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Journal of the Physical Society of Japan

Phase Coherent States for Enhancing the Performance of Continuous Variable Quantum Key Distribution

Ahmed Becir, Mohamed Ridza Wahiddin

Published Mar 9, 2012• 5 citations
Publisher

Abstract

In this paper, we present three continuous variable quantum key distribution protocols (named two-, four-, and eight-state protocols) based on the discrete modulation of phase coherent states. We investigate the security of these protocols against the collective eavesdropping attacks and show their performance in a realistic setup by taking into consideration the realistic lossy noisy quantum channels, and the imperfection of detection. We show that the use of phase coherent states gives better performance in terms of noise tolerance, key rate, and achievable distance than that of common linear coherent states.

DOI
10.1143/jpsj.81.034005

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In this paper, we present three continuous variable quantum key distribution protocols (named two-, four-, and eight-state protocols) based on the discrete modulation of phase coherent states. We investigate the security of these protocols against the collective eavesdropping attacks and show their performance in a realistic setup by taking into consideration the realistic lossy noisy quantum channels, and the imperfection of detection. We show that the use of phase coherent states gives better performance in terms of noise tolerance, key rate, and achievable distance than that of common linear coherent states.
We establish a lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with a discrete modulation of coherent states. The bound is valid against collective attacks and is obtained by formulating the problem as a semidefinite program. We illustrate our general approach with the quadrature phase-shift keying (QPSK) modulation scheme and show that distances over 100 km are achievable for realistic values of noise. We also discuss the application to more complex quadrature amplitude modulations (QAM) schemes. This work is a major step towards establishing the full security of continuous-variable protocols with a discrete modulation in the finite-size regime and opens the way to large-scale deployment of these protocols for quantum key distribution.
In this paper, we first study a generalized protocol of discrete modulation for continuous-variable quantum key distribution with N coherent states in a Gaussian lossy and noisy channel and investigate its performance against collective attacks. We find that discrete modulation protocols with more than eight states do not perform better than the eight-state protocol. Then, we study the improvement of this protocol by using a nondeterministic noiseless linear amplifier (NLA) on Bob's detection stage. Results indicate that a NLA with gain g can extend the maximum transmission distance by 50 log10g2km and can increase the maximal tolerable excess noise. With the reconciliation efficiency β, we find the gain of NLA has a maximal value defined as gmaxand by adjusting the gain to about βgmaxone can have the best improvement on secret key rate.
Security of continuous-variable quantum key distribution with four-state protocol based on discrete modulation of noisy coherent states is analyzed. Combing discrete modulation and reverse reconciliation, this protocol can be used for long distance cryptography. There is a small difference between the state Alice sends in the discrete modulation protocol and the Gaussian modulation protocol, and it can be treated as excess noise and loss in the channel. As Alice cannot do a precise modulation, she will induce noise to the coherent state. We look this noise as the source noise and derive a lower bound to the secure key rate assuming the eavesdropper cannot benefit from the noise in the source. For avoiding the fast and random phase locking between the signal and local oscillator in experiment, we also analyze the security of four-state protocol using no-switching scheme.
We consider discrete-modulation protocols for continuous-variable quantum key distribution (CV-QKD) that employ a modulation constellation consisting of a finite number of coherent states and that use a homodyne- or a heterodyne-detection receiver. We establish a security proof for collective attacks in the asymptotic regime, and we provide a formula for an achievable secret-key rate. Previous works established security proofs for discrete-modulation CV-QKD protocols that use two or three coherent states. The main constituents of our approach include approximating a complex, isotropic Gaussian probability distribution by a finite-size Gauss-Hermite constellation, applying entropic continuity bounds, and leveraging previous security proofs for Gaussian-modulation protocols. As an application of our method, we calculate secret-key rates achievable over a lossy thermal bosonic channel. We show that the rates for discrete-modulation protocols approach the rates achieved by a Gaussian-modulation protocol as the constellation size is increased. For pure-loss channels, our results indicate that in the high-loss regime and for sufficiently large constellation size, the achievable key rates scale optimally, i.e., proportional to the channel's transmissivity.