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arXiv:2501.10278 · Photonics Research

Finite-size security of continuous-variable quantum key distribution with imperfect heterodyne measurement

Adnan A. E. Hajomer, Akash nag Oruganti, Ivan D. Derkach, Ulrik L. Andersen, Vladyslav C. Usenko, Tobias Gehring

Published Sep 23, 2025• 1 citation
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Abstract

Continuous-variable quantum key distribution (CV-QKD) using coherent states and heterodyne detection enables secure quantum communication based on technology that has large similarities to coherent optical telecommunication. Yet, practical implementations of coherent receivers used in both technologies encounter device imperfections, which for CV-QKD are often not addressed in security proofs. Here, we present a theoretical framework that rigorously accounts for imperfect heterodyne measurements arising from phase and amplitude imbalances in the coherent (heterodyne) receiver. Focusing on collective attacks, we establish a finite-size security proof that reveals how measurement imperfections limit the distance over which a positive key rate is achievable. To mitigate these effects, we propose a local transformation during classical post-processing. We validate our approach experimentally on a CV-QKD system with an imperfect coherent receiver, underscoring its potential for scalable, cost-effective CV-QKD with photonic integrated receivers in which phase and amplitude imbalances naturally appear through manufacturing tolerances.

DOI
10.1364/prj.559136

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Continuous-variable quantum key distribution (CV-QKD) using coherent states and heterodyne detection enables secure quantum communication based on technology that has large similarities to coherent optical telecommunication. Yet, practical implementations of coherent receivers used in both technologies encounter device imperfections, which for CV-QKD are often not addressed in security proofs. Here, we present a theoretical framework that rigorously accounts for imperfect heterodyne measurements arising from phase and amplitude imbalances in the coherent (heterodyne) receiver. Focusing on collective attacks, we establish a finite-size security proof that reveals how measurement imperfections limit the distance over which a positive key rate is achievable. To mitigate these effects, we propose a local transformation during classical post-processing. We validate our approach experimentally on a CV-QKD system with an imperfect coherent receiver, underscoring its potential for scalable, cost-effective CV-QKD with photonic integrated receivers in which phase and amplitude imbalances naturally appear through manufacturing tolerances.
Abstract In comparison to conventional discrete-variable (DV) quantum key distribution (QKD), continuous-variable (CV) QKD with homodyne/heterodyne measurements has distinct advantages of lower-cost implementation and affinity to wavelength division multiplexing. On the other hand, its continuous nature makes it harder to accommodate to practical signal processing, which is always discretized, leading to lack of complete security proofs so far. Here we propose a tight and robust method of estimating fidelity of an optical pulse to a coherent state via heterodyne measurements. We then construct a binary phase modulated CV-QKD protocol and prove its security in the finite-key-size regime against general coherent attacks, based on proof techniques of DV QKD. Such a complete security proof is indispensable for exploiting the benefits of CV QKD.
In comparison to conventional discrete-variable (DV) quantum key distribution (QKD), continuous-variable (CV) QKD with homodyne/heterodyne measurements has distinct advantages of lower-cost implementation and affinity to wavelength division multiplexing. On the other hand, its continuous nature makes it harder to accommodate to practical signal processing, which is always discretized, leading to lack of complete security proofs so far. Here we propose a tight and robust method of estimating fidelity of an optical pulse to a coherent state via heterodyne measurements. We then construct a binary phase modulated CV-QKD protocol and prove its security in the finite-key-size regime against general coherent attacks, based on proof techniques of DV QKD. Such a complete security proof is indispensable for exploiting the benefits of CV QKD.
Continuous variable-quantum key distribution (CV-QKD) based on coherent detection has potential to be a cost-effective solution to deploy secure keys over optical networks. This is due to its compatibility with mature optical communication technologies, such as homodyne and heterodyne detection schemes. Nevertheless, realistic implementations of CV-QKD systems suffer from imperfections in the practical devices used to generate and detect the coherent states. Those imperfections deteriorate the performance of the CV-QKD system and can be exploited by an eavesdropper to steal key information without being detected. In this work, we discuss the impact of practical device imperfections on the excess noise of the CV-QKD and on the achievable transmission distance. We also analyze the impact of detection imperfections on the measurement of the shot noise, and its impact on CV-QKD system security.
Continuous variable quantum key distribution via discrete modulations ensures information-theoretic security using standard telecom technologies, providing affordable and scalable quantum communications with simplified classical postprocessing. However, existing security proofs against general attacks often rely on restrictive assumptions, such as a bounded dimension for coherent states, or require impractically large block sizes. In this work, we develop a finite-size security analysis that removes these limitations while incorporating realistic experimental features. Our approach combines an optimization under infinite dimensions, a security proof against general attacks, a trusted detector model accounting for the receiver imperfections, and postselection strategies to reuse discarded measurement outcomes. We report positive key rates in the finite-size regime for relevant block sizes of the order of 10710^7107. These results contribute to narrowing the gap between theoretical security proofs and practical implementations of discrete-modulated continuous variable quantum key distribution protocols.
Abstract Continuous-variable quantum key distribution (CV-QKD) has many practical advantages including compatibility with current optical communication technology. Implementation using heterodyne measurements is particularly attractive since it eliminates the need for active phase locking of the remote pair of local oscillators, but the full security of CV QKD with discrete modulation was only proved for a protocol using homodyne measurements. Here we propose an all-heterodyne CV-QKD protocol with binary modulation and prove its security against general attacks in the finite-key regime. Although replacing a homodyne measurement with a heterodyne measurement would be naively expected to incur a 3-dB penalty in the rate-distance curve, our proof achieves a key rate with only a 1-dB penalty.
Continuous-variable quantum key distribution (CV-QKD) has emerged as a promising approach for secure quantum communication, offering advantages such as high key generation rates, compatibility with standard telecommunication infrastructure, and potential for integration on photonic chips. This review provides an accessible introduction to the theory of CV-QKD, aimed at researchers entering this rapidly developing field. We focus on fundamental concepts, key protocols, and security analysis essential for understanding CV-QKD systems, with a special emphasis on prepare-and-measure protocols using coherent states under asymptotic security conditions. We explain their equivalence to entanglement-based protocols and detail the security proof framework against collective attacks, encompassing both Gaussian and discrete modulation schemes. We also briefly address more advanced topics, including measurement-device-independent CV-QKD and finite-size security analysis. This work is motivated by Brazil's growing investment in quantum communication technologies. By presenting a clear learning path from basic concepts to advanced topics, this work aims to equip newcomers with the essential tools to engage with current research in CV-QKD, thereby supporting the training of a new generation of researchers in this strategic field.