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Optics Express

Parameter estimation calibration of discretized polar modulation continuous-variable quantum key distribution

Tianyi Wang, Ming Li, Xu Wang, Lei Hou

Published May 30, 2023• 4 citations
Publisher

Abstract

In experimental setups of continuous-variable quantum key distribution (CV-QKD), the ideal Gaussian modulation will suffer from discretization and degrade into discretized polar modulation (DPM), which deteriorates the accuracy of parameter estimation and results in an overestimation of excess noise. We demonstrate that in the asymptotic case, the DPM-induced estimation bias is determined exclusively by the modulation resolutions and can be modeled as a quadratic function. To obtain an accurate estimation, a calibration on the estimated excess noise is implemented based on the closed-form expression of the quadratic bias model, while statistical analysis of the model residuals defines the upper bound of estimated excess noise and the lower bound of secret key rate. Simulation results show that when modulation variance is 25 and excess noise is 0.02, the proposed calibration scheme can eliminate an estimation bias of 14.5%, thus enhancing the efficiency and feasibility of DPM CV-QKD.

DOI
10.1364/oe.492426

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In experimental setups of continuous-variable quantum key distribution (CV-QKD), the ideal Gaussian modulation will suffer from discretization and degrade into discretized polar modulation (DPM), which deteriorates the accuracy of parameter estimation and results in an overestimation of excess noise. We demonstrate that in the asymptotic case, the DPM-induced estimation bias is determined exclusively by the modulation resolutions and can be modeled as a quadratic function. To obtain an accurate estimation, a calibration on the estimated excess noise is implemented based on the closed-form expression of the quadratic bias model, while statistical analysis of the model residuals defines the upper bound of estimated excess noise and the lower bound of secret key rate. Simulation results show that when modulation variance is 25 and excess noise is 0.02, the proposed calibration scheme can eliminate an estimation bias of 14.5%, thus enhancing the efficiency and feasibility of DPM CV-QKD.
In experimental setups of Gaussian modulation continuous-variable quantum key distribution (CV-QKD), the amplitude and phase of coherent states will be modulated and discretized due to the finite resolution of voltages that drive electro-optical modulators, resulting in Gaussian modulation discretization. We demonstrate that the impact of discretization on CV-QKD can be described as a multiplicative coefficient experimentally and modeled as a preparation noise imposed on ideal Gaussian modulation theoretically. To obtain an accurate estimation of quadrature fluctuation induced by discretized polar modulation, a data-filtering process named preselection can be applied before state transmission. Numerical results show that when amplitude resolution is 0.25 and phase resolution is 0.02, discretized polar modulation enables a transmission distance of 69 km under homodyne detection, reaching 89% of ideal Gaussian modulation. As for heterodyne detection, 55 km and 80% are achieved.
In experimental setups of continuous-variable quantum key distribution (CVQKD) independently modulating the amplitude and phase of coherent states, the ideal Gaussian modulation will be degraded into discretized polar modulation (DPM) due to the finite resolution of the driving voltages of electro-optical modulators. To compensate for the performance degradation induced by the joint effect of amplitude and phase discretization, linear optics cloning machine (LOCM) can be introduced at the receiver side to reduce the impact of channel excess noise. Implemented by linear optical elements, homodyne detection and controlled displacement, LOCM introduces extra noise that can be transformed into an advantageous one to combat channel excess noise by dynamically adjusting the relevant parameters into a suitable range. In this paper, the prepare-and-measure version of LOCM DPM-CVQKD is presented, where the incoming signal state enters a tunable LOCM before being measured by the nonideal heterodyne detector. The equivalent entanglement-based model is also established to perform security analysis, where the LOCM is reformulated into combining the incoming signal state and a thermal state on a beam splitter. The composable secret key rate is derived to investigate the security of LOCM DPM-CVQKD. Simulation results demonstrate that the secret key rate is closely related to the tuning gain and the transmittance of LOCM. Once the two parameters are set to appropriate values, LOCM allows the promotion of the secret key rate of DPM-CVQKD, as well as its resistance to excess noise. Meanwhile, taking finite-size effect into consideration, LOCM can also effectively reduce the requirement for the block size of the exchanged signals, which is beneficial to the feasibility and practicability of CVQKD. Since the performance of LOCM DPM-CVQKD is heavily reliant on the calibrate selection of relevant parameters, further research may concentrate on the optimization of LOCM in experimental implementations, where machine learning related methods may be exploited.
