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Abstract
We propose a rigorous calibration method for homodyne detection efficiency, which combines all the factors that affect detection efficiency to calibrate together through the actual homodyne detection. With this method, the transmittance converted from electronic noise in the one-time calibration method of the shot noise can be attributed to the detection inefficiency. Thus, a trusted detection noise-free model for continuous-variable quantum key distribution (CV-QKD) can be established, which simplifies the calibration of shot noise while having the same performance as the trusted detection noise model. We demonstrate this calibration method with a balanced detector based on a transimpedance amplifier. Experimental results show that detection efficiency will be overestimated if the integration factor of the detector is overlooked. The overestimation of the detection efficiency leads to an underestimation of modulation variance and excess noise when the modulation variance is monitored by the balanced detector, which opens security loopholes. Our method may prove a necessary method in the calibration of detection efficiency for CV-QKD.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a family of quantum key distribution (QKD) protocols, continuous-variable (CV) QKD has its unique advantages in allowing two communication parties to share secure keys [1]. The similarities between CV-QKD and coherent optical communication enable the former to be realized with commercial optical communication components. Due to the highly effective filtering of the local oscillator (LO) used in the coherent detection, CV-QKD can tolerate more incoherent channel noise than discrete-variable QKD. Thus, it has good compatibility with conventional optical fiber networks [2,3]. The Gaussian-modulated coherent state (GMCS) protocol [4,5] is the most widely studied protocol in CV-QKD. Theoretically, its security against collective attacks has already been proven [6–8], while its composable security against general attacks has also been well established [9,10]. Experimentally, the GMCS protocol has been demonstrated over long transmission distances [11–13], with high repetition rate systems [14–16], and in field fiber [17–21].
Although CV-QKD has made remarkable achievements in both theory and experiment, there are still crucial steps to be taken to reduce the gap between theory and practice implementation. This gap opens security loopholes to an eavesdropper, who may launch various attacks [22–28]. For a practical CV-QKD system, the detection imperfections such as detection inefficiency and electronic noise of the detector should be considered. In actual homodyne detection, detection efficiency is affected by various factors [29]. It is inconvenient to calibrate these factors one by one, which requires exploring all the factors that affect detection efficiency. The detection efficiency will be overestimated if any factors are overlooked.
In this paper, we propose a method to calibrate the detection efficiency, which combines all the factors to calibrate together. All known or unknown factors are included in the calibrated detection efficiency to achieve rigorous calibration of detection efficiency. When the shot noise is calibrated using the one-time calibration method [30], the transmittance converted from electronic noise can be included in the detection efficiency and calibrated together with other factors. Thus, a trusted detection noise-free model can be established. A balanced detector based on a transimpedance amplifier is used in actual homodyne detection to demonstrate this method. The experimental results show that the detection efficiency of this type of detector will be affected by the integration factor, which represents the incomplete integration of the optical pulse by the detector. The detection efficiency will be overestimated if the integration factor of the detector is overlooked. Hence the calibration method we propose here should be used in CV-QKD to close the security loopholes caused by overestimating detection efficiency.
The rest of the paper is organized as follows. In Section 2, the rigorous calibration method of detection efficiency is given and combined with the one-time calibration method of shot noise, thereby establishing a trusted detection noise-free model in Section 3. In Section 4, the calibration method is demonstrated experimentally. Based on the experimental results, we analyze the practical security of CV-QKD with overestimated detection efficiency in Section 5, and draw the conclusions in Section 6.
