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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.
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1. Introduction
Quantum key distribution (QKD) guarantees the secure distribution of symmetric key between two distant legitimate parties through fundamental principles of quantum mechanics [1–3]. The past two decades have witnessed continuous-variable QKD (CV-QKD), compatible with off-the-shelf commercial telecom components, emerging and prospering as a cost-effective option in metropolitan networks. Recent progresses in CV-QKD include revolutionary advances in both theoretically refined security proofs [4,5], as well as experimentations of long-distance transmission [6–11]. Furthermore, practical issues in CV-QKD implementations have been studied extensively, including shot-noise calibration [12], modulation leakage [13] and carrier recovery [14].
Thanks to the convenience in security analysis, Gaussian modulation on the quadratures of coherent states is favored for its intrinsic convenience in security analysis. However, due to the finite resolution of electro-optical modulators, an exact Gaussian modulation is impracticable in experimental implementations and has to be approximated with a discrete and bounded one [15,16]. Various schemes of discrete constellations have been proposed to approximate the ideal Gaussian distribution, among which are Gauss-Hermite quadrature [17], binomial distribution [18,19] and discrete Gaussian distribution [18,20].
Since Gaussian modulation is implemented by polar modulation experimentally, discretized polar modulation (DPM) CV-QKD has been proposed to precisely represent the discretization effect [21]. In DPM CV-QKD, the Rayleigh-distributed amplitude and the uniformly distributed phase will be sampled and discretized before modulated onto coherent states separately and independently. It is shown that the combined effect of amplitude and phase discretization results in fluctuations in the modulated quadratures, which can be modeled as a preparation noise imposed on the ideal Gaussian modulation.
In DPM CV-QKD, the reliability of parameter estimation needs to be clarified, which plays a critical role in acquiring the knowledge of insecure quantum channel [22]. Due to the quadrature deviations induced by modulation discretization, the accuracy of parameter estimation may be degraded. For DPM CV-QKD, to guarantee that the parameters of quantum channel are accurately estimated, it requires carefully selecting appropriate signal states that are insusceptible to modulation discretization, whose amplitude and phase exactly match the coarse-grained outputs of modulators. However, if the signal states used for parameter estimation are chosen randomly, the quadrature fluctuation due to modulation discretization will be attributed to the estimated parameters, thus degrading the accuracy of parameter estimation and resulting in an underestimation to the secret key rate.
The countermeasure to enhance the parameter estimation of DPM CV-QKD is eliminating the accuracy degradation by calibrating the estimated parameters. We investigate the impact of DPM on parameter estimation and demonstrate that if the normal linear model and the maximum likelihood estimators proposed in [22] are applied, DPM exhibits no degradation on the accuracy of channel transmittance, but leads to an overestimation of excess noise with an additional bias item related to modulation precision. Furthermore, it is verified that once the modulation variance is specified, the induced bias is solely related to the modulation resolution, regardless of the transmission distance and the true value of excess noise.
To calibrate the estimated excess noise, the characteristics of the induced bias is analyzed numerically to obtain a quadratic model of both amplitude and phase resolutions. Taken the residuals of the fitted model into consideration, the estimated excess noise can be calibrated to a result more close to the true value. We show that the proposed calibration scheme promotes not only the precision of parameter estimation to get a more accurate evaluation of secret key rate, but also the practicability of DPM CV-QKD, enabling the implementation of parameter estimation with arbitrary signal states without serious loss of precision.
This paper is organized as follows: the principles of DPM-CV-QKD are presented in Sect. 2 and the proposed calibration scheme is elaborated in Sect. 3. Conclusion is drawn in Sect. 4.
2. DPM CV-QKD protocol
In CV-QKD experimentation, an ideal Gaussian modulation will be degenerated into DPM [21]. Since the quadratures constitute a two-dimensional Gaussian variable $(x_A, p_A)$, it can be derived straightly that the amplitude $A = \sqrt {x_A^2 + p_A^2}$ and phase $\theta = \arctan (p_A / x_A)$ of the signal state will follow the Rayleigh distribution and the uniform distribution, respectively. Then the amplitude-phase tuple $(A, \theta )$ will be modulated onto the signal state polarly by electro-optical modulators.
