Abstract

Continuous-variable quantum key distribution (CVQKD) provides an approach for secure communication in optical fiber communication systems. However, its practical implementation has been hindered by low secret key bit rates that are usually limited to several bits/s to hundreds of kbits/s at distances of more than 25 kilometers. In this paper, we use a pair of optical frequency combs (OFCs) for both multiple parallel transmission and coherent reception, which assign multiple sub-channels involving multiple independent secret keys in a single fiber to increase the key bit rate. The first and last sub-channels are selected for propagating phase references to compensate the phase offset between two free-running combs. We analyze possible excess noise caused by dispersive walk-off in the transmission, imperfect phase compensation in the reception and photon leakage from the phase references. Compared to the previous single-channel CVQKD method, simulation results show more than a factor of 20 increase in the secret key rate at a transmission distance of $35$ km and the number of comb lines of $35$.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum key distribution (QKD) is one of the most famous applications of quantum cryptography that allows two distant parties to establish a common secret key in an untrusted environment [14]. Both discrete-variable (DV) QKD based on single photon detection [57] and continuous-variable (CV) QKD based on coherent detection [810] have achieved remarkable accomplishments. The former encodes the key information in properties of single photon pulses, which may be limited by the detection speed and efficiency of the single photon detectors [11]. Fortunately, CVQKD can get rid of this limitation by encoding the information on continuous variables, such as the quadratures of a mode of the electromagnetic field [12,13]. One well-known CVQKD protocol is the Gaussian-modulated coherent state (GMCS) CVQKD protocol [1416]. In this protocol, the sender Alice encodes the key information by modulating the quadratures $x$ and $p$ of few-photon coherent states with a independent Gaussian distribution, while the receiver Bob measures either one of the two quadratures randomly using homodyne detection [8,17], or both quadratures simultaneously by heterodyne detection [15,18]. It has been demonstrated that using Gaussian modulation can reach the theoretically optimal secret key rate, and it is secure against collective attacks [1921] and coherent attacks [22,23].

However, owing to the fiber loss and the excess noise occurring naturally in the channel, the secret key rate of numerous existing CVQKD protocols is still low in practice, which has a negative influence on its practical implementation [2427]. In recent years, researchers have proposed many schemes to increase the key rate [28,29]. A possible way to improve the secret key bit rate is to use multiplexing technologies to carry out parallel transmission, such as orthogonal frequency division multiplexing (OFDM) [3032] and wavelength division multiplexing (WDM) [33,34]. OFDM has the advantages of strong anti-multipath fading and high spectral efficiency, but most of the current OFDM need fast Fourier transform (FFT) or equivalent functions for implementing the multiplexing and demultiplexing process, which is restricted by the processing speed of digital signal processors. WDM systems are known for the ability of parallel transmitting hundreds of carriers with different wavelengths in a single fiber, however, using these systems requires stacks of laser sources in the transmitter and complex digital signal processing (DSP) for frequency and phase estimation. In recent years, OFC has gained extensive attention because of its high frequency stability and strong coherence between frequency components. A comb consists of a series of ultrashort optical pulses at a equal frequency intervals, and can be completely represented by two parameters of center frequency $f_c$ and repetition rate $f_r$. Unlike separate lasers are free-running and independent, the comb lines of different frequencies of an OFC are essentially phase-locked to each other, which enables denser packed channels and simpler carrier recovery process for WDM systems than independent lasers. Therefore, the stability in repetition rate and the broadband in phase coherence make the OFC becomes an attractive light source for WDM technologies [35]. Suggested in [3638], comb-based WDM transmission has been experimentally demonstrated to generate terabit/s super-channel with chip-scale devices and conventional setups. Besides, the employment of a pair of synchronized OFCs as light source for WDM transmission and as multi-wavelength LO at the receiver has been demonstrated in experiment [39].

In this paper, we propose a multichannel parallel CVQKD protocol that transmits secret key using a pair of OFCs for parallel transmission and coherent reception. The first and last comb lines of the comb are selected as pilot lines to estimate the phase offset of different frequency components between the two combs in transmitter and receiver. We consider the excess noise caused by dispersive walk-off, imperfect phase compensation, and leakage from pilot lines, analyze the parameters that influence the security and evaluate the performance of this scheme in the asymptotic limit for both each sub-channel and the whole system. Results show that although the maximum transmission distance is slightly shortened, the secret key rate is greatly improved by using the OFC-based CVQKD scheme.

This paper is organized as follows. In Sec. 2., we describe the principle of the OFC-based CVQKD scheme and show the process of phase compensation. In Sec. 3., we analyze the excess noise originates from measurement inaccuracy, dispersion and photon leakage of pilot lines. In Sec 4., we demonstrate the security of this proposal in terms of the secret key bit rate. Finally, we give a brief conclusion in Section 5..

2. Scheme description

2.1 OFC-based CVQKD protocol

As shown in Fig. 1, Alice generates an OFC with center frequency of $f_0^s$ and repetition rate of $f_r^s$ as a multi-wavelength source that can be expressed as

$$\hat{s}(t)= \sum^{n_{max}}_{n=n_{min}}\hat{a}_n{\exp}\{j[-\varphi(t)+2{\pi}f_n^st]\},$$
where $\hat {a}_n$ is the dimensionless complex amplitude operator corresponding to the mode representing the $n$-th comb lines of the frequency $f_n^s = f_0^s+ nf_r^s$, which can be disassembled by two quadratures as $\hat {a}_n = X^A_n+iP^A_n$. The random function $\varphi (t)$ represents the phase noise in the comb, $n_{max}$ and $n_{min}$ denote the outermost two lines of the OFC with $n_{min}<0$ and $n_{max}>0$. We denote $n_{max}-n_{min}=N-1$, where $N$ is the number of comb lines. The OFC is first passed through a demultiplexer to form $N$ sub-channels, where the number of sub-channels is equal to the number of comb lines. The sub-channel $k$, which varies from $n_{min}+1$ to $n_{max}-1$, is independently amplitude modulated by $V_k$ and phase modulated by $\Phi _k$, where $V_k$ subjects to the Rayleigh distribution with the probability density function as
$$V_k{\sim}\textrm{Ra}(\sigma) = \frac{V_k}{\sigma^2}e^{-\frac{{V_k}^2}{2\sigma^2}},$$
and $\Phi _k$ corresponds to the uniform distribution on $[0,2\pi ]$. Expressed in terms of two quadratures, $X_k^A$ and $P_k^A$ are subject to the Gaussian distribution centered at zero and of variance $V_a$, where $k\in \{n_{min}+1, n_{min}+2, \ldots , 0, \ldots , n_{max}-2, n_{max}-1\}$ and $V_a=\sigma ^2$ (here $V_k$, $\Phi _k$ and $V_a$ are expressed in shot noise units). While the remaining two sub-channels $n_{min}$ and $n_{max}$ are modulated to $X^A_r$ and $P^A_r$ for phase compensation, where $r\in \{n_{min}, n_{max}\}$. Afterwards, all the sub-channels are combined together by a frequency multiplexer and transmitted to Bob. After receiving the multi-frequency signal sent by Alice, Bob first employs two frequency demultiplexers to separate the received OFC and a locally generated comb into $N$. The LO comb with center frequency of $f_0^L$ and repetition rate of $f_r^L$ that are set equal to Alice. Then the outermost two pilot lines of the signal comb and the corresponding LO lines are selected and sent to the phase estimation setup. The remaining pairs of signal lines and LO lines are directed to independent detectors for homodyne detection. To select different basis in different sub-channels, Bob use phase modulators to switch the quadratures before detection. Finally, after a phase rotation that based on the result of the phase estimation setup, Alice and Bob perform classical postprocessing which includes four steps. First, Bob announces the quadratures measured by him in signal sub-channels and Alice keeps the corresponding Gaussian variables. Then, Alice and Bob disclose part of the raw keys to evaluate the parameters of the transmission in each signal sub-channel. Subsequently, they carry out information reconciliation which includes two parts such as multidimensional reconciliation and error correction. Finally, an universal hash function is applied on their corrected keys so that the output sequence is the perfectly secure key shared by them and Eve has nearly zero information on this sequence.