2. Rigorous calibration of detection efficiency
Figure 1 depicts the homodyne detection implementation. An optical signal pulse with a certain pulse width is used to prepare the desired coherent state. When homodyne detection is performed, it interferes with the LO. The interference result is converted into an electric pulse by a balanced detector and then sampled by an ADC. For fiber-based homodyne detection, the detection efficiency can be written as [29]
where $\eta _m$ is the interferometer mode-matching efficiency, $\eta _t$ is the transmittance of a fiber coupler, $\eta _d$ is the quantum efficiency of the photodiodes. Here, three factors need to be calibrated, though more factors should be considered in some cases. For the balanced detector based on a transimpedance amplifier, if the pulse width of the optical signal pulse is much larger than the response time of the photodiodes used in detector, multiple sampling points should be collected and integrated since the quadrature of the coherent state is proportional to the area under the electric pulse [31]. To simplify data collection, the prepared optical pulse should be narrow enough to be completely integrated by the detector. Then the quadrature is proportional to the peak value of the electric pulse, which can taken as the measurement result [32–34]. However, the optical pulse width prepared in CV-QKD is usually on the order of nanoseconds [14,35,36], which should not be assumed to be completely integrated by the detector. In such a situation, the integration factor representing the completeness of the integration should be calibrated. Otherwise, the detection efficiency will be overestimated.In order to avoid overestimating the detection efficiency, we propose a method to calibrate the detection efficiency $\eta$ rigorously. First, it is required to prepare $m$ coherent states with the same amplitude and four different phases, $\left \{| {\alpha _\mathrm {A} e^{i(\theta +\varphi _k)}}\right \rangle \}_{k=1 \ldots m}$. Here, $\theta$ is the initial phase of each coherent state, and $\varphi _k\in \lbrace 0, \mathrm{\pi} /2, \mathrm{\pi}, 3\mathrm{\pi} /2\rbrace$ is the phase that modulates on the coherent state. The number of coherent states modulating the four phases is the same. These coherent states directly enter Bob’s detection device without passing through the quantum channel. Bob performs homodyne detection on these coherent states. Due to the homodyne detection inefficiency, the relation between the measurement voltage-output result of the quadrature $q_{U,k}$ and the coherent state $| {\alpha _\mathrm {A} e^{i(\theta +\varphi _k)}} \rangle$ prepared can be written as
Here, $N_0$ is the shot noise, $N_0=\tilde {N}_0-V_\text {el}$, which can be calibrated in two steps in the conventional method. The first step is to obtain the total detection noise $\tilde {N}_0$, which can be measured as the output-voltage variance of the detector with only LO light input. The second step is to obtain electronic noise $V_\text {el}$, which can be measured as the output-voltage variance of the detector without signal and LO light input. The measurement noise $\delta _k$ follows a normal distribution with a mean of zero and a variance of $\sigma _0^2=1+v_\text {el}$, where $v_\text {el}=\left. V_\text {el}\right /N_0$. Here, the estimator of $\eta$ is given by
Fig. 1. Implementation of homodyne detection. Sig: signal pulse; LO: local oscillator; BS: beam splitter; BD: balanced detector; ADC: analog-to-digital converter.
In Ref. [30], a one-time calibration method of shot noise is proposed, where only $\tilde {N}_0$ is obtained. Then $\left.q_{U,k}\right /\sqrt {\tilde {N}_0}=\sqrt {\eta _e}\left. q_{U,k}\right /\sqrt {N_0}$, where $\eta _e=\left. N_0\right /\tilde {N}_0=1-\left. V_\text {el}\right /\tilde {N}_0$. Here, $V_\text {el}$ is converted to $\eta _e$, which is not calibrated and attributed to the channel loss in Ref. [30]. By using the calibration method we proposed, $\eta _e$ can be attributed to the detection inefficiency. In this way, the homodyne detection efficiency should be replaced by
with the estimator of $\eta '$ written asSimilarly, the approximate estimator of $\eta '$ can be written as
for the coherent state with $\alpha _\mathrm {A}^2>1000$.3. Trusted detection noise-free model
In the prepare-and-measure (PM) scheme of the GMCS protocol based on homodyne detection, Alice prepares a coherent state $| {\frac {q_\mathrm {A}+\mathrm {i} p_\mathrm {A}}{2}} \rangle $ and transmits it to Bob through the quantum channel. The two independent and identically distributed (i.i.d.) random numbers $(q_\mathrm {A}, p_\mathrm {A})$ are drawn from a zero-mean bi-dimensional Gaussian distribution with a variance of $V_\mathrm {A}$. Bob performs practical homodyne detection to randomly measure either the quadrature $Q$ or quadrature $P$, and the measurement result is $y=q_{\mathrm {B}_2}$ or $y=p_{\mathrm {B}_2}$.