For any realistic electro-optical modulator, the discrete values of driving voltage will result in discrete outputs, so both $A$ and $\theta$ will be quantized during amplitude and phase modulation, respectively. Assuming the resolution of amplitude modulator is $\delta _a$, the amplitude range will be binned into a set of $N$ equally spaced intervals of size $\delta _a$, and any amplitude $A$ located in the $j-$th interval $[j\delta _a, (j + 1)\delta _a)$ will be projected onto $A^* = (j + 1 /2)\delta _a$ due to discretization. Similarly, assuming the resolution of phase modulator is $\delta _p$, the phase range will be binned into a set of $M$ equally spaced intervals of size $\delta _p$, and any phase $\theta$ located in the $k-$th interval $[k\delta _p, (k + 1)\delta _p)$ will be projected onto $\theta ^* = (k + 1 /2)\delta _p$. In this sense, the whole phase space will be partitioned into a set of small sector-shaped regions, where every region will be represented by the discrete amplitude-phase tuple $A^*, \theta ^*$, resulting in a set of polar discretized modulation outputs, as shown in Fig. 1.
Fig. 1. The modulator-induced discretization in CV-QKD setup: (a) schematic of ideal Gaussian modulation, (b) schematic of DPM.
To quantitively describe the quadrature fluctuation induced by modulation discretization, the modulated quadratures $(x^*_A, p^*_A)$ are related to $(x_A, p_A)$ via random variables $T_x$ and $T_p$, which can be expressed as
From. Equation (1), it can be inferred that smaller $x_A$s and $p_A$s are more susceptible to the noise induced by modulation discretization, and thus are not suitable for secret key generation. To eliminate the signal states severely deteriorated by modulation discretization, a procedure named preselection is adopted in DPM CV-QKD. In preselection, Alice sifts Gaussian random numbers that are larger than a threshold $\eta _{th}$ as raw data for further post-processing and key generation, while those smaller than $\eta _{th}$ are still modulated and distributed through quantum channel to preserve the ensemble Gaussianity, but not used to generate secret key. It has been demonstrated that preselection enables a more precise characterization of $T_x$ and $T_p$, which is advantageous for evaluating the impact of DPM on secret key rate [21].
After preselection, the discretely modulated states $(x^*_A, p^*_A)$ prepared by Alice will be sent to Bob through the quantum channel, which is totally controlled by an eavesdropper Eve and characterized by its transmittance $T$ and excess noise $\epsilon$. Upon receiving the state from the quantum channel, Bob randomly measures the state with homodyne detection or heterodyne detection. In this paper, we assume that Bob utilizes homodyne detection, where no extra noise will be introduced during detection. Without regard to the imperfections of detection, Bob’s measurement result can be expressed as
where $x_N$ and $p_N$ denotes the noise induced in the quantum channel imposed on both quadratures, following a Gaussian distribution $\mathcal {N} (0, \epsilon )$. Then the measurements of the states that survived preselection at Alice’s side are used for secret key generation by performing parameter estimation, reverse reconciliation and privacy amplification.To implement security analysis of DPM CV-QKD, the modulation discretization described above is reformulated into an entanglement-based (EB) scheme, where every Gaussian-modulated coherent state is displaced from $(x_A, p_A)$ to $(T_x x_A, T_p p_A)$, then the displaced mode $B_1$ is sent to Bob through the quantum channel. Since $T_x$ and $T_p$ are identically distributed, the covariance matrix of the ensemble state after DPM can be rewritten as [21]
After DPM, the modulated states are then sent into the quantum channel totally controlled by Eve, where she can apply an entangling cloner attack. After propagating through a quantum channel featured by a transmittance $T$ and an excess noise $\epsilon$, the covariance matrix of the output state can be expressed as
From an information-theoretic perspective, in the asymptotic case, the secret key that Alice and Bob can distill from reverse reconciliation under collective attacks is defined as
where $p_{ps}$ is the percentage of signal states that are preselected and used for secret key generation, $I_{AB}$ is the Shannon mutual information between Alice and Bob, $S_{BE}$ is the Holevo bound for the mutual information shared by Eve and Bob, and $\beta < 1$ is the reconciliation efficiency, which can achieve approximately 98% [11].In this paper, we focus on the case of homodyne detection, where the classical mutual information $I_{AB}$ can be derived directly from the covariance matrix $\gamma _{AB}$
Meanwhile, $S_{BE}$ can be calculated with the facts that Eve purifies the system $AB$ to maximize her information and that Bob’s projective measurement purifies the system $AE$
where $S( \cdot )$ denotes the von Neumann entropy and $S( \cdot |b)$ denotes the conditional von Neumann entropy on Bob’s measurements. The von Neumann entropies are determined by the symplectic eigenvalues $\lambda _{1,2}$ of the covariance matrix $\gamma _{AB}$ and the symplectic eigenvalue $\lambda _3$ of the conditional matrix $\gamma _{A|b}$, which can be derived using where $\gamma _A = \sigma ^2_A \mathbb {I}, \gamma _B = \sigma ^2_B \mathbb {I}$ and $\sigma _{AB} = c_{AB} \sigma _z$, MP stands for Moore-Penrose inverse of a matrix and $X =$ diag $(1, 0)$. Then the maximized mutual information between Eve and Bob can be calculated by3. Parameter estimation calibration for DPM CV-QKD
Without any loss of generality, we assume that Bob measures the $x$-quadrature of the signal state. To calculate the secret key rate, Alice and Bob have to estimate quantum channel parameters $(T, \epsilon )$ according to the correlated data $(x_A^i, x_B^i), i = 1, \ldots, N$, where $N$ is the number of signals used for parameter estimation. In this section, we focus on the asymptotic case of parameter estimation to highlight the DPM-induced accuracy degradation, where $N = \infty$. Assuming the quantum channel is linear and the maximum-likelihood estimation of normal linear model is applied, the constructed estimators can be expressed as
The estimators above are asymptotically unbiased for an ideal Gaussian modulation. However, in DPM CV-QKD, since the quadrature values will be discretized in a polar manner before transmission, the fluctuation induced by modulation discretization will also be attributed to channel loss, thus leading to a potential deviation from real channel parameters in estimation result. It indicates that if the signal states used for parameter estimation are randomly chosen, legitimate users will obtain more pessimistic channel parameters and underestimate the secret key rate, thus sacrificing the system performance.