 

Fig. 1. The OFC-based multichannel parallel CVQKD system. Alice splits an OFC with center frequency of $f_0^s$ and repetition rate of $f_r^s$, generating $N$ sub-channels followed by independent Gaussian modulation. Bob detects the received comb by a locally generated OFC with the same parameters as Alice. The outermost two sub-channels are selected as pilot channel for phase compensation.

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2.2 Phase compensation

There are many advantages in implementation of key distribution with locally generated LO, the most important one is to close any potential security loophole related to LO. In addition, the dependency of LO intensity on propagation distance is eliminated, which means sufficient LO power can be guaranteed in detection no matter how long the distance is. However, a main challenge in these schemes is the lack of phase reference between free-running lasers for coherent detection, which results in inaccurate measuring results in Bob side. In our OFC-based CVQKD scheme, although the center frequency and repetition rate of the two combs are set to equal, the indeterminable phase offset occurs during transmission causing a phase decorrelation between the two parties. Unlike classical communication, quantum signals are extremely weak, as such the carrier phase cannot be recovered directly from the quantum signals. Besides, CVQKD systems are more sensitive to phase noise than classical communication systems since the transmission rate is too low. Fortunately, there are still several related solutions, such as transmission of reference frames to perform phase compensation, and the security of GMCS protocols using locally LO has been demonstrated [4044]. The afore-mentioned schemes are useful and easy to implement but also with some impairments. Firstly, they are rely on time-multiplexed quantum signal with phase reference, leading to reduction of the spectral efficiency and secret key bit rate. In particular, when the phase offset is as fast as the signal pulse rate, the phase reference needs to be replaced frequently. Secondly, measuring the quantum signal and reference signal in the same balanced detector requires that the receiver has sufficiently high saturation limit and low electronic noise. It’s hard to achieve in practice. Therefore, we employ frequency division multiplexing technology by selecting the first and last sub-channels to transmit two pilot-lines, establishing a common phase reference between Alice and Bob.

In each round of key distribution, the quantum signal in the $k$-th comb line can be expressed as a coherent state $|\alpha _k\rangle =|X_k^A+iP_k^A\rangle$, where $X_k^A$ and $P_k^A$ are both distributed as $\mathcal {N}(0, V_a)$. The reference signal in the pilot line $r$ is also a coherent state $|\alpha _r\rangle =|X_r^A+iP_r^A\rangle$ whose quadrature values are disclosed. The amplitude $|\alpha _r|$ of the pilot line is fixed and may be several times lager than the amplitude $|\alpha _k|$ of quantum signal but far less than the typical LO. After Bob receives the multi-frequency signal, he first performs a heterodyne detection to measure both of the quadratures $X_r^B$ and $P_r^B$ of the two reference signals and one of the quadratures, $X_k^B$ or $P_k^B$ of the quantum signals. Accordingly, Bob can estimate the phase offset $\theta _r$ between the pilot line $r$ and its corresponding LO line using the public quadratures $(X_r^A, P_r^A)$ and his measured values $(X_r^B,P_r^B)$, which satisfies the constraints

$$\left(\begin{array}{c}X_r^B \\ P_r^B\end{array}\right) = \left( \begin{array}{cc} \cos{\hat{\theta}_r} & -\sin{\hat{\theta}_r} \\ \sin{\hat{\theta}_r} & \cos{\hat{\theta}_r} \end{array} \right) \left(\begin{array}{c}X_r^A \\ P_r^A\end{array}\right).$$
Subsequently, the phase offset $\theta _r$ can be derived as
$$\hat{\theta}_r=\tan^{{-}1}\left(\frac{P_r^BX_r^A-X_r^BP_r^A}{X_r^BX_r^A+P_r^BP_r^A}\right).$$
Since the outermost two sub-channels are selected for transmitting reference signals, we can obtain $\hat {\theta }_{n_{max}}$ and $\hat {\theta }_{n_{min}}$ via Eq. (4). Due to the phase coherence of the OFC, knowing the phase offset of the two pilot lines, we can obtain the phase offset in the $k$-th quantum sub-channel, as [35]
$$\hat{\theta}_k=\hat{\theta}_{n_{min}}+\frac{k-n_{min}}{n_{max}-n_{min}}(\hat{\theta}_{n_{max}}-\hat{\theta}_{n_{min}}).$$
Finally, Alice can correct the phase offset in each sub-channel through adjusting her values $(X_k^A,P_k^A)$ with $\hat {\theta }_k$, to obtain the estimation values of Bob’s measurement, given by
$$\left(\begin{array}{c}\hat{X}_k^B \\ \hat{P}_k^B\end{array}\right) = \left( \begin{array}{cc} \cos{\hat{\theta}_k} & -\sin{\hat{\theta}_k} \\ \sin{\hat{\theta}_k} & \cos{\hat{\theta}_k} \end{array} \right) \left(\begin{array}{c}X_k^A \\ P_k^A\end{array}\right).$$
The schematic presentation of the phase compensation process is shown in Fig. 2. As an alternative approach, phase offset of any two sub-channels can be used to estimate phase offset of other channels instead of using the outermost two tributaries. Considering the influence of pilot lines on quantum signals, here we choose the first and last two sub-channels to transmit reference signals.

 

Fig. 2. Schematic presentation of the phase compensation process. $f^s_{n_{min}}$ and $f^s_{n_{max}}$ are the outermost two comb lines of the signal comb that sent by Alice, $f^L_{n_{min}}$ and $f^L_{n_{max}}$ are the outermost two comb lines of the LO comb that generated by Bob.

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3. Excess noise model

3.1 Noise caused by inaccurate measurement

The requirement for perfect phase compensation is that the estimation of $\theta _k$ is as close as possible to the real phase offset in the $k$-th sub-channel. However, because of the quantum uncertainty and experimental noise of the system, the estimation values $\hat {\theta }_{n_{min}}$ and $\hat {\theta }_{n_{max}}$ can’t be measured precisely, resulting in an extra phase noise. Given that the measurement inaccuracy of the reference signals are

$$\Delta\theta_{n_{min}}=\hat{\theta}_{n_{min}}-\theta_{n_{min}},\quad \Delta\theta_{n_{max}}=\hat{\theta}_{n_{max}}-\theta_{n_{max}}.$$
According to Eq. (5), the estimation error of the $k$-th sub-channel can be described by
$$\begin{aligned} \Delta\theta_k^{error}=\hat{\theta}_k-\theta_k=\frac{k-n_{min}}{n_{max}-n_{min}}(\Delta\theta_{n_{max}}-\Delta\theta_{n_{min}})+\Delta\theta_{n_{min}}. \end{aligned}$$
Then, the phase noise variance of the $k$-th sub-channel due to measurement inaccuracy is give by
$$V^{error}_k=V^{error}_{n_{min}}+\left(\frac{k-n_{min}}{n_{max}-n_{min}}\right)^2(V^{error}_{n_{max}}+V^{error}_{n_{min}}),$$
where $V^{error}_{n_{max}}$ and $V^{error}_{n_{min}}$ are the variance of $\Delta \theta _{n_{max}}$ and $\Delta \theta _{n_{min}}$, respectively. They represent the measurement inaccuracy noise of the outermost two pilot sub-channels and expressed as
$$V^{error}_{n_{max}} = \frac{\chi_{n_{max}}+1}{|\alpha_{n_{max}}|^2}, \quad V^{error}_{n_{min}} = \frac{\chi_{n_{min}}+1}{|\alpha_{n_{min}}|^2},$$
where $|\alpha _{n_{max}}|$ and $|\alpha _{n_{min}}|$ are the amplitude of reference signals in sub-channels $n_{max}$ and $n_{min}$ respectively. $\chi _{n_{max}}$ and $\chi _{n_{min}}$ are the total noise imposed on the reference signals which are given by
$$\chi_{n_{max}} = \chi_{n_{min}} = \frac{2-{\eta}T}{{\eta}T}+\frac{2v_{el}}{{\eta}T}+\varepsilon,$$
where $\eta$ and $V_{el}$ represent the detector efficiency and the electronic noise of the detector, respectively. $T$ denotes the transmittance of the channel. The first term of Eq. (11) is the loss-induced vacuum noise, the second term is the electronic noise of the heterodyne detection, and $\varepsilon$ is the excess noise in the channel. We assume that the detectors used to measure two pilot sub-channels have the same parameters and the transmittance of each tributary is identical since all of the sub-channels are propagated in the same fiber and the wavelengths of the comb lines are close so their losses can be considered as a constant.