The entanglement-based (EB) scheme of GMCS protocol based on homodyne detection is shown in Fig. 2, which is equivalent to the PM scheme [37] . Alice prepares the two-mode squeezed vacuum (EPR) state $\rho _\mathrm {A'A}$. She keeps mode $\mathrm {A'}$ of each EPR state and sends mode $\mathrm {A}$ to Bob through the quantum channel, which is characterized by transmittance $T$, and total channel added noise $\chi =1/T-1+\xi$. Here, $\xi$ is the excess noise, and all noises are expressed in shot-noise units. The quantum state $\rho _\mathrm {A'A}$ becomes $\rho _\mathrm {A'B_1}$, which can be described by the covariance matrix (CM),
Fig. 2. Entanglement-based scheme of GMCS protocol with a trusted detection noise model. Alice prepares the EPR states. Mode $\mathrm {A'}$ of each EPR state is kept for heterodyne detection, while mode $\mathrm {A}$ is transmitted to Bob through the quantum channel controlled by Eve. Bob performs homodyne detection with a practical detector whose detection efficiency and electronic noise have been calibrated.
In actual homodyne detection, detection efficiency $\eta$ and electronic noise $V_\text {el}$ should be calibrated accurately. In the trusted detection noise model [37], as shown in Fig. 2, a beam splitter with transmittance $\eta$ is used to model the detector efficiency. An EPR state of variance $v=1+\left.v_\text {el}\right /(1-\eta )$ is used to model the electronic noise. When the one-time calibration method of shot noise and the rigorous calibration method of detection efficiency are used, the electronic noise is converted to the detection inefficiency rather than the channel loss in the Ref. [30], which is calibrated together with detection efficiency. In this way, a trusted detection noise-free model can be established, in which the detection efficiency is modeled by a beam splitter with transmittance $\eta '=\eta \eta _e$, as shown in Fig. 3.
Fig. 3. Entanglement-based scheme of GMCS protocol with a trusted detection noise-free model. The electronic noise is converted to detection inefficiency. The detection efficiency is modeled by a beam splitter with transmittance $\eta \eta _e$.
We follow the derivation process in the Refs. [30,37] to obtain the asymptotic secret key rate of this model under collective attacks, which can be expressed as
where $\beta$ is the reconciliation efficiency, $I_\mathrm {AB}$ is the mutual information between Alice and Bob, where $\chi _t=\chi +\frac {1-\eta '}{T\eta '}=\frac {1}{T\eta '}-1+\xi$. The Holevo bound can be written asHere, $m_\mathrm {B_3}$ represents the measurement result of Bob, $p(m_\mathrm {B_3})$ is the probability density of the measurement result. The system $\mathrm {E}$ of Eve purifies $\mathrm {A'B_1}$, so that $S(\rho _\mathrm {E})=S(\rho _\mathrm {A'B_1})= g(\lambda _1)+ g(\lambda _2)$, where $g(\lambda )=\left (\frac {\lambda +1}{2}\right ) \log _2\left (\frac {\lambda +1}{2}\right )-\left (\frac {\lambda -1}{2}\right ) \log _2\left (\frac {\lambda -1}{2}\right )$, $\lambda _1$ and $\lambda _2$ are the symplectic eigenvalues of $\mathbf {V}_{\mathrm {A'B_1}}$, given by
Bob’s measurement purifies the system $\mathrm {A'EH}$, thus $S(\rho _\mathrm {E}^{m_\mathrm {B_3}})=S(\rho _\mathrm {A'H}^{m_\mathrm {B_3}})= g(\lambda _3)+ g(\lambda _4)$, where $\lambda _{3,4}$ are the symplectic eigenvalues of $\mathbf {V}_\mathrm {A'H|B_3}$. The CM $\mathbf {V}_\mathrm {A'H|B_3}$ is obtained by performing homodyne detection on mode $\mathrm {B_3}$ of $\mathbf {V}_\mathrm {A'HB_3}$, which can be expressed as
The symplectic eigenvalues $\lambda _{3,4}$ are given by
Compared with the trusted detection noise model, this model replaces $\eta$ with $\eta \eta _e$ and sets $v_\text {el}=0$. Although the final key rate of this model remains the same as that of the trusted detection noise model, the calibration procedure of shot noise is simplified.