To characterize the deviation originated from modulation discretization, the results of Eq. (10) and Eq. (11) in DPM CV-QKD under different channel transmittance are plotted in Fig. 2. It is illustrated in Fig. 2 that once $\delta _a$ and $\delta _p$ are known, the estimation of $T$ remains unbiased, while the estimation of $\epsilon$ is biased with a constant deviation independent of $T$. The results demonstrate that while modulation discretization almost exhibits no degradation on the estimation of $T$, it does lead to a bias on the estimation of $\epsilon$. Fortunately, the overestimation in excess noise is independent from the transmission distance, which indicates that it is related to the configurations of Alice exclusively. Hence, to accurately calculate the secret key rate of DPM CV-QKD, the estimated excess noise must be calibrated.
Fig. 2. The normalized estimated channel parameters versus the channel transmittance. The curves are plotted for modulation variance $V_A = 25$, channel excess noise $\epsilon = 0.02$, amplitude modulation resolution $\delta _a = 0.25$, phase modulation resolution $\delta _p = 0.01$ and preselection threshold $\eta _{th} = 2\delta _a$.
Since the main impact of modulation discretization on parameter estimation is the induced bias in the estimation of excess noise, the quantitive relationship between the estimation bias and modulation resolutions is further investigated, as shown in Fig. 3. It is demonstrated that under the coarse-grained resolution regime, the bias can be as large as 50%. In order to remove the bias and get an accurate estimation, a quantitive description of $\hat \epsilon$ is required for calibration. Based on the shape of the plotted contour in Fig. 3, different regression models are applied to fit the data and obtain a closed-form expression of the desired quantitive relationship. Among chosen models are exponential model and polynomial model, both of which exhibit super-linear nonlinearity, as well as multivariate linear model.
Fig. 3. The contour plot of the estimated excess noise by Eq. (11) versus $\delta _a$ and $\delta _p$. The curves are plotted for modulation variance $V_A$ = 25, channel excess noise $\epsilon$ = 0.02 and preselection threshold $\eta _{th} = 2\delta _a$.
In Table 1, the goodness of the chosen models is shown for evaluation. In the table, the root mean square error (RMSE) measures the ensemble closeness between the output of the fitted model and the corresponding estimation result, which decreases as the output approaches the estimation. The $R$-squared, also known as coefficient of determination, measures the proportion of variability in the estimation result that can be explained by the fitted model, which should be equal to 1 if the variability is perfectly explained by the model. The adjusted $R$-squared is the modified version of the $R$-squared that adjusts for the number of variables in the fitted model, a large value of which indicates a better generalization performance. Based on the interpretations above, it can be inferred that the quadratic model provides the most precise quantitive description of the biased estimation, since it is featured by the smallest RMSE and the greatest $R$-squared.