In experimental setups of continuous-variable quantum key distribution (CVQKD) independently modulating the amplitude and phase of coherent states, the ideal Gaussian modulation will be degraded into discretized polar modulation (DPM) due to the finite resolution of the driving voltages of electro-optical modulators. To compensate for the performance degradation induced by the joint effect of amplitude and phase discretization, linear optics cloning machine (LOCM) can be introduced on the receiver side. Implemented by linear optical elements, heterodyne detection and controlled displacement, LOCM introduces extra noise that can be transformed into an advantageous one to combat channel excess noise by dynamically adjusting the relevant parameters into a suitable range. In this paper, the prepare-and-measure version of LOCM DPM-CVQKD is presented, where the incoming signal state enters a tunable LOCM before being measured by the nonideal heterodyne detector. The equivalent entanglement-based model is also established to perform security analysis, where the LOCM is reformulated into combination of the incoming signal state and a thermal state on a beam splitter. The composable secret key rate is derived to investigate the security of LOCM DPM-CVQKD. Simulation results demonstrate that the composable secret key rate and transmission distance are closely related to the tuning gain and the transmittance of LOCM. Once these two parameters are set to appropriate values, LOCM can improve the secret key rate and transmission distance of DPM-CVQKD, as well as its resistance to excess noise. Meanwhile, taking finite-size effect into consideration, the LOCM can also effectively reduce the requirement for the block size of the exchanged signals, which is beneficial to the feasibility and practicability of CVQKD. Owing to the fact that the performance of LOCM DPM-CVQKD is largely reliant on the calibration selection of relevant parameters, further research may concentrate on the optimization of LOCM in experimental implementations, where machine learning related methods may be utilized.
In continuous-variable quantum key distribution, the loss and excess noise of the quantum channel are key parameters that determine the secure key rate and the maximal distribution distance. We investigate the imperfect quantum state preparation in Gaussian modulation coherent-state protocol both theoretically and experimentally. We show that the Gaussian distribution characteristic of the prepared states in phase space is broken due to the incorrect calibration of the working parameters for the amplitude modulator and phase modulator. This further causes a significant increase of the excess noise and misestimate of the channel loss. To ensure an accurate estimate of the quantum channel parameters and achieve a reliable quantum key distribution, we propose and demonstrate two effective schemes to calibrate the working parameters of the modulators.
Continuous variable quantum key distribution (CV QKD) is a promising candidate for the deployment of quantum cryptography. At present, the longest distance is limited to ∼100 km at fiber-based quantum channel. We investigated in depth the realistic rate-distance limit (RDL) of CV QKD, considering reconciliation efficiency, finite-size effect, and realistic excess noise under collective attack. It is shown that the excess noise generated on Bob's side degrades significantly the transmission distance and we verify it in experiment. The improvement in RDL by reconciliation efficiency depends on the excess noise level, considerable increase of RDL by improving the reconciliation efficiency occurs only for relative large excess noises. A convergence modulation variance, useful in calculation simplification, is found. Furthermore, we restudy the finite-size analysis and eliminates a loophole arising from the Holevo-bound information monotonicity and a safe RDL is guaranteed. Based on the revised finite-size analysis, the optimum ratio determining the amount of data used for parameter estimation is analyzed.
Continuous variable quantum key distribution protocols (CV-QKD) with discrete modulation have been intensively investigated to bridge the gap between ideal Gaussian modulation and modern coherent optical communication systems. To mitigate the penalty of discrete modulation, probabilistic constellation shaping (PCS) is applied to the modulation format and is typically performed by distribution matching (DM) algorithms. In this paper, we address the application of DM algorithms to perform PCS in CV-QKD protocols. We investigate the impact of approximating optimized Maxwell-Boltzman distributions with DM algorithms based on Huffman (HDM) and constant composition (CCDM) codes on the protocol's secret key rate (SKR) and tolerance to excess noise. Our results show that specifically symbol-by-symbol HDM degrades the SKR by at least 30\%, whereas CCDM matches the optimal SKR with code length of 10310^3103 or more symbols. Furthermore, we also provide a statistical analysis of symbol dependence for both approaches, showing that CCDM must operate with blocks of at least 10510^5105 symbols for the correlations become negligible. Finally, we propose an algorithm to generate independent symbols following near-optimal distributions.