Asymmetry effects in homodyne and heterodyne measurements: Positive operator-valued measures and asymptotic security of Gaussian-continuous-variable quantum key distribution
A. S. Naumchik, Roman K. Goncharov, Alexei D. Kiselev
quant-phSubmitted Dec 27, 2025• Announced Dec 30, 2025• 0 citations• Revised Jul 20, 2026 (v3)
We use the Gaussian approximation describing photocount statistics for both the homodyne and the double homodyne (heterodyne) measurements to study asymmetry effects arising from imbalance of the beam splitters and variations in quantum efficiencies of the photodetectors. After computing the $Q$ symbols of the positive operator-valued measures (POVMs) of noisy measurements that take into account the asymmetry effects, the operator representations for the POVMs are obtained in the form that assumes applying the additive noise quantum channel to the POVMs of noiseless (ideal) measurements. For double homodyne detection, it was found that the noiseless measurements should generally be expressed in terms of the projectors onto squeezed-states and the corresponding squeezed-state operator representation of POVM along with the measurement noise channel depend on the squeezing parameter that lies in the interval dictated by the condition for the excess noise covariance matrix to be positive semi-definite. The analytical results are used to perform analysis of the asymptotic security of the Gaussian-modulated continuous variable quantum key distribution (CV-QKD) protocol in the untrusted-noise scenario where the measurement noise is assumed to be accessible to an adversary. The inherent non-uniqueness of the operator representation for the double-homodyne POVM manifests itself in the squeezing dependent Holevo information that needs to be additionally optimized. For both types of the measurements, the mutual information, the Holevo information and the asymptotic secret fraction are sensitive to asymmetry effects leading to degraded performance of the protocol.
We use the Gaussian approximation describing photocount statistics for both the homodyne and the double homodyne (heterodyne) measurements to study asymmetry effects arising from imbalance of the beam splitters and variations in quantum efficiencies of the photodetectors. After computing the Q symbols of the positive operator-valued measures (POVMs) of noisy measurements that take into account the asymmetry effects, the operator representations for the POVMs are obtained in the form that assumes applying the additive noise quantum channel to the POVMs of noiseless (ideal) measurements. For double homodyne detection, it was found that the noiseless measurements should generally be expressed in terms of the projectors onto squeezed-states and the corresponding squeezed-state operator representation of POVM along with the measurement noise channel depend on the squeezing parameter that lies in the interval dictated by the condition for the excess noise covariance matrix to be positive semi-definite. The analytical results are used to perform analysis of the asymptotic security of the Gaussian-modulated continuous variable quantum key distribution (CV-QKD) protocol in the untrusted-noise scenario where the measurement noise is assumed to be accessible to an adversary. The inherent non-uniqueness of the operator representation for the double-homodyne POVM manifests itself in the squeezing dependent Holevo information that needs to be additionally optimized. For both types of the measurements, the mutual information, the Holevo information and the asymptotic secret fraction are sensitive to asymmetry effects leading to degraded performance of the protocol.
Abstract We address the crucial role of noisy squeezing in security and performance of continuous-variable (CV) quantum key distribution (QKD) protocols. Squeezing has long been recognized for its numerous advantages in CV QKD, such as enhanced robustness against channel noise and loss, and improved secret key rates. However, the excess noise of the squeezed states, that unavoidably originates already from optical loss as well as other imperfections in the source, raises concerns about its potential exploitation by an eavesdropper. For the widely adopted trust assumption on the excess noise in the signal states, we confirm the stability of the protocol against the noisy squeezing in both purely attenuating as well as noisy channels in the asymptotic limit, which implies perfect parameter estimation. In the finite-size regime we show that this stability largely holds at up to 107 data points using optimal biased homodyne detection for key distribution and parameter estimation. Untrusted assumption on the noisy squeezing, on the other hand, introduces additional security bounds on the squeezing excess noise already in the asymptotic regime, which is further enforced by the finite-size effects. Additionally, we show the critical negative role of noisy squeezing in the case of atmospheric free-space channels, already in the trusted-noise and asymptotic assumptions, which emphasizes the importance of squeezing purity in the free-space quantum channels. Our results pave the way towards practical realization of squeezed-state CV QKD protocols in both fibre and free-space channels.
Abstract We address the crucial role of noisy squeezing in security and performance of continuous-variable (CV) quantum key distribution (QKD) protocols. Squeezing has long been recognized for its numerous advantages in CV QKD, such as enhanced robustness against channel noise and loss, and improved secret key rates. However, the excess noise of the squeezed states, that unavoidably originates already from optical loss as well as other imperfections in the source, raises concerns about its potential exploitation by an eavesdropper. For the widely adopted trust assumption on the excess noise in the signal states, we confirm the stability of the protocol against the noisy squeezing in both purely attenuating as well as noisy channels in the asymptotic limit, which implies perfect parameter estimation. In the finite-size regime we show that this stability largely holds at up to 107 data points using optimal biased homodyne detection for key distribution and parameter estimation. Untrusted assumption on the noisy squeezing, on the other hand, introduces additional security bounds on the squeezing excess noise already in the asymptotic regime, which is further enforced by the finite-size effects. Additionally, we show the critical negative role of noisy squeezing in the case of atmospheric free-space channels, already in the trusted-noise and asymptotic assumptions, which emphasizes the importance of squeezing purity in the free-space quantum channels. Our results pave the way towards practical realization of squeezed-state CV QKD protocols in both fibre and free-space channels.
