# Explicit asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation

Aurélie Denys1, Peter Brown2, and Anthony Leverrier1

1Inria, France
2ENS Lyon, France

### Abstract

We establish an analytical lower bound on the asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation of coherent states. Previously, such bounds were only available for protocols with a Gaussian modulation, and numerical bounds existed in the case of simple phase-shift-keying modulations. The latter bounds were obtained as a solution of convex optimization problems and our new analytical bound matches the results of Ghorai $\textit{et al.}$ (2019), up to numerical precision. The more relevant case of quadrature amplitude modulation (QAM) could not be analyzed with the previous techniques, due to their large number of coherent states. Our bound shows that relatively small constellation sizes, with say 64 states, are essentially sufficient to obtain a performance close to a true Gaussian modulation and are therefore an attractive solution for large-scale deployment of continuous-variable quantum key distribution. We also derive similar bounds when the modulation consists of arbitrary states, not necessarily pure.

Quantum key distribution (QKD) allows two distant agents to generate a shared secret key using an untrusted quantum channel and classical communication. It is a promising near-term application of quantum technologies, enabling information-theoretically secure communication. QKD schemes that operate using continuous variable (CV) systems are particularly interesting in this regard as they can likely be integrated into existing telecom networks. However, analyzing CV QKD protocols is difficult due to the infinite dimensional nature of the underlying Fock space and a pressing open problem is how to obtain reasonably tight bounds on the secret key rates for general protocols.
In this work, we provide a solution to this problem by deriving an explicit analytical lower bound on the asymptotic secret key rate of any standard one-way CV QKD protocol. Our analytical results allow us to account for imperfections in the state preparation and also straightforwardly to optimize the preparation constellations, further improving performance of the protocols.

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### Cited by

[1] Florian Kanitschar and Christoph Pacher, "Postselection Strategies for Continuous-Variable Quantum Key Distribution Protocols with Quadrature Phase-Shift Keying Modulation", arXiv:2104.09454.

[2] Wen-Bo Liu, Chen-Long Li, Yuan-Mei Xie, Chen-Xun Weng, Jie Gu, Xiao-Yu Cao, Yu-Shuo Lu, Bing-Hong Li, Hua-Lei Yin, and Zeng-Bing Chen, "Homodyne Detection Quadrature Phase Shift Keying Continuous-Variable Quantum Key Distribution with High Excess Noise Tolerance", arXiv:2104.11152.

[3] Cosmo Lupo and Yingkai Ouyang, "Quantum key distribution with non-ideal heterodyne detection", arXiv:2108.00428.

[4] Min-Gang Zhou, Zhi-Ping Liu, Wen-Bo Liu, Chen-Long Li, Jun-Lin Bai, Yi-Ran Xue, Yao Fu, Hua-Lei Yin, and Zeng-Bing Chen, "Machine learning for secure key rate in continuous-variable quantum key distribution", arXiv:2108.02578.

[5] Ignatius William Primaatmaja, Cassey Liang, Gong Zhang, Jing Yan Haw, Chao Wang, and Charles Ci-Wen Lim, "Discrete-variable quantum key distribution with homodyne detection", arXiv:2109.00492.

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