4. Experimental demonstration
As shown in Fig. 4, we present the experimental setup to calibrate homodyne detection efficiency. A narrow-linewidth $ {1550}\,\textrm{nm }$ fiber laser is used to generate continuous-wave coherent light, which passes through a variable optical attenuator (VOA). Then, it is divided into signal light and LO light by a beam splitter (BS). The purpose of placing a VOA here is to adjust the LO light to a suitable intensity and prevent the detector from being saturated. Another VOA is used to adjust the intensity of the signal light to the required value, i.e., $ {-53.87}\,\textrm{dBm}$ after passing through a Sagnac loop (SL) and a phase modulator (PM). Before the SL, a circulator (CIR) blocks light from reflection. The SL converts the continuous wave into signal pulses with a pulse width of $w= {5}\,\textrm{ns }$ and a repetition rate of $ {20}\,\textrm{MHz }$. It can be achieved by loading a $ {10}\,\textrm{MHz }$ square wave signal on the PM. The optical fiber delay difference on both sides of the PM in the SL is $ {5}\,\textrm{ns }$. It should be noted that two adjacent signal pulses generated in this way will have an additional $\mathrm{\pi}$ phase difference. These signal pulses then are phase modulated by a PM, with the modulated phases being $0$, $0$, $0$, $\mathrm{\pi} /2$, $\mathrm{\pi} /2$, $\mathrm{\pi} /2$, $\mathrm{\pi} /2$, and $0$ in turn. Thus, the total phases of the signal pulses modulated by the SL and the PM are $0$, $\mathrm{\pi}$, $0$, $3\mathrm{\pi} /2$, $\mathrm{\pi} /2$, $3\mathrm{\pi} /2$, $\mathrm{\pi} /2$, and $\mathrm{\pi}$, respectively. In this way, a series of coherent states with four phases and an average photon number of 1600 are prepared.
In order to perform homodyne detection well, two VOAs and a variable optical delay line (VODL) are used to control the insertion loss and delay between the BS and the balanced detector (BD) with $ {200}\,\textrm{MHz }$ bandwidth. When signal light pulses enter the input port of BS, the polarization controller (PC) is adjusted to minimize the indication value of the power sensor (PS) so that the polarization of the signal light and the LO light are as consistent as possible. An isolator (ISO) is inserted here to block the LO light reflected from the detector. In the experiment, the balanced detector used is based on a transimpedance amplifier and the peak value of the electric pulse is taken as the measurement result.
The experimental results are given in Table 1. The detection efficiency $\hat {\eta }_R=0.263$ and $\hat {\eta }'_R=0.247$ are obtained through rigorous calibration. The reduced detection efficiency of $\hat {\eta }'_R$ compared to $\hat {\eta }_R$ corresponds to the $\eta _e$. By using the conventional calibration method, $\hat {\eta }_C \approx \eta _t \eta _d=0.557$ is also obtained. Clearly, $\hat {\eta }_C$ and $\hat {\eta }_R$ are quite different. It can be inferred that the prepared optical pulse is too wide to be completely integrated by the detector, resulting in a smaller measured quadrature than expected. This conclusion can be further verified by preparing the signal pulses with the same average photon number and wider pulse width. The wider the prepared optical pulse, the more incomplete the integration of the detector will be. When the pulse width increases from $ {5}\,\textrm{ns }$ to $ {10}\,\textrm{ns }$, the detection efficiency $\hat {\eta }_R$ is approximately reduced by half. As the pulse width increases, the detection efficiency $\hat {\eta }_R$ is further reduced. Indeed, the integration factor will affect the detection efficiency of this type of detector.
