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.

中文翻译：

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连续变量量子密钥分配的离散调制协议的渐近安全性

我们考虑连续可变量子密钥分配（CV-QKD）的离散调制协议，该协议采用由有限数量的相干态组成的调制星座，并使用零差或外差检测接收器。我们为渐进体制中的集体攻击建立了安全证明，并提供了可实现的秘密密钥率的公式。先前的工作为使用两个或三个相干状态的离散调制CV-QKD协议建立了安全性证明。我们方法的主要组成部分包括通过有限大小的Gauss-Hermite星座图近似复杂的各向同性高斯概率分布，应用熵连续性边界以及为高斯调制协议利用先前的安全证明。作为我们方法的一种应用，我们计算了在有损耗的热玻色子通道上可达到的密钥率。我们显示，随着星座图大小的增加，离散调制协议的速率接近高斯调制协议的速率。对于纯损耗信道，我们的结果表明，在高损耗情况下，对于足够大的星座图，可实现的关键码率最佳缩放，即与信道的透射率成比例。