3.2 Noise caused by dispersive walk-off

In a OFC-based system, dispersive walk-off during fiber transmission is a major factor to affect the phase coherence, which causes a temporal decorrelation between the sub-channels [35]. In our system, the delay between two sub-channels $i$ and $j$ caused by dispersion is given by [45]

$$\tau_{ij} = DL\Delta\lambda_{ij},$$
where $D$ is the fiber chromatic dispersion parameter which is about 16 ps/km${\cdot }$nm in a standard single-mode fiber. L is the transmission distance and $\Delta \lambda _{ij}$ is the wavelength offset between the two sub-channels. We assume that the quantum signal in sub-channel $k$ arrives at Bob’s receiver at time $t$, the two pilot sub-channels $n_{max}$ and $n_{min}$ arrive at Bob’s receiver at times $t-\tau _{kn_{max}}$ and $t+\tau _{kn_{min}}$, respectively. $\tau _{kn_{min}}$ and $\tau _{kn_{max}}$ are the delay caused by dispersion, as shown in Fig. 3. In the ideal case where there is no dispersion, the quantum and reference signals will arrive Bob’s receiver at the same time $t$, the phase offsets of the two pilot sub-channels can be expressed as
$$\hat{\theta}_{n_{max}}=\varphi^B_{n_{max}}(t)-\varphi^A_{n_{max}}(t_A),\quad \hat{\theta}_{n_{min}}=\varphi^B_{n_{min}}(t)-\varphi^A_{n_{min}}(t_A),$$
where $\varphi ^A_{n_{max}}(t_A)$ and $\varphi ^A_{n_{min}}(t_A)$ are the phase of the reference signals of sub-channels $n_{max}$ and $n_{min}$ at Alice side, $\varphi ^B_{n_{max}}(t)$ and $\varphi ^B_{n_{min}}(t)$ are the phase when they arriving Bob’s receiver at time $t$. When we take account into the influence of dispersion, the different arrive times between the pilot and quantum sub-channels will introduce a phase noise in each tributary. To be specific, due to the delay between sub-channels, the phase offset $\hat {\theta }_{n_{max}}$ and $\hat {\theta }_{n_{min}}$ should be rewritten as
$$\hat{\theta}'_{n_{max}}=\varphi^B_{n_{max}}(t-\tau_{kn_{max}})-\varphi^A_{n_{max}}(t_A), \quad \hat{\theta}'_{n_{min}}=\varphi^B_{n_{min}}(t+\tau_{kn_{min}})-\varphi^A_{n_{min}}(t_A).$$
The phase of reference signals at different times are related by [41]
$$\begin{aligned} & \varphi^B_{n_{max}}(t) = \varphi^B_{n_{max}}(t-\tau_{kn_{max}})+2{\pi}f_{n_{max}}\tau_{kn_{max}}+N_{n_{max}}, \\ & \varphi^B_{n_{min}}(t+\tau_{kn_{min}}) = \varphi^B_{n_{min}}(t)+2{\pi}f_{n_{min}}\tau_{kn_{min}}+N_{n_{min}}. \end{aligned}$$
where $f_{n_{max}}$ and $f_{n_{min}}$ are the central frequencies of the respective signals, $N_{n_{max}}$ and $N_{n_{min}}$ are independent Gaussian noises. As a result, the phase offset in the $k$-th signal sub-channel is actually calculated by
$$\hat{\theta}'_k=\varphi^B_{n_{min}}(t+\tau_{kn_{min}})-\varphi^A_{n_{min}}(t_A) +\frac{k-n_{min}}{n_{max}-n_{min}}[\varphi^B_{n_{max}}(t-\tau_{kn_{max}}) -\varphi^B_{n_{min}}(t+\tau_{kn_{min}})].$$
For simplicity we assume that $\varphi ^A_{n_{max}}(t_A) = \varphi ^A_{n_{min}}(t_A) = \varphi ^A_k(t_A)$ since all of the sub-channels are came from the same OFC and carried out the same process at Alice side. Then, we can obtain the variance of the deviation $\hat {\theta }'_k-\hat {\theta }_k$ caused by dispersion as
$$\begin{aligned} V_k^{disp}={<}(\Delta\varphi(\tau_{kn_{min}}))^2>{+}\left(\frac{k-n_{min}}{n_{max}-n_{min}}\right)^2 (<(\Delta\varphi(\tau_{kn_{max}}))^2>{+}<(\Delta\varphi(\tau_{kn_{min}}))^2>), \end{aligned}$$
where $<(\Delta \varphi (\tau _{kn_{max}}))^2>$ and $<(\Delta \varphi (\tau _{kn_{min}}))^2>$ are the variances of $N_{n_{max}}$ and $N_{n_{min}}$ repectively, which can be expressed as [41]
$$<(\Delta\varphi(\tau_{kn_{max}}))^2>{=}\frac{2\tau_{kn_{max}}}{t^c_{n_{max}}}, \quad <(\Delta\varphi(\tau_{kn_{min}}))^2>{=}\frac{2\tau_{kn_{min}}}{t^c_{n{min}}},$$
where $t^c_{n_{max}}$ and $t^c_{n_{min}}$ are the coherence times which are related to the linewidth of the OFC ${\Delta }\nu$ by
$$t^c_{n_{max}}=t^c_{n_{min}}=\frac{1}{\pi{\Delta}\nu}.$$
A feasible way to avoid this kind of noise is to estimate the time difference between each quantum sub-channel and the two pilot sub-channels before detection, then a delay line can be applied to compensate for the time difference so as to offset the deviation between $\varphi _r^B(t)$ and $\varphi _r^B(t\pm \tau _{kr})$, where $r\in \{n_{min}, n_{max}\}$. This approach requires accurate estimation of the delay and additional delay line on each sub-channel, which slightly increases the complexity of reception. In order to analyze the possible noise under normal conditions, this paper we consider the simpler receiving method without delay compensation.

 

Fig. 3. Temporal decorrelation between quantum and reference signals.

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3.3 Noise caused by photon leakage

In a CVQKD system that transmits quantum signal with high-power LO in the same fiber, the leakage from the strong LO into the quantum signal path may cause an additional noise. In this OFC-based CVQKD scheme, although no LO exists, the leaked photons from the pilot sub-channels to the weak quantum sub-channels still need to be considered because frequency comb generation is always constrained by a finite extinction ratio $R_e$ in realistic system. Extinction ratio $R_e$ is the ratio of high level and low level of optical pulses, which is determined by the technologies of light pulses generation. The photon leakage noise caused by leaked photons from the two pilot sub-channels is given by [9]

$$\varepsilon_{LE} = \frac{2\langle\hat{N}^{Alice}_{n_{max}}\rangle}{R_e}+ \frac{2\langle\hat{N}^{Alice}_{n_{min}}\rangle}{R_e}$$
where $\langle \hat {N}^{Alice}_{n_{max}}\rangle$ represents the power of the reference signal in sub-channel $n_{max}$, $\langle \hat {N}^{Alice}_{n_{min}}\rangle$ denotes the power of the reference signal in sub-channel $n_{min}$.