Fig. 4. Experimental setup to calibrate homodyne detection efficiency. CW: continuous-wave laser; VOA: variable optical attenuator; BS: beam splitter; CIR: circulator; PM: phase modulator; SL: Sagnac loop; PC: polarization controller; ISO: isolator; PBS: polarization beam splitter; PS: power sensor; VODL: variable optical delay line; BD: balanced detector; ADC: analog-to-digital converter.
Table 1. Experimental results. The data block size is $m=2\times 10^8$.
The detection efficiency will be overestimated if the integration factor is overlooked. Therefore, the rigorous calibration method should be adopted to prevent overestimating the detection efficiency. For the balanced detector based on a transimpedance amplifier, if ultrafast pulses cannot be prepared, taking the peak value as the measurement result decreases the burden of data processing at the price of reducing the detection efficiency. In order to pursue high detection efficiency, the balanced detector based on a charge amplifier is more suitable for homodyne detection of nanosecond optical pulses [38].
5. Practical security analysis
After $N$ rounds of quantum transmission, Bob announces which quadrature he measured for each received coherent state, and Alice reserves the corresponding $x=q_\mathrm {A}$ or $x=p_\mathrm {A}$. Alice and Bob obtain a set of correlated raw keys $(x_i,y_i)_{i=1 \ldots N}$. Then, they reveal a portion of raw keys to perform parameter estimation. If only the homodyne detection efficiency is overestimated, $\hat {\eta }>\eta$, and the modulation variance $V_\mathrm {A}$ is correctly obtained, the estimated values and the actual values of the channel parameters are linked through
In practice, however, it is convenient to use a calibrated homodyne detector to determine $V_\mathrm {A}$. In Ref. [28], $V_\mathrm {A}$ is monitored by the homodyne detector to prevent the reduced optical attenuation attack. In such a situation, Eq. (15) should be replaced by
Fig. 5. Secret key rate versus transmission distance. Simulation parameters are as follows: reconciliation efficiency $\beta =0.95$, fiber attenuation $\gamma = {0.2}\,\textrm{dB/km}$, data block size $N=2\times 10^8$, detection efficiency $\hat {\eta }_R=0.263$, $\hat {\eta }'_R=0.247$, $\hat {\eta }_C=0.557$, electronic noise $v_\text {el}=0.067$, excess noise $\xi =0.005$ in (a), $\xi =0.05$ in (b). The modulation variances for different calibration methods are optimized in the range 1 to 1000 at each distance to achieve the maximum key rate. The data block size used for parameter estimation is $N/2$. The blue and black solid curves represent the key rates when the detection efficiency is estimated with $\hat {\eta }_R$ in the asymptotic and the finite-size regimes, respectively. The cyan and green dashed curves represent the key rates when the detection efficiency is estimated with $\hat {\eta }'_R$ in the asymptotic and the finite-size regimes, respectively. The red curves represent the key rates when the detection efficiency is overestimated with $\hat {\eta }_C$, while the magenta curves represent the corresponding secure key rates. The solid and dashed curves represent the key rates in the asymptotic and the finite-size regime, respectively.
6. Conclusion
In conclusion, we have proposed a rigorous method to calibrate the homodyne detection efficiency, preventing overestimating the detection efficiency due to overlooking some unknown factors that affect detection efficiency. We have established a trusted detection noise-free model by combining this method with the one-time calibration method of shot noise. While this model simplifies the calibration of shot noise, it still has the same performance as the trusted detection noise model. We have experimentally demonstrated this method and found that for the balanced detector based on a transimpedance amplifier, the influence of the integration factor on the detection efficiency should be considered. Otherwise, the detection efficiency will be overestimated. If the modulation variance is monitored by the homodyne detector, the overestimation of detection efficiency will leads to an underestimation of the modulation variance and excess noise, thus opening security loopholes of CV-QKD. Therefore, applying our calibration method to the practical CV-QKD implementation is necessary.
Funding
China State Railway Group Co., Ltd Scientific and Technological Research Project (K2019G062); National Natural Science Foundation of China (61875182); Anhui Initiative in Quantum Information Technologies and Fundamental Research Funds for the Central Universities (WK2340000083).
Acknowledgments
M. Z. thanks Hao Qin for fruitful discussion and Prof. Yongmin Li for construction advice.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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