Abstract
In continuous-variable quantum key distribution (CV-QKD), the key information are encoded on quadratures of the optical field, which are measured via balanced homodyne detector (BHD). The bandwidth of the BHD is one of key parameters for precise characterization of quantum states. We establish a theoretical model to analyze the impact of the BHD bandwidth and signal modulation patterns on the channel parameters estimation of CV-QKD systems. Based on the proposed model, the secure key rate of a practical CV-QKD system under different BHD bandwidths and signal modulation patterns are investigated. Our results show that insufficient BHD bandwidth will result in wrong estimate of the transmission loss and excess noise, which significantly affects the performance of CV-QKD systems. Given the BHD bandwidth, there exists an optimal signal repetition rate that maximizes the secure key rate. The BHD bandwidth requirement of the QKD system increases with the transmission distance for large duty cycle pulse. Furthermore, the root raised-cosine pulse signal modulation performs better than the square pulse signal modulation in general.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD)[1–3] enables two legitimate communication parties (Alice and Bob) to share information-theoretically secure key through an insecure quantum channel and an authenticated classical channel. Any eavesdropping behaviours aiming at the QKD system can be detected and the security is guaranteed by the laws of quantum mechanics rather than the complexity of mathematical algorithms. When combine with one-time pad encryption, secure and private communication can be realized. Various QKD protocols have been proposed and demonstrated during the past decades [4,5]. Many of them utilize discrete-variable (DV) of light fields as the carrier of the key information, such as the polarization or phase of single photon. The other protocols encode key information in continuous-variable (CV) of light fields, such as the amplitude and phase quadratures of quantized optical mode [6–8]. Due to its good compatibility with the standard optical telecommunication technologies and potential high secure key rate over metropolitan areas, significant progress have been seen in continuous-variable quantum key distribution (CV-QKD) [9–38].
For CV-QKD protocol based on Gaussian-modulated coherent states (GMCS) and homodyne detection [11], Alice prepares a train of Gaussian distribution coherent states and sends them to Bob. Bob randomly measures either amplitude or phase quadrature of the received states using the balanced homodyne detector (BHD). Based on the estimated secure key rate, Alice and Bob extract the secret key by using data error correction and privacy amplification. Recently, the composable security of the GMCS CV-QKD protocol has been proved against general attacks [28,39–41].
The GMCS CV-QKD protocol is considered to be a promising alternative in practical applications over metropolitan area secure networks. To date, a number of works have been implemented from laboratory system to field test. And the transmission distance has been extended to 202 km using ultralow-loss optical fiber [42]. Given the transmission distance, the secure key rate of the QKD system is mainly limited by the system excess noise and signal repetition rate. The former is caused by the imperfect state preparation [43], the Eve’s eavesdropping, and the inaccurate measurement of the quantum state, etc. The latter is mainly limited by the BHD bandwidth and the speed of the classical data post-processing.
The design and construction of a wideband homodyne detector for time-domain quantum measurements has been reported [44]. By assuming a pulsed local oscillator (LO) with the pulse width much shorter than the time resolution of the detector, effective efficiency due to the electronic noise is derived. To get high secure key rate, Chi et al. discussed the induced excess noise by the electrical pulses overlap in time-domain BHD [45]. In [46], Tang et al. analyzed various noise sources at different clock rates and evaluated the performance of CV-QKD with fixed BHD bandwidth. Recently, the impact of receiver imbalances on the security of a heterodyne based CV-QKD system was reported in [47]. Although the finite bandwidth of the receiver is involved in the analysis, it remains constant. At present, the systematic investigation about the impact of the BHD bandwidth and signal modulation patterns on the performance of the CV-QKD system is still missing.
Since the design and fabrication of shot noise limited BHDs with very high bandwidth is technically challenging. In order to fully exploit the bandwidth of BHD and achieve high performance CV-QKD, in this paper, a theoretical model for the impact of the BHD bandwidth and signal modulation patterns on the channel parameter estimation of a CV-QKD system is proposed. In our model, we keep the pulse duration, pulse repetition period, and the detection bandwidth as independent variables, and choose the integration limits in a reasonable manner. Then, we analyze the performance of the CV-QKD system under the conditions of different BHD bandwidths and signal modulation patterns. For the fixed BHD bandwidth and transmission distance, the optimal signal repetition rates are derived.
The paper is organized as follows. In Sec. 2, we model the parameter estimation of the CV-QKD system with a finite-bandwidth BHD. In Sec. 3, we derive the secure key rate under finite-bandwidth BHD, and analyze the influences of the bandwidth on practical performance of a CV-QKD system. In Sec. 4, we analyze the parameter estimation of the CV-QKD system with a root raised-cosine pulse signal modulation and finite-bandwidth BHD. Finally, we present the conclusion in Sec. 5.