3.4 Total excess noise

Based on the analysis in the above subsections, the total excess noise $\varepsilon _k$ in the $k$-th sub-channel can be defined as

$$\varepsilon_k = V_a(V^{error}_k+V^{disp}_k)+\varepsilon_{LE}.$$
The first term of Eq. (21) is the phase noise from imperfect phase compensation and dispersive walk-off, the second term is the noise originating from leaked photons of the pilot sub-channels. For security, although all of these noise comes from QKD devices, we still consider the most pessimistic situation that they can be generated and controlled by Eve.

It is worth noting that the power of the reference signals should be limited in order to suppress the photon leakage noise, but too weak reference signals will lead to a large measurement inaccuracy. Figure 4(a) shows the excess noise of each sub-channel in terms of the ratio $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ when $N=7$. Here we assume that $\langle \hat {N}^{Alice}_{n_{max}}\rangle =\langle \hat {N}^{Alice}_{n_{min}}\rangle$ for simplicity. We find that the optimal range of the ratio $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is between $500$ and $700$ at a transmission distance of 30 km with $R_e = 60$ dB. When the value of the ratio smaller than $500$ or larger than $700$, the excess noise of each sub-channel apparently increases. In Fig. 4(b), we plot the excess noise in each sub-channel when the extinction ratio $Re=\{40,\ 50,\ 60\}$ dB, the number of sub-channel $N=15$. We can see that the extinction ratio is an important parameter as the larger $Re$ can effectively suppress the photon leakage noise. Other parameters used in Fig. 4 are $V_a=8$, $V_{el} = 0.01$, $\eta = 0.7$ (all in shot-noise units), $\Delta \nu =100$ kHz and $f_r^s=10$ GHz.

 

Fig. 4. (a) The the excess noise of each sub-channel in terms of the ratio $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ when $N=7$. $k$ is the channel index. (b) The excess noise in each sub-channel when the extinction ratio $Re=\{40,50,60\}$ dB. The number of sub-channel is set as $N=15$.

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4. Security analysis

4.1 Possible attacks and countermeasures

An important assumption in this paper is that the phase offset between each sub-channel is correlated, which is valid when there is no attack against this correlation. Therefore, we analyze the phase offset related attacks that Eve might initiate and discuss the security of the system under such attacks in this subsection.

First, we consider the situation that Eve attacks the quantum sub-channel $k$ with a general Gaussian collective attack and changes the phase offset $\theta _k$ into $\theta _k^{attack}$ to break the phase coherence. Alice and Bob don’t realize this change and still use $\hat {\theta }_k$ to correct Alice’s quadrature values. Then the total excess noise in the $k$-th sub-channel should be rewritten as

$$\varepsilon^{attack}_k = V_a(\tilde{V}^{error}_{k}+V^{disp}_k)+\varepsilon_{LE}+\varepsilon_{Eve},$$
where $\varepsilon _{Eve}$ is attributed to Eve’s attack on the quantum signal and $\tilde {V}^{error}_k = Var(\hat {\theta }_k-\theta _k^{attack})$ represents the measurement inaccuracy noise after Eve changes the phase offset. To hide her attack successfully, Eve needs to make the increased excess noise caused by the attack on quantum signal be compensated by a reduction of measurement inaccuracy noise. Only in this way can the total excess noise keep unchanged and the legitimate parties won’t discover the attack in parameter estimation process. Therefore, we can get a relation as
$$V_a\tilde{V}^{error}_{k}+\varepsilon_{Eve}\leq{V_a}V^{error}_{k}.$$
That is to say, the changed phase offset $\theta _k^{attack}$ must satisfy the following relation as
$$Var(\hat{\theta}_k-\theta_k^{attack})<Var(\hat{\theta}_k-\theta_k).$$
This inequality can’t be realized by simply making $\theta _k^{attack}<\theta _k$ or $\theta _k^{attack}>\theta _k$, Eve should calculate the value of $\hat {\theta }_k$ first. However, as described in Eq. (5), the value of $\hat {\theta }_k$ is based on the values of $\hat {\theta }_{n_{max}}$ and $\hat {\theta }_{n_{min}}$, which means that Eve must measure the quadrature values of the reference signals. This process will increase the uncertainty of the reference sub-channels and introduce extra excess noise to the system [46]. Therefore, we can conclude that Eve is unlikely to launch such an attack.

Second, we consider the situation that Eve attacks the quantum sub-channel $k$ with a general Gaussian collective attack and changes the phase offset $\theta _{n_{max}}$ and $\theta _{n_{min}}$ of pilot sub-channels into $\theta ^{attack}_{n_{max}}$ and $\theta ^{attack}_{n_{min}}$ to break the phase coherence. Alice and Bob obtain an estimated value of $\hat {\theta }^{attack}_k$, which is also known by Eve because she can estimate it using the phase coherent property of the comb. Then the measurement inaccuracy noise can be expressed as $\tilde {V}^{error}_k = Var(\hat {\theta }^{attack}_k-\theta _k)$ and we can get a relation as

$$Var(\hat{\theta}^{attack}_k-\theta_k)<Var(\hat{\theta}_k-\theta_k).$$
It means that Eve must calculate the real phase offset $\theta _k$ in order to satisfy the above inequality. However, the phase offset of weak quantum signal is hard to determine and can only be shared between co-transmitted carriers from emitter to receiver [40]. Therefore, we can also conclude that Eve is unlikely to launch such an attack.

Finally, we consider the situation that Eve attacks the quantum signal in sub-channel $k$ with a general Gaussian collective attack and separates the pilot sub-channels and quantum sub-channels into two paths transmitting to Bob. One of the path containing quantum information is transmitted through standard single mode fiber (SMF) while the another path involving reference information is transmitted through low loss fiber. This kind of attack is the most possible way for Eve to hide herself and was analyzed in [46]. The purpose of this attack is to elevate the amplitude of the reference signals at Bob side to obtain a smaller measurement inaccuracy noise. Because Alice and Bob only estimate the total excess noise but overlook the variation of individual noise, Eve could gain more information from quantum signal than usual without being discovered. [46] also proposed the countermeasure against this kind of attack, monitoring the instantaneous amplitude of the reference signals and calibrating phase noise in real time enables the legitimate parties to estimate information of Eve more accurately.

4.2 System performance

According to the above analysis, we show the performance of the OFC-based multichannel parallel CVQKD protocol for reverse reconciliation in this section. We assume the attacks of Eve on each channel is independent since the quadrature information of each sub-channel is independent of each other. Therefore the total secret key bit rate can be expressed as the sum of key rate of each sub-channel given by

$$R_{tot}=\sum_kR_k,$$
where $k$ is the index of the quantum sub-channels which varies from $n_{min}+1$ to $n_{max}-1$, for the outermost two sub-channels $n_{min}$ and $n_{max}$ are used to transmit the reference signals without involving any key information. The notation $R_k=f_{rep}K_k$ is the secret key bit rate of sub-channel $k$ against the optimal collective attack, where $K_k$ is described as
$$K_k = {\beta}I_{AB}-\chi_{BE}.$$
Here $f_{rep}$ denotes the system repetition rate, $\beta$ is the reverse reconciliation efficiency assumed to be constant for each channel, $I_{AB}$ represents the mutual information between Alice and Bob and $\chi _{BE}$ represents the actual Holevo quantity for Eve’s maximum accessible information, which are detailed in the Appendix.

