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arXiv:quant-ph/0204127

Reverse reconciliation protocols for quantum cryptography with continuous variables

Frédéric Grosshans, Philippe Grangier

Published Apr 22, 2002• 61 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0204127

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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 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 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.
A big challenge in continuous-variable quantum key distribution is to prove security against arbitrary coherent attacks including realistic assumptions such as finite-size effects. Recently, such a proof has been presented in [Phys. Rev. Lett. 109, 100502 (2012)] for a two-mode squeezed state protocol based on a novel uncertainty relation with quantum memories. But the transmission distances were fairly limited due to a direct reconciliation protocol. We prove here security against coherent attacks of a reverse-reconciliation protocol under similar assumptions but allowing distances of over 16 km for experimentally feasible parameters. We further clarify the limitations when using the uncertainty relation with quantum memories in security proofs of continuous-variable quantum key distribution.
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.
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.
Most quantum key distribution (QKD) protocols can be classified as either a discrete-variable (DV) protocol or continuous-variable (CV) protocol, based on how classical information is being encoded. We propose a protocol that combines the best of both worlds – the simplicity of quantum state preparation in DV-QKD together with the cost-effective and high-bandwidth of homodyne detectors used in CV-QKD. Our proposed protocol has two highly practical features: (1) it does not require the honest parties to share the same reference phase (as required in CV-QKD) and (2) the selection of decoding basis can be performed after measurement. We also prove the security of the proposed protocol in the asymptotic limit under the assumption of collective attacks. Our simulation suggests that the protocol is suitable for secure and high-speed practical key distribution over metropolitan distances.