Table 1. Goodness of the fitted excess noise estimation results
To further verify the precision of all the models, their residuals are plotted in Fig. 4. In curve fitting, the residual is defined as the difference between the real value and the prediction of the fitted model. It is shown that the residuals of the quadratic model are lower in magnitude and distributed evenly around zero plane, thus validating the optimality of quadratic model gained from Table 1. Solving the coefficients $a_{ij}$ through the least-square method, the fitted quadratic model reads
As shown in Eq. (12), the estimator $\hat \epsilon$ consists of two independent parts: the true value of excess noise $\epsilon$ ($\epsilon = 0.02$ in Fig. 3) and an item of estimation bias constituted by quadratic items of $\delta _a$ and $\delta _p$. The results imply that the full characteristics of the estimation bias is exclusively determined by the configurations at the transmitting side and entirely unrelated to the insecure quantum channel. In this case, the calibration of excess noise will be enabled: once the modulation variance $V_A$ is determined and $\delta _a$ and $\delta _p$ are measured, the coefficients in Eq. (12) can be determined through a training process by sending known data through an unknown quantum channel and applying standard parameter estimation. Then the obtained coefficients can be applied to calibrate the $\hat \epsilon$ of real quantum by eliminating the additional bias. With the help of Eq. (12), the parameter estimation in DPM CV-QKD can be implemented with arbitrary modulated signal states.
Although the model in Eq. (12) describes the general characteristics of excess noise estimation, there still exists an inevitable difference between the calibrated output and the real value of excess noise, which originates from the residuals of the quadratic model. To analyze the statistics of the residuals, the probability distribution of the residuals of Eq. (12) is shown in Fig. 5. It can be observed that the residual can be approximated as a Gaussian distribution with zero mean. According to the Gaussianity of the residuals, the probability that the calibrated excess noise do not belong to a default confidence interval can be calculated. Since the residuals follow a zero-centered Gaussian distribution, the confidence interval of $\epsilon$ will be $[\epsilon - 6.5 \sigma, \epsilon + 6.5 \sigma ]$ at a probability of $10^{-10}$, where $\sigma$ denotes the standard deviation of the residual.
Fig. 5. The fitted distribution of the residuals of the quadratic model and the reference Gaussian distribution with the same standard deviation as the residuals.
Based on the characteristics of the residuals, an upper bound of excess noise can be obtained by adding the largest residual possible to the calibrated estimation output, leading to a lower bound of secret key rate to guarantee the safety under inaccurate parameter estimation. The estimated excess noise $\hat \epsilon$ will be calibrated according to Eq. (12) to eliminated the bias related to the modulation resolutions, then the calibrated output $\epsilon$ will be further corrected to $\epsilon + 6.5 \sigma$ to obtain an upper bound of calibrated excess noise and a lower bound of secret key rate.
Under the upper bound of the estimated excess noise, the lower bound of secret key rate is presented in Fig. 6. It is shown that when the inaccuracy of parameter estimation is not taken into consideration, DPM scheme enables a maximum channel loss of 14.36 dB. However, in the practical configurations, the biased estimation of $\epsilon$ will induce a decrease of 0.4 dB in the absence of calibration, exhibiting a tolerance of 13.95 dB. Once the calibration according to the quadratic model is implemented, the maximum loss tolerance increase to 14.3 dB correspondingly. The results indicate that although DPM scheme introduces an ineluctable bias in parameter estimation of excess noise, a properly fitted model can characterize the bias, and minimize its impact on security analysis.
Fig. 6. Secret key rate of DPM CV-QKD under parameter estimation results with and without calibration as a function of channel loss. The curves are plotted for modulation variance $V_A = 25$, channel excess noise $\epsilon = 0.02$, amplitude modulation resolution $\delta _a = 0.25$, phase modulation resolution $\delta _p = 0.01$ and preselection threshold $\eta _{th} = 2\delta _a$.
4. Conclusion
In CV-QKD experimentations, ideal Gaussian modulation will be discretized and degraded into DPM. In this paper, a calibration scheme is proposed to improve the accuracy of parameter estimation in DPM CV-QKD in the asymptotic case. While transmissivity estimation is immune to modulation discretization, the overestimation in channel excess noise can be calibrated by removing the bias item from the estimation result. Data fitting results show that the bias item is a quadratic function of the modulation resolutions, and residual analysis of the fitted function gives the upper bound of estimated excess noise and the lower bound of secret key rate. Numerical results demonstrate that under the modulation variance of 25 and the excess noise of 0.02, the proposed scheme can correct the estimated excess noise value from 0.0229 to real value, verifying the feasibility of calibration of parameter estimation for DPM CV-QKD.
Funding
Guizhou Provincial Science and Technology Projects (ZK[2021]304); Training Program of Guizhou University ([2021]56); Scientific Research Foundation for Talent Introduced in Guizhou University ([2019]37).
Acknowledgment
This work was supported in part by Guizhou Provincial Science and Technology Projects, in part by the Training Program of Guizhou University, and in part by the Scientific Research Foundation for Talent Introduced in Guizhou University.
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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