We perform numerical simulations with the parameters listed in Table 1, corresponding to the current state-of-the-art experimental technology. In order to determine the impact of the crucial parameters on the secret key rate and transmission distance, we plot the expected total secret key bit rate of the proposed scheme for different values of the linewidth ${\Delta }\nu$ in Fig. 5(a), and for different values of OFC repetition rate $f_r^s$ in Fig. 5(b), respectively. The number of sub-channel and the extinction ratio are set as $N=15, Re=60$ dB. We find that the increased linewidth sharply reduces the transmission distance because the increased ${\Delta }\nu$ results in a increased dispersion noise in each sub-channel. From Fig. 5(b) we note that the security distance is impressionable with the repetition rate $f_r^s$ because the changed repetition rate results in a changed wavelength offset between the signal and pilot sub-channels, thereby changing the delay caused by dispersion and finally affecting the excess noise, as analyzed in Eqs. (12), (17) and (18). A small OFC repetition rate contributes to a high spectral efficiency and a decreased excess noise, but it’s value is restricted by the generation technology of frequency comb. The secret key bit rate of each sub-channel when $N=15$ is shown in Fig. 6(a), we find that the first and last sub-channels have the maximal and minimal transmission distances, because the excess noise is an increasing function of the channel index $k$ as shown in Fig. 4(b). Figure 6(b) shows the secret key bit rate for the first and last sub-channels when $N={7,15,35}$. For each case the first sub-channel always performs best and with the $N$ increases, the transmission distance of the first sub-channel increases slightly but the transmission distance of the last sub-channel decreases significantly. The three-dimensional diagram of total secret key bit rate $R_{tot}$, modulation variance $V_a$ and transmission distance $L$ is shown in Fig. 7(a). Figure 7(b) shows a top view of the surface in Fig. 7(a) when $N=35$. We find that an optimization of $V_a$ can effectively improve the security distance as well as the total key rate. To achieve a higher secret key rate and a longer transmission distance, the best value range of $V_a$ is between 5 and 10. It is worth noting that although the transmission distance is the largest when the value of $V_a$ is around 3, the key rate at that time is not the highest (different colors represent different values of key rate). Figure 8(a) shows the secret key rate for $N=\{7, 15, 35\}$, comparing with a single-channel CVQKD with the identical parameters as those in the proposed scheme. It demonstrates that with the $N$ increases, the total secret key bit rate is significantly improved, whereas the maximum transmission distance is slightly shortened at the same time. The reason is that the increased number of tributaries results in the increased delay time between quantum sub-channels and pilot sub-channels, thereby increasing the phase noise in QKD process.

 

Fig. 5. (a) The secret key rate of the OFC-based CVQKD system against collective attacks in the asymptotic limit with the variable parameter ${\Delta }\nu =\{10^4,10^5,10^6,10^7\}$ Hz. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate and the repetition rate of the comb is set as $f_r^s=10$ GHz. (b) The secret key rate of the OFC-based CVQKD system against collective attacks in the asymptotic limit with the variable parameter $f_r^s=\{10,20,30,40\}$ GHz. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate and the linewidth is given by $\Delta \nu =10^5$ Hz.

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Fig. 6. (a) The secret key bit rate of each sub-channel when $N=15$. (b) The secret key bit rate for the first and last sub-channels when $N={7,15,35}$. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate. Other parameters are $f^s_r=10$ GHz, $\Delta \nu =100$ kHz, and $Re=60$ dB.

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Fig. 7. (a) Secret key bit rate $R_{tot}$ as a function of modulation variance $V_a$ and transmission distance $L$. The number of sub-channels are $N=7$, $N=15$, and $N=35$, respectively. (b) The top view of the surface in Fig. 7(a) when $N=35$. The black solid line represents the bound between positive and negative secret key rate. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is set as 500. Other parameters are $f^s_r=10$ GHz, $\Delta \nu =100$ kHz, and $Re=60$ dB.

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Fig. 8. (a) The secret key bit rate as a function of the transmission distance. Curves from top to bottom are $N=35$, $N=15$, $N=7$, and the single channel case, respectively. (b) The multichannel gain of the system. Curves from top to bottom are $N=35$, $N=15$, and $7$, respectively. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate. Other parameters are $f^s_r=10$ GHz, $\Delta \nu =100$ kHz, and $Re=60$ dB.

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Tables Icon

Table 1. Numerical Parameters

In order to demonstrate the improvement of the CVQKD system in terms of the secret key bit rate, we take multichannel gain $G$ to describe the tendency, which can be expressed as

$$G=\frac{R_{tot}}{R_{s}},$$
where $R_{s}$ represents the secret key bit rate in the single channel CVQKD with the identical parameters shown in Table 1. Figure 8(b) shows the evolution of $G$ as a function of the transmission distance $L$. With the lengthened $L$, the value of $G$ decreases and finally falls to zero when approaching the maximum transmission distance. We also find that the multichannel gain of $N=35$ is higher than those of $N=15$ and $N=7$ when the distance is less than 47 km.

For purpose of illustrate the improvement of the OFC-based CVQKD system in numerical simulations, we take the modules as ideal devices, such as the light source, modulator, demultiplexer and multiplexer. In practice, the implementation imperfections of these modules may reduce the improvement, which depends on the actual performance of the devices. Demultiplexer is a crucial device related to the sub-channel numbers and frequency spacing. A arrayed-waveguide gratings (AWG) demultiplexer with 32 output and 5-GHz channel spacing has been available since 2009 [47]. Besides, AM and PM are required for each tributary, which inevitably increases hardware complexity, especially when the number of tributary is large. Moreover, the extinction ratio and the repetition rate of the comb, the number of comb lines also have a great influence on the performance of the system. We assume that the OFCs employed are densely spaced with high extinction ratio and narrow linewidth. However, it is difficult to produce a narrow linewidth comb with a wide bandwidth and high repetition rate using the current OFC generation techniques. A tunable gain-switched OFC with a repetition rate of $10$ GHz, extinction ratio in excess of $50$ dB and narrow linewidth (¡100kHz) has been demonstrated in 2009 [48], a $20$-line OFC with a frequency spacing of $10$ GHz and flatness of $0.7$ dB has been demonstrated in experiment [49]. Consequently, the improvement of the OFC-based CVQKD system could be seen as the upper bound, future improved devices and technologies will make the practical performance close to the ideal estimation.

5. Conclusion

We proposed the OFC-based CVQKD protocol for the parallel multichannel system, which enables the increased secret key bit rate without LO transmission. The phase compensation can be performed by selecting the first and last two comb lines to transmit reference signals. We studied the excess noise resulting from unperfect phase compensation, dispersive walk-off and photon leakage from pilot lines, analyzed the crucial parameters that related to the excess noise and the security of the system. We evaluated the increment on the secret key bit rate of the proposed system against collective attacks. Numerical simulations shown that the OFC-based multichannel parallel CVQKD system can considerably increase the key rate at the cost of slightly shortening the maximal transmission distance.

