Kaszlikowski, Hui Khoon Ng, Wee Kang Chua, Řeháček, Jaroslav, Janet Anders

Published Dec 9, 2004• 24 citations
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Abstract

We introduce the Singapore protocol, a qubit protocol for quantum key distribution that is fully tomographic, more efficient than other tomographic protocols, and very robust. Under ideal circumstances the efficiency is log_2(4/3)=0.415 key bits per qubit sent. This is 25% more than the efficiency of 1/3=0.333 for the standard six-state protocol, which sets the benchmark. We describe a simple two-way communication scheme that extracts 0.4 key bits per qubit and thus gets close to the information-theoretical limit. The noise thresholds that we report for a hierarchy of eavesdropping attacks demonstrate the robustness of the protocol: A secure key can be extracted if there is less than 38.9% noise.

DOI
10.48550/arxiv.quant-ph/0412075

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arXiv:0806.0272•Physical Review A•Oct 31, 2008•65 citations
We introduce the Singapore protocol, a qubit protocol for quantum key distribution that is fully tomographic, more efficient than other tomographic protocols, and very robust. Under ideal circumstances the efficiency is log_2(4/3)=0.415 key bits per qubit sent. This is 25% more than the efficiency of 1/3=0.333 for the standard six-state protocol, which sets the benchmark. We describe a simple two-way communication scheme that extracts 0.4 key bits per qubit and thus gets close to the information-theoretical limit. The noise thresholds that we report for a hierarchy of eavesdropping attacks demonstrate the robustness of the protocol: A secure key can be extracted if there is less than 38.9% noise.
The new qubit protocol for quantum key distribution presented here is fully tomographic and more efficient than other tomographic protocols. Under ideal circumstances the efficiency is log2(4/3)=0.415\\log_2(4/3)=0.415log2​(4/3)=0.415 key bits per qubit sent, which is 25% more than the efficiency of 1/3=0.3331/3=0.3331/3=0.333 for the standard 6-state protocol. We describe a simple two-way communication scheme that extracts 0.4 key bits per qubit and thus gets close to the information-theoretical limit, and report noise thresholds for secure key bit generation in the presence of unbiased noise.
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.
We thoroughly analyse the novel quantum key distribution protocol introduced recently in quant-ph/0412075, which is based on minimal qubit tomography. We examine the efficiency of the protocol for a whole range of noise parameters and present a general analysis of incoherent eavesdropping attacks with arbitrarily many steps in the iterative key generation process. The comparison with the tomographic 6-state protocol shows that our protocol has a higher efficiency (up to 20%) and ensures the security of the established key even for noise parameters far beyond the 6-state protocol's noise threshold.
Semiquantum key distribution protocols are designed to allow two parties to establish a shared secret key, secure against an all-powerful adversary, even when one of the users is restricted to measuring and preparing quantum states in one single basis. While interesting from a theoretical standpoint, these protocols have the disadvantage that a two-way quantum communication channel is necessary which generally limits their theoretical efficiency and noise tolerance. In this paper, we construct a semiquantum key distribution (SQKD) protocol which actually takes advantage of this necessary two-way channel, and, after performing an information theoretic security analysis against collective attacks, we show it is able to tolerate a channel noise level higher than any prior SQKD protocol to date. We also compare the noise tolerance of our protocol to other two-way fully quantum protocols, along with BB84 with classical advantage distillation. We also comment on some practical issues involving semiquantum key distribution (in particular, concerning the potential complexity in physical implementation of our protocol as compared with other standard QKD protocols). Finally, we develop techniques that can be applied to the security analysis of other (S)QKD protocols reliant on a two-way quantum communication channel.
Traditionally, quantum information is formulated and implemented in terms of elementary quantum systems e.g. qubits. Now, it has been shown that certain appealing properties of entangled systems, like for instance nonlocality, are more robust against noise when they are realized at the level of higher dimensional systems, like qutrits (or more generally qudits with d strictly larger than 2). It has also been shown, in the case of the generalized Ekert protocol, (a so-called entanglement-based protocol) that confidentiality is enhanced when Quantum Key Distribution is realized with carriers of information of dimensionality strictly higher than 2 (this is because optimal cloning is less effective when the dimension d increases). This brought us to consider the possible generalization of the so-called Singapore protocol, a full tomographic qubit protocol, to dimensions strictly higher than 2. As we shall show, dimension 3 is, like dimension 2, privileged regarding the connections of the protocol with Wigner tomography. We also show that, although the qutrit version of the Singapore protocol is, just like Ekert's protocol, more robust against noise than its qubit counterpart, it is less appealing for what concerns the effective information rate (e.g. the number of qudits to be sent in average in order to exchange one classical bit of the key).
Tomography of the two-qubit density matrix shared by Alice and Bob is an essential ingredient for guaranteeing an acceptable margin of confidentiality during the establishment of a secure fresh key through the quantum key distribution scheme. We show how the Singapore protocol for key distribution is optimal from this point of view, due to the fact that it is based on so-called symmetric informationally complete positive-operator-valued measure (SIC POVM) qubit tomography which allows the most accurate full tomographic reconstruction of an unknown density matrix on the basis of a restricted set of experimental data. We illustrate with the help of experimental data the deep connections that exist between SIC POVM tomography and discrete Wigner representations. We also emphasize the special role played by Bell states in this approach and propose a protocol for quantum key distribution during which a third party is able to concede or to deny a posteriori to the authorized users the ability to build a fresh cryptographic key.