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arXiv:quant-ph/0306141 · Quantum Information and Computation

Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables

Frédéric Grosshans, Nicolas J. Cerf, Jérôme Wenger, Rosa Tualle-Brouri, Philippe Grangier

Published Oct 1, 2003• 226 citations
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Abstract

We discuss quantum key distribution protocols using quantum continuous variables. We show that such protocols can be made secure against individual gaussian attacks regardless the transmission of the optical line between Alice and Bob. %while other ones require that the line transmission is larger than 50%. This is achieved by reversing the reconciliation procedure subsequent to the quantum transmission, that is, using Bob's instead of Alice's data to build the key. Although squeezing or entanglement may be helpful to improve the resistance to noise, they are not required for the protocols to remain secure with high losses. Therefore, these protocols can be implemented very simply by transmitting coherent states and performing homodyne detection. Here, we show that entanglement nevertheless plays a crucial role in the security analysis of coherent state protocols. Every cryptographic protocol based on displaced gaussian states turns out to be equivalent to an entanglement-based protocol, even though no entanglement is actually present. This equivalence even holds in the absence of squeezing, for coherent state protocols. This ``virtual'' entanglement is important to assess the security of these protocols as it provides an upper bound on the mutual information between Alice and Bob if they had used entanglement. The resulting security criteria are compared to the separability criterion for bipartite gaussian variables. It appears that the security thresholds are well within the entanglement region. This supports the idea that coherent state quantum cryptography may be unconditionally secure.

DOI
10.26421/qic3.s-6

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We discuss quantum key distribution protocols using quantum continuous variables. We show that such protocols can be made secure against individual gaussian attacks regardless the transmission of the optical line between Alice and Bob. %while other ones require that the line transmission is larger than 50%. This is achieved by reversing the reconciliation procedure subsequent to the quantum transmission, that is, using Bob's instead of Alice's data to build the key. Although squeezing or entanglement may be helpful to improve the resistance to noise, they are not required for the protocols to remain secure with high losses. Therefore, these protocols can be implemented very simply by transmitting coherent states and performing homodyne detection. Here, we show that entanglement nevertheless plays a crucial role in the security analysis of coherent state protocols. Every cryptographic protocol based on displaced gaussian states turns out to be equivalent to an entanglement-based protocol, even though no entanglement is actually present. This equivalence even holds in the absence of squeezing, for coherent state protocols. This ``virtual'' entanglement is important to assess the security of these protocols as it provides an upper bound on the mutual information between Alice and Bob if they had used entanglement. The resulting security criteria are compared to the separability criterion for bipartite gaussian variables. It appears that the security thresholds are well within the entanglement region. This supports the idea that coherent state quantum cryptography may be unconditionally secure.
We introduce new quantum key distribution protocols using quantum continuous variables, that are secure against individual attacks for any transmission of the optical line between Alice and Bob. In particular, it is not required that this transmission is larger than 50 %. Though squeezing or entanglement may be helpful, they are not required, and there is no need for quantum memories or entanglement purification. These protocols can thus be implemented using coherent states and homodyne detection, and they may be more efficient than usual protocols using quantum discrete variables.
We propose several methods for quantum key distribution (QKD) based on the generation and transmission of random distributions of coherent or squeezed states, and we show that they are secure against individual eavesdropping attacks. These protocols require that the transmission of the optical line between Alice and Bob is larger than 50%, but they do not rely on "sub-shot-noise" features such as squeezing. Their security is a direct consequence of the no-cloning theorem, which limits the signal-to-noise ratio of possible quantum measurements on the transmission line. Our approach can also be used for evaluating various QKD protocols using light with Gaussian statistics.
Quantum key distribution (QKD) protocols allow two trusted parties to distribute a cryptographic key which they can further use for the unconditionally secure classical communication. During the last decades the field has grown mature with the commercial prototypes being available. They mostly rely on the faint laser pulses implementing the protocols on the basis of single qubits and qubit pairs. Alternatively, the continuous-variable (CV) protocols based on the multi-photon states of light were developed recently. They are typically built using the quadrature modulation of the coherent or squeezed states and subsequent homodyne detection. In the case of Gaussian modulation the security of the protocols is based on the extremality of Gaussian states, which allows estimating bounds on the leaked information. The Gaussian CV QKD protocols were shown secure for any degree of channel attenuation, but are restricted by the channel noise. Moreover, the applicability of Gaussian CV QKD protocols is limited by the effectiveness of classical post-processing algorithms, used by the trusted parties to process the measured data and distill the cryptographic key. On the other hand, the squeezedstate CV QKD protocols were previously considered ineffective under limited degrees of squeezing, while strong squeezing remains experimentally challenging. In the present work we distinguish between the classical and quantum resources in the Gaussian CV QKD and address the role of nonclassicality in the Gaussian protocols. We demonstrate, that by properly combining squeezed resource and coherent modulation, trusted parties are able to decouple eavesdropper from the channel, thus being able to establish the secure key from any amount of the classical mutual information. Our result shows a very promising path towards the long-distance CV QKD.
We address the role of the phase-insensitive trusted preparation and detection noise in the security of a continuous-variable quantum key distribution, considering the Gaussian protocols on the basis of coherent and squeezed states and studying them in the conditions of Gaussian lossy and noisy channels. The influence of such a noise on the security of Gaussian quantum cryptography can be crucial, even despite the fact that a noise is trusted, due to a strongly nonlinear behavior of the quantum entropies involved in the security analysis. We recapitulate the known effect of the preparation noise in both direct and reverse-reconciliation protocols, as well as the detection noise in the reverse-reconciliation scenario. As a new result, we show the negative role of the trusted detection noise in the direct-reconciliation scheme. We also describe the role of the trusted preparation or detection noise added at the reference side of the protocols in improving the robustness of the protocols to the channel noise, confirming the positive effect for the coherent-state reverse-reconciliation protocol. Finally, we address the combined effect of trusted noise added both in the source and the detector.
In this paper, we propose a continuous variable quantum key distribution protocol based on entanglement of a squeezed coherent state and reverse reconciliation. We present a rigorous security proof against individual and collective attacks with realistic lossy, noisy quantum channels, imperfect detector efficiency and detector electronic noise. This protocol is promising for convenient, high-speed operation at link distances to over 50 km. The setup can be further improved in terms of key generation rate and distance by inserting a noiseless phase-sensitive amplifier at the legitimate receiver Bob's input in order to compensate for the inherent imperfection of heterodyne detection.
A novel quantum key distribution (QKD) protocol is proposed and demonstrated. It is based on the transmission of Gaussian-modulated coherent states that are laser pulses containing a few hundred photons .Shot-noise limited coherent (homodyne) detection is used, which makes it possible to achieve high secret-key rates. It remains, in principle, efficient for any value of the line transmission by use of a "reverse reconciliation" technique, which is shown to be secure against individual attacks based on entanglement and quantum memories. The basic idea is to use a reverse reconciliation protocol, that is, Alice attempts to guess what was received by Bob rather than Bob guessing what was sent by Alice. A reverse protocol always gives Alice an advantage over a potential eavesdropper Eve, regardless the line loss. In addition, a specific advantage of using quantum continuous variables is that the high dimensionality of the phase space may be exploited by modulating the field quadratures with a large dynamics, allowing the encoding of several key bits per pulse. This, together with the fact that fast modulation and detection can be achieved, results in a high-rate secret key distribution.