Appendix. Calculation of the secret key rate

In this appendix, we provide the details of secret key rate calculation, which are based on the analysis of [50]. The secret key under the optimal collective attack, in the case of reverse reconciliation, is given by Eq. 27. The mutual information between Alice and Bob, is derived from Bob’s measured variance $V_B={\eta }T(V+\chi _{tol})$ and the conditional variance $V_{B|A}={\eta }T(1+\chi _{tol})$ using Shannon’s equation,

$$I_{AB}=\frac{1}{2}\log_2\frac{V_B}{V_{B|A}}=\frac{1}{2}\log_2\frac{V+\chi_{tol}}{1+\chi_{tol}},$$
where $\chi _{tol}=\chi _{line}+\chi _{h}/T$ denotes the total noise referred to the channel input, $\chi _{line}=1/T+\varepsilon -1$ denotes the channel-added noise referred to the channel input, $\chi _{h}$ denotes the detection-added noise, which can be given by the expressions $\chi _{hom}=[(1-\eta )+v_{el}]/\eta$ and $\chi _{het}=[1+(1-\eta )+2v_{el}]/\eta$ for homodyne and heterodyne detection, respectively. $\chi _{BE}$ is the maximum information available to Eve on Bob’s key, with the form
$$\chi_{BE}=S(\rho_E)-{\int}d_{m_B}p(m_B)S(\rho_E^{m_B}),$$
where $m_B$ denotes the measurement of Bob, $p(m_B)$ is the probability density of the measurement, $\rho _E^{m_B}$ is the eavesdropper’s state conditional on Bob’s measurement, and $S$ represents the Von Neumann entropy of the state. In the case of Gaussian attack, Eq. 30 can be simplified to
$$\chi_{BE}=\sum_{i=1}^2G\left(\frac{\lambda_i-1}{2}\right)-\sum_{i=3}^5G\left(\frac{\lambda_i-1}{2}\right),$$
where $G(x)=(x+1)\log _2(x+1)-x\log _2(x)$, the symplectic eigenvalues $\lambda _{1,2}$ are given by
$$\lambda_{1,2}^2=\frac{1}{2}\left(A\pm\sqrt{A^2-4B}\right),$$
with
$$\begin{array}{c} A=V^2+T^2(V+\chi_{line})^2+2T(1-V^2),\\ B=T^2(1+V\chi_{line})^2, \end{array}$$
the symplectic eigenvalues $\lambda _{3,4}$ are given by
$$\lambda_{3,4}^2=\frac{1}{2}\left(C\pm\sqrt{C^2-4D}\right),$$
with
$$\begin{array}{c}C=\frac{A\chi_{hom}+V\sqrt{B}+T(V+\chi_{line})}{T(V+\chi_{tot})},\\ D=\frac{\sqrt{B}V+B\chi_{hom}}{T(V+\chi_{tot})}, \end{array}$$
and the last eigenvalue is $\lambda _5=1$. Based on the above formulas, we can obtain the upper bound of $\chi _{BE}$ and get the final secret key rate.

Funding

Fundamental Research Funds for the Central Universities (2019zzts278); Natural Science Foundation of Hunan Province (2019JJ40352); National Natural Science Foundation of China (61379153, 61572529).

Disclosures

The authors declare no conflicts of interest.

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References

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  1. H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
    [Crossref]
  2. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
    [Crossref]
  3. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
    [Crossref]
  4. R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable qkd with intense dwdm classical channels,” New J. Phys. 17(4), 043027 (2015).
    [Crossref]
  5. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theor. Comput. Sci. 560, 7–11 (2014).
    [Crossref]
  6. F. Xu, M. Curty, B. Qi, L. Qian, and H. K. Lo, “Discrete and continuous variables for measurement-device-independent quantum cryptography,” Nat. Photonics 9(12), 772–773 (2015).
    [Crossref]
  7. Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
    [Crossref]
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    [Crossref]
  10. Y. Guo, Q. Liao, Y. Wang, D. Huang, P. Huang, and G. Zeng, “Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction,” Phys. Rev. A 95(3), 032304 (2017).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. M. Navascués, F. Grosshans, and A. Acin, “Optimality of gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97(19), 190502 (2006).
    [Crossref]
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    [Crossref]
  22. R. Renner and J. I. Cirac, “de Finetti representation theorem for infinite-dimensional quantum systems and applications to quantum cryptography,” Phys. Rev. Lett. 102(11), 110504 (2009).
    [Crossref]
  23. F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
    [Crossref]
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2019 (2)

2018 (5)

T. Wang, P. Huang, Y. Zhou, W. Liu, and G. Zeng, “Pilot-multiplexed continuous-variable quantum key distribution with a real local oscillator,” Phys. Rev. A 97(1), 012310 (2018).
[Crossref]

H. Zhang, Y. Mao, D. Huang, J. Li, L. Zhang, and Y. Guo, “Security analysis of orthogonal-frequency-division-multiplexing–based continuous-variable quantum key distribution with imperfect modulation,” Phys. Rev. A 97(5), 052328 (2018).
[Crossref]

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, J. Schröder, and P. Andrekson, “Frequency comb-based WDM transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
[Crossref]

P. Huang, J. Huang, Z. Zhang, and G. Zeng, “Quantum key distribution using basis encoding of gaussian-modulated coherent states,” Phys. Rev. A 97(4), 042311 (2018).
[Crossref]

2017 (3)

Y. Guo, Q. Liao, Y. Wang, D. Huang, P. Huang, and G. Zeng, “Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction,” Phys. Rev. A 95(3), 032304 (2017).
[Crossref]

A. Marie and R. Alléaume, “Self-coherent phase reference sharing for continuous-variable quantum key distribution,” Phys. Rev. A 95(1), 012316 (2017).
[Crossref]

K. Qu, S. Zhao, X. Li, Z. Zhu, D. Liang, and D. Liang, “Ultra-flat and broadband optical frequency comb generator via a single mach–zehnder modulator,” IEEE Photonics Technol. Lett. 29(2), 255–258 (2017).
[Crossref]

2016 (4)

J. N. Kemal, J. Pfeifle, P. Marin-Palomo, M. D. G. Pascual, S. Wolf, F. Smyth, W. Freude, and C. Koos, “Multi-wavelength coherent transmission using an optical frequency comb as a local oscillator,” Opt. Express 24(22), 25432–25445 (2016).
[Crossref]

H. Qin, R. Kumar, and R. Alléaume, “Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution,” Phys. Rev. A 94(1), 012325 (2016).
[Crossref]

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
[Crossref]

D. Huang, P. Huang, H. Li, T. Wang, Y. Zhou, and G. Zeng, “Field demonstration of a continuous-variable quantum key distribution network,” Opt. Lett. 41(15), 3511–3514 (2016).
[Crossref]

2015 (9)

S. Bahrani, M. Razavi, and J. A. Salehi, “Orthogonal frequency-division multiplexed quantum key distribution,” J. Lightwave Technol. 33(23), 4687–4698 (2015).
[Crossref]

J. Pfeifle, V. Vujicic, R. T. Watts, P. C. Schindler, C. Weimann, R. Zhou, W. Freude, L. P. Barry, and C. Koos, “Flexible terabit/s Nyquist-WDM super-channels using a gain-switched comb source,” Opt. Express 23(2), 724–738 (2015).
[Crossref]

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Ultrahigh count coherent WDM channels transmission using optical parametric comb-based frequency synthesizer,” J. Lightwave Technol. 33(3), 694–699 (2015).
[Crossref]

D. Huang, D. Lin, C. Wang, W. Liu, S. Fang, J. Peng, P. Huang, and G. Zeng, “Continuous-variable quantum key distribution with 1 Mbps secure key rate,” Opt. Express 23(13), 17511–17519 (2015).
[Crossref]

F. Xu, M. Curty, B. Qi, L. Qian, and H. K. Lo, “Discrete and continuous variables for measurement-device-independent quantum cryptography,” Nat. Photonics 9(12), 772–773 (2015).
[Crossref]

R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable qkd with intense dwdm classical channels,” New J. Phys. 17(4), 043027 (2015).
[Crossref]

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5(4), 041009 (2015).
[Crossref]

D. B. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5(4), 041010 (2015).
[Crossref]

D. Huang, P. Huang, D. Lin, C. Wang, and G. Zeng, “High-speed continuous-variable quantum key distribution without sending a local oscillator,” Opt. Lett. 40(16), 3695–3698 (2015).
[Crossref]

2014 (6)

T. Shao, E. Martin, P. M. Anandarajah, C. Browning, V. Vujicic, R. Llorente, and L. P. Barry, “Chromatic dispersion-induced optical phase decorrelation in a 60 GHz OFDM-RoF system,” IEEE Photonics Technol. Lett. 26(20), 2016–2019 (2014).
[Crossref]

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theor. Comput. Sci. 560, 7–11 (2014).
[Crossref]