2. Influence mechanism of the BHD bandwidth on channel parameters estimation
2.1 BHD output under finite bandwidth
In Fig. 1, we show a typical schematic of the BHD with finite bandwidth. A signal mode is interfered with a phase reference mode (local oscillator, LO) at a 50:50 beam splitter. The two output fields emerging from the beam splitter are incident upon two photodiodes and the resulting photocurrents are subtracted. For BHD with infinite bandwidth, the photocurrent difference is proportional to the instantaneous quadrature of the signal field $\hat x\left ( {t,\theta } \right ) = \hat a_s^\dagger (t){e^{i\theta }} + {\hat a_s}(t){e^{ - i\theta }}$, where $\hat a_s^\dagger (t)$ and ${\hat a_s}(t)$ denote creation and annihilation operators of the signal field, $\theta$ is the relative phase between the signal field and LO. Assume that the LO and signal have the same spatial mode, and their pulses duration and repetition period are ${{s}_{0}}$ and $s$, respectively, then the measured quadrature of the single pulsed signal mode $\hat x\left ( \theta \right )$ defined by the spatial and temporal mode of the LO is given by
For a practical BHD with finite bandwidth, we introduce a response function $h(t)$ [48,49], so that the measured quadrature of the single pulsed signal mode has the form
2.2 Estimation of the channel parameters in CV-QKD with finite-bandwidth BHD
In GMCS CV-QKD, Alice prepares a series of coherent states and their quadratures obey a centered bivariate Gaussian distribution in phase space. The amplitude quadrature of the signal mode is written as
where ${\hat x_{v1}}(t)$ is the quadrature of a vacuum field, and ${x_A}(t)$ denotes the Gaussian modulation signal with modulation variance ofBy introducing a noise operator ${\hat x_N}(t)$, Eq. (6) can be rewritten as
where ${\hat x_N}(t)$ is defined asIn the above analysis, we have assumed that Bob performs measurement with an ideal BHD that has an infinite bandwidth. However, the practical BHDs always have a finite bandwidth, which inevitably distort the measurement outcomes and result in inaccurate parameter estimation. In the following part, the effect of the finite BHD bandwidth on the estimate of the quantum channel parameters is investigated.
Considering that the Butterworth filter is wildly used on the BHD, we assume that a fifth-order Butterworth low-pass filter is exploited hereafter, which has transfer function of the form [50]
In the case of infinite-bandwidth BHD, the normalized electronic noise keeps constant. Although the electronic noise is estimated without quantum signal and LO, when the BHD bandwidth decreases from 5 GHz to around 1 GHz, the electronic noise power of current period declines due to the filtering effect, whereas the crosstalk of the adjacent pulse increases. The overall effect results the rise of the electronic noise. When the BHD bandwidth decreases further, the crosstalk of the adjacent pulse cannot compensate the rapid degradation of the current electronic noise, therefore the electronic noise starts to drop quickly. Above effect results in that the electronic noise has a peak value around 1 GHz and is dependent of duty cycle, as shown in Fig. 2(a).
Notice that estimation of the channel parameters is less sensitive for smaller duty cycle ${{{s}_{0}}}/{s}$ when BHD bandwidth is larger than l.5 GHz (the repetition rate of the QKD system is 1 GHz). The transmittance can be accurately determined at BHD bandwidth above 3.5, 2 and 1.5 GHz for duty cycles of 0.8, 0.5 and 0.2, respectively. On the other hand, the excess noise can be accurately estimated when the BHD bandwidth is higher than 3.0, 2.0 and 2.0 GHz for duty cycles of 0.8, 0.5 and 0.2, respectively.