Continuous Variable (CV) quantum key distribution (QKD) is a promising candidate for practical implementations due to its compatibility with the existing communication technology. A trusted device scenario assuming that an adversary has no access to imperfections such as electronic noises in the detector is expected to provide significant improvement in the key rate, but such an endeavor so far was made separately for specific protocols and for specific proof techniques. Here, we develop a simple and general treatment that can incorporate the effects of Gaussian trusted noises for any protocol that uses homodyne/heterodyne measurements. In our method, a rescaling of the outcome of a noisy homodyne/heterodyne detector renders it equivalent to the outcome of a noiseless detector with a tiny additional loss, thanks to a noise-loss equivalence well-known in quantum optics. Since this method is independent of protocols and security proofs, it is applicable to Gaussian-modulation and discrete-modulation protocols, to the finite-size regime, and to any proof techniques developed so far and yet to be discovered as well.
Continuous-variable quantum key distribution (CV-QKD) is a leading technology for secure communication, whose security guarantees fundamentally rely on the quantum nature of the transmitted states and the measurement process. However, real-world detectors invariably suffer from limitations, such as inefficiency, which affects the overall security and performance of the system. Here, we investigate the impact of detector efficiency on remote state preparation in the context of entanglement-based CV-QKD. Specifically, by using the Wigner function formalism and starting from a two-mode squeezed vacuum state, we derive the analyticalform of the quantum state of one mode conditioned on a homodyne measurement performed on the other mode. Our analysis explicitly contrasts the ideal case of unit efficiency (η=1), which prepares a pure, displaced squeezed state, with the non-ideal scenario. For η < 1, we demonstrate that the preparation result is a mixedstate characterized by increased noise and degraded squeezing. We calculate the exact analytical expressions for the conditional state’s variance and mean displacement, quantifying their dependence on both the efficiency η and the initial squeezing parameter.
In a continuous-variable quantum key distribution (CV-QKD) protocol, which is based on heterodyne detection at the receiver, the application of a noiseless linear amplifier (NLA) on the received signal before the detection can be emulated by the postselection of the detection outcome. Such a postselection, which is also called a measurement-based NLA, requires a cutoff on the amplitude of the heterodyne-detection outcome to produce a normalizable filter function. Increasing the cutoff with respect to the received signals results in a more faithful emulation of the NLA and nearly Gaussian output statistics at the cost of discarding more data. While recent works have shown the benefits of postselection via an asymptotic security analysis, we undertake an investigation of such a postselection utilizing a composable security proof in the realistic finite-size regime, where this tradeoff is extremely relevant. We show that this form of postselection offers only a small fraction of the asymptotic improvement in the finite-size regime. This postselection can improve the secure range of a CV-QKD over lossy thermal channels if the finite block size is sufficiently large and the optimal value for the filter cutoff is typically in the non-Gaussian regime. The relatively modest improvement in the finite-size regime as compared to the asymptotic case highlights the need for new tools to prove the security of non-Gaussian cryptographic protocols. These results also represent a quantitative assessment of a measurement-based NLA with an entangled-state input in both the Gaussian and non-Gaussian regime.
Continuous-variable quantum key distribution protocol using coherent states and heterodyne detection, called the no-switching protocol, is widely used in practical systems due to the simple experimental setup without basis switching and easy assessment to phase information. The security of an ideal no-switching protocol has been proved against general attacks in the finite-size regime and composable security framework, whose heterodyne detector consists of a beam splitter with transmittance of $50\mathrm{%}$ and two ideal homodyne detectors. However, the transmittance of a beam splitter is inaccurate and the two detectors always have different quantum efficiency and electronic noise, which introduce asymmetry into the heterodyne detection, and further lead to the mismatch between the ideal protocol and practical systems, thereby overestimating the secret key rate and resulting in a practical security loophole. In this paper, we close this loophole by proposing a modified no-switching protocol with biased quadrature detection, where the asymmetry of the heterodyne detection is modeled to match the practical systems, and the security of the protocol is analyzed in asymptotic and finite-size regimes. Further, an optimization strategy is proposed to achieve the optimal secret key rate by adjusting the transmittance of the beam splitter. Simulation results show the necessity of considering the asymmetry in heterodyne detection and the effectiveness of the optimization, which provides a promising way to realize a practical secure and high-performance no-switching system.