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

Z. Li, Y. C. Zhang, F. Xu, X. Peng, and H. Guo, “Continuous-variable measurement-device-independent quantum key distribution,” Phys. Rev. A 89(5), 052301 (2014).
[Crossref]

X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer in multiuser OFDM systems,” IEEE Trans. Wirel. Commun. 13(4), 2282–2294 (2014).
[Crossref]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014).
[Crossref]

2013 (1)

P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7(5), 378–381 (2013).
[Crossref]

2012 (3)

F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

K. Yoshino, M. Fujiwara, A. Tanaka, S. Takahashi, Y. Nambu, A. Tomita, S. Miki, T. Yamashita, Z. Wang, M. Sasaki, and A. Tajima, “High-speed wavelength-division multiplexing quantum key distribution system,” Opt. Lett. 37(2), 223–225 (2012).
[Crossref]

2011 (1)

2010 (1)

A. Leverrier and P. Grangier, “Simple proof that gaussian attacks are optimal among collective attacks against continuous-variable quantum key distribution with a gaussian modulation,” Phys. Rev. A 81(6), 062314 (2010).
[Crossref]

2009 (3)

R. Renner and J. I. Cirac, “de Finetti representation theorem for infinite-dimensional quantum systems and applications to quantum cryptography,” Phys. Rev. Lett. 102(11), 110504 (2009).
[Crossref]

J. Z. Zhang, A. B. Wang, J. F. Wang, and Y. C. Wang, “Wavelength division multiplexing of chaotic secure and fiber-optic communications,” Opt. Express 17(8), 6357–6367 (2009).
[Crossref]

S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At., Mol. Opt. Phys. 42(11), 114014 (2009).
[Crossref]

2007 (1)

B. Qi, L. L. Huang, L. Qian, and H. K. Lo, “Experimental study on the gaussian-modulated coherent-state quantum key distribution over standard telecommunication fibers,” Phys. Rev. A 76(5), 052323 (2007).
[Crossref]

2006 (2)

R. Garcia-Patron and N. J. Cerf, “Unconditional optimality of gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97(19), 190503 (2006).
[Crossref]

M. Navascués, F. Grosshans, and A. Acin, “Optimality of gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97(19), 190502 (2006).
[Crossref]

2004 (1)

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum cryptography without switching,” Phys. Rev. Lett. 93(17), 170504 (2004).
[Crossref]

2003 (1)

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref]

2002 (2)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
[Crossref]

Acin, A.

M. Navascués, F. Grosshans, and A. Acin, “Optimality of gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97(19), 190502 (2006).
[Crossref]

Alic, N.

Alléaume, R.

A. Marie and R. Alléaume, “Self-coherent phase reference sharing for continuous-variable quantum key distribution,” Phys. Rev. A 95(1), 012316 (2017).
[Crossref]

H. Qin, R. Kumar, and R. Alléaume, “Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution,” Phys. Rev. A 94(1), 012325 (2016).
[Crossref]

R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable qkd with intense dwdm classical channels,” New J. Phys. 17(4), 043027 (2015).
[Crossref]

R. Kumar, H. Qin, and R. Alléaume, “Experimental demonstration of the coexistence of continuous-variable quantum key distribution with an intense DWDM classical channel,” in CLEO: QELS_Fundamental Science (Optical Society of America, 2014), pp. FM4A–1.

Anandarajah, P.

Anandarajah, P. M.

T. Shao, E. Martin, P. M. Anandarajah, C. Browning, V. Vujicic, R. Llorente, and L. P. Barry, “Chromatic dispersion-induced optical phase decorrelation in a 60 GHz OFDM-RoF system,” IEEE Photonics Technol. Lett. 26(20), 2016–2019 (2014).
[Crossref]

Andrekson, P.

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, J. Schröder, and P. Andrekson, “Frequency comb-based WDM transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
[Crossref]

Ataie, V.

Bahrani, S.

Barry, L. P.

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theor. Comput. Sci. 560, 7–11 (2014).
[Crossref]

Berta, M.

F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
[Crossref]

Bettelli, S.

H. H. Brunner, L. C. Comandar, F. Karinou, S. Bettelli, D. Hillerkuss, F. Fung, D. Wang, S. Mikroulis, Q. Yi, M. Kuschnerov, A. Poppe, C. Xie, and M. Peev, “A low-complexity heterodyne cv-qkd architecture,” in 2017 19th International Conference on Transparent Optical Networks (ICTON), (2017), pp. 1–4.

Bobrek, M.

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5(4), 041009 (2015).
[Crossref]

Bowen, W. P.

C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum cryptography without switching,” Phys. Rev. Lett. 93(17), 170504 (2004).
[Crossref]

Brasch, V.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theor. Comput. Sci. 560, 7–11 (2014).
[Crossref]

Brif, C.

D. B. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5(4), 041010 (2015).
[Crossref]

Brouri, R.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref]

Browning, C.

T. Shao, E. Martin, P. M. Anandarajah, C. Browning, V. Vujicic, R. Llorente, and L. P. Barry, “Chromatic dispersion-induced optical phase decorrelation in a 60 GHz OFDM-RoF system,” IEEE Photonics Technol. Lett. 26(20), 2016–2019 (2014).
[Crossref]

Brunner, H. H.

H. H. Brunner, L. C. Comandar, F. Karinou, S. Bettelli, D. Hillerkuss, F. Fung, D. Wang, S. Mikroulis, Q. Yi, M. Kuschnerov, A. Poppe, C. Xie, and M. Peev, “A low-complexity heterodyne cv-qkd architecture,” in 2017 19th International Conference on Transparent Optical Networks (ICTON), (2017), pp. 1–4.

Camacho, R. M.

D. B. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5(4), 041010 (2015).
[Crossref]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

R. Garcia-Patron and N. J. Cerf, “Unconditional optimality of gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97(19), 190503 (2006).
[Crossref]

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
[Crossref]

Chen, C.

Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
[Crossref]

Cirac, J. I.

R. Renner and J. I. Cirac, “de Finetti representation theorem for infinite-dimensional quantum systems and applications to quantum cryptography,” Phys. Rev. Lett. 102(11), 110504 (2009).
[Crossref]

Coles, P. J.

D. B. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5(4), 041010 (2015).
[Crossref]

Comandar, L. C.

H. H. Brunner, L. C. Comandar, F. Karinou, S. Bettelli, D. Hillerkuss, F. Fung, D. Wang, S. Mikroulis, Q. Yi, M. Kuschnerov, A. Poppe, C. Xie, and M. Peev, “A low-complexity heterodyne cv-qkd architecture,” in 2017 19th International Conference on Transparent Optical Networks (ICTON), (2017), pp. 1–4.

Curty, M.

F. Xu, M. Curty, B. Qi, L. Qian, and H. K. Lo, “Discrete and continuous variables for measurement-device-independent quantum cryptography,” Nat. Photonics 9(12), 772–773 (2015).
[Crossref]

H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

Debuisschert, T.

S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At., Mol. Opt. Phys. 42(11), 114014 (2009).
[Crossref]

Diamanti, E.

P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7(5), 378–381 (2013).
[Crossref]

S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At., Mol. Opt. Phys. 42(11), 114014 (2009).
[Crossref]

Fang, S.

Fossier, S.

S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At., Mol. Opt. Phys. 42(11), 114014 (2009).
[Crossref]

Franz, T.

F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
[Crossref]

Freude, W.

Fujiwara, M.

Fung, F.

H. H. Brunner, L. C. Comandar, F. Karinou, S. Bettelli, D. Hillerkuss, F. Fung, D. Wang, S. Mikroulis, Q. Yi, M. Kuschnerov, A. Poppe, C. Xie, and M. Peev, “A low-complexity heterodyne cv-qkd architecture,” in 2017 19th International Conference on Transparent Optical Networks (ICTON), (2017), pp. 1–4.