3. Performance analysis of the CV-QKD system with the finite-bandwidth BHD
The performance of a practical CV-QKD system is mainly characterized by the secure key rate and maximal transmission distance. Based on the estimated experimental parameters of the system ${V_{\rm {A}}}$, ${T_1}$, ${\varepsilon _1}$, $\eta$, and ${\upsilon _{el}'}$, Alice and Bob can estimate the secure key rate, which is expressed as [51]
In the finite-size regime, the estimated channel parameters in Eqs. (28) and (29) are no longer the true channel parameters, but the expected values. We need to replace these parameters with their confidence intervals and find their boundary values that are most favorable for Eve. According to [51], the worst estimation of the unknown parameters ${T_1}$ and ${\varepsilon _1}$ are given by
Considering the statistic fluctuations, $I_{AB}^{{{\epsilon }_{PE}}}$ is given by
Figure 3(c) and 3(d) show the secure key rate $K{\rm {\ =\ }}R \cdot {K^L}$ (in unit of Mbits per second) as a function of signal repetition rate of the QKD system. Here $R$ is the signal repetition rate of the QKD system, and the bandwidth of the BHD is 1 GHz. Three different transmission distances of 25, 35 and 50 km are investigated. We can see that the signal repetition rate exists an optimal value at each transmission distance, at this point, the secure key rate is maximized. When the signal repetition rate $R$ is much smaller than the BHD bandwidth, the signal quantum states can be precisely measured and the secure key rate ${K^L}$ approaches to its maximum value. However, the small $R$ inevitably decreases $K$. In contrast, when the signal repetition rate is close to the BHD bandwidth, although larger signal repetition rate is beneficial to the secure key rate $K$, the finite BHD bandwidth decreases the estimate value of the transmittance and increases the estimate value of the excess noise (Fig. 2). In this case, the value of secure key rate ${K^L}$ drops quickly to zero, which results in that the secure key rate $K$ drops to zero. Therefore, there exists an optimal $R$ to maximize the secure key rate $K$. From Fig. 3(c), when the transmission distances are 25, 35 and 50 km, the optimal repetition rates are close and has the value of 0.82 GHz. From Fig. 3(d), when the transmission distances are 25, 35 and 50 km, the optimal repetition rates are 0.38, 0.36 and 0.34 GHz, respectively. We find that the large duty cycle pulse has a smaller optimal signal repetition rate, and the longer the transmission distances, the smaller the optimal signal repetition rate. The reason is that the secure key rate $K$ at longer transmission distances is more sensitive to the excess noise and transmission distance of the quantum channel. Smaller signal repetition rate $R$ results a lower estimate excess noise and higher estimate transmission distance.
In Fig. 3(c) and 3(d), the shape of the secure key rate lines is different for duty cycles of 0.5 and 0.8. For a duty cycle of 0.8, the secure key rate increases linearly with the repetition rate until it reaches a maximum. Nevertheless, we do not observe the same behaviour for a duty cycle of 0.5. The main reason is due to the different variations of the excess noise versus the BHD bandwidth for duty cycles of 0.5 and 0.8. As shown in Fig. 2(b), for duty cycle of 0.8, the excess noise monotonically increases as the BHD bandwidth reduces. However, for duty cycle of 0.5, the excess noise exhibits fluctuation as the BHD bandwidth decreases. In particular, the excess noise has an explicit peak at BHD bandwidth of around 1.5 GHz, which results in a valley point in secure key rate at the repetition rate of around 0.67 GHz in Fig. 3(c).
4. Performance of the CV-QKD system with a root raised-cosine pulse signal modulation and finite-bandwidth BHD
In the above sections, we have analyzed the performance of the CV-QKD system with the finite-bandwidth BHD. We have assumed that the signal modulation profile in each signal period is a square pulse, which is simple to implement in practice and has the form
It is possible to utilize other signal modulation pattern to suppress the crosstalk between adjacent signal pulses in the QKD system. Here we assumed that a root raised-cosine pulse signal modulation pattern [47,52] with roll-factor $\gamma =1$ [53] is employed for the Gaussian signal modulationNotice that the improvement with the root raised-cosine pulse signal modulation is more evident for the large duty cycle pulse signal modulation (see Fig. 4(b) and 4(d)). For duty cycle of 0.8 and repetition period of 1 ns, the spectrum range of the root raised-cosine pulse almost cuts off at 2.5 GHz, however, the spectrum range of the square pulse is unbounded. As the BHD bandwidth is larger than 1.8 GHz, the root raised-cosine pulse can be measured with high fidelity and little crosstalk compared to the square pulse. In this case, the estimated excess noise and transmittance are almost unaffected.
Although the improvement of parameters estimation with root raised-cosine pulse signal modulation is not significant for large BHD bandwidth, it is useful for other CV-QKD protocols such as four-state modulation protocol and measurement-device-independent CV-QKD protocol, which are very sensitive to the channel excess noises.
5. Conclusion
In summary, we investigate the influence mechanism of the finite BHD bandwidth and signal modulation patterns on the performance of practical CV-QKD systems. To this end, the dependence model of the channel parameters estimation in CV-QKD on the BHD bandwidth and signal modulation patterns is established. Then, we derive the secure key rate in the finite-size regime under the condition of finite-bandwidth BHD. The results show that insufficient BHD bandwidth can significantly affect the channel parameters estimation and therefore the performance of the CV-QKD. We also find that the degree of influence depends on the exploited signal modulation patterns due to their different frequency spectrum distribution to be employed. The requirement of the BHD bandwidth for a CV-QKD system increases with the transmission distance for the large duty cycle pulse. The method developed is useful to analyze the impact of the homodyne bandwidth on other types of CV-QKD protocols. Our results can provide guidance for the design and optimization of high speed practical CV-QKD systems and may find other applications in the fields of quantum optics and quantum information.
Funding
National Key Research and Development Program of China (2016YFA0301403); National Natural Science Foundation of China (No. 11904219, No. 62175138); Key Research and Development Program of Guangdong Province (2020B0303040002); Aeronautical Science Foundation of China (20200020115001); Shanxi 1331KSC.
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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