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Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
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D. B. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5(4), 041010 (2015).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
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D. B. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-referenced continuous-variable quantum key distribution protocol,” Phys. Rev. X 5(4), 041010 (2015).
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F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
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J. Pfeifle, V. Vujicic, R. T. Watts, P. C. Schindler, C. Weimann, R. Zhou, W. Freude, L. P. Barry, and C. Koos, “Flexible terabit/s Nyquist-WDM super-channels using a gain-switched comb source,” Opt. Express 23(2), 724–738 (2015).
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Wang, J. F.

Wang, T.

T. Wang, P. Huang, Y. Zhou, W. Liu, and G. Zeng, “Pilot-multiplexed continuous-variable quantum key distribution with a real local oscillator,” Phys. Rev. A 97(1), 012310 (2018).
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D. Huang, P. Huang, H. Li, T. Wang, Y. Zhou, and G. Zeng, “Field demonstration of a continuous-variable quantum key distribution network,” Opt. Lett. 41(15), 3511–3514 (2016).
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Y. Guo, Q. Liao, Y. Wang, D. Huang, P. Huang, and G. Zeng, “Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction,” Phys. Rev. A 95(3), 032304 (2017).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
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J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014).
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J. Pfeifle, V. Vujicic, R. T. Watts, P. C. Schindler, C. Weimann, R. Zhou, W. Freude, L. P. Barry, and C. Koos, “Flexible terabit/s Nyquist-WDM super-channels using a gain-switched comb source,” Opt. Express 23(2), 724–738 (2015).
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F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
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Wolf, S.

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Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
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H. H. Brunner, L. C. Comandar, F. Karinou, S. Bettelli, D. Hillerkuss, F. Fung, D. Wang, S. Mikroulis, Q. Yi, M. Kuschnerov, A. Poppe, C. Xie, and M. Peev, “A low-complexity heterodyne cv-qkd architecture,” in 2017 19th International Conference on Transparent Optical Networks (ICTON), (2017), pp. 1–4.

Xu, F.

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Yi, Q.

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W. Jiang, K. Okamoto, F. M. Soares, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “5 GHz channel spacing InP-based 32-channel Arrayed-Waveguide Grating,” in 2009 Conference on Optical Fiber Communication-incudes post deadline papers (2009), pp. 1–3.

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Yu, Y.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
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P. Huang, J. Huang, Z. Zhang, and G. Zeng, “Quantum key distribution using basis encoding of gaussian-modulated coherent states,” Phys. Rev. A 97(4), 042311 (2018).
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T. Wang, P. Huang, Y. Zhou, W. Liu, and G. Zeng, “Pilot-multiplexed continuous-variable quantum key distribution with a real local oscillator,” Phys. Rev. A 97(1), 012310 (2018).
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Y. Guo, Q. Liao, Y. Wang, D. Huang, P. Huang, and G. Zeng, “Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction,” Phys. Rev. A 95(3), 032304 (2017).
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D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
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D. Huang, P. Huang, H. Li, T. Wang, Y. Zhou, and G. Zeng, “Field demonstration of a continuous-variable quantum key distribution network,” Opt. Lett. 41(15), 3511–3514 (2016).
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D. Huang, D. Lin, C. Wang, W. Liu, S. Fang, J. Peng, P. Huang, and G. Zeng, “Continuous-variable quantum key distribution with 1 Mbps secure key rate,” Opt. Express 23(13), 17511–17519 (2015).
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H. Zhang, Y. Mao, D. Huang, J. Li, L. Zhang, and Y. Guo, “Security analysis of orthogonal-frequency-division-multiplexing–based continuous-variable quantum key distribution with imperfect modulation,” Phys. Rev. A 97(5), 052328 (2018).
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Zhang, J. Z.

Zhang, L.

H. Zhang, Y. Mao, D. Huang, J. Li, L. Zhang, and Y. Guo, “Security analysis of orthogonal-frequency-division-multiplexing–based continuous-variable quantum key distribution with imperfect modulation,” Phys. Rev. A 97(5), 052328 (2018).
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X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer in multiuser OFDM systems,” IEEE Trans. Wirel. Commun. 13(4), 2282–2294 (2014).
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Zhang, Y. C.

Z. Li, Y. C. Zhang, F. Xu, X. Peng, and H. Guo, “Continuous-variable measurement-device-independent quantum key distribution,” Phys. Rev. A 89(5), 052301 (2014).
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Zhang, Z.

P. Huang, J. Huang, Z. Zhang, and G. Zeng, “Quantum key distribution using basis encoding of gaussian-modulated coherent states,” Phys. Rev. A 97(4), 042311 (2018).
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Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
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Zhao, S.

K. Qu, S. Zhao, X. Li, Z. Zhu, D. Liang, and D. Liang, “Ultra-flat and broadband optical frequency comb generator via a single mach–zehnder modulator,” IEEE Photonics Technol. Lett. 29(2), 255–258 (2017).
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T. Wang, P. Huang, Y. Zhou, W. Liu, and G. Zeng, “Pilot-multiplexed continuous-variable quantum key distribution with a real local oscillator,” Phys. Rev. A 97(1), 012310 (2018).
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D. Huang, P. Huang, H. Li, T. Wang, Y. Zhou, and G. Zeng, “Field demonstration of a continuous-variable quantum key distribution network,” Opt. Lett. 41(15), 3511–3514 (2016).
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K. Qu, S. Zhao, X. Li, Z. Zhu, D. Liang, and D. Liang, “Ultra-flat and broadband optical frequency comb generator via a single mach–zehnder modulator,” IEEE Photonics Technol. Lett. 29(2), 255–258 (2017).
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Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
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Appl. Sci. (1)

L. Lundberg, M. Karlsson, A. Lorences-Riesgo, M. Mazur, J. Schröder, and P. Andrekson, “Frequency comb-based WDM transmission systems enabling joint signal processing,” Appl. Sci. 8(5), 718 (2018).
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IEEE Photonics Technol. Lett. (2)

T. Shao, E. Martin, P. M. Anandarajah, C. Browning, V. Vujicic, R. Llorente, and L. P. Barry, “Chromatic dispersion-induced optical phase decorrelation in a 60 GHz OFDM-RoF system,” IEEE Photonics Technol. Lett. 26(20), 2016–2019 (2014).
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K. Qu, S. Zhao, X. Li, Z. Zhu, D. Liang, and D. Liang, “Ultra-flat and broadband optical frequency comb generator via a single mach–zehnder modulator,” IEEE Photonics Technol. Lett. 29(2), 255–258 (2017).
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IEEE Trans. Wirel. Commun. (1)

X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer in multiuser OFDM systems,” IEEE Trans. Wirel. Commun. 13(4), 2282–2294 (2014).
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J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

J. Phys. B: At., Mol. Opt. Phys. (1)

S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At., Mol. Opt. Phys. 42(11), 114014 (2009).
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Nat. Photonics (4)

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator kerr frequency combs,” Nat. Photonics 8(5), 375–380 (2014).
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P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7(5), 378–381 (2013).
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H. K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
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F. Xu, M. Curty, B. Qi, L. Qian, and H. K. Lo, “Discrete and continuous variables for measurement-device-independent quantum cryptography,” Nat. Photonics 9(12), 772–773 (2015).
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Nature (1)

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003).
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R. Kumar, H. Qin, and R. Alléaume, “Coexistence of continuous variable qkd with intense dwdm classical channels,” New J. Phys. 17(4), 043027 (2015).
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Opt. Express (5)

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A. Leverrier and P. Grangier, “Simple proof that gaussian attacks are optimal among collective attacks against continuous-variable quantum key distribution with a gaussian modulation,” Phys. Rev. A 81(6), 062314 (2010).
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H. Zhang, Y. Mao, D. Huang, J. Li, L. Zhang, and Y. Guo, “Security analysis of orthogonal-frequency-division-multiplexing–based continuous-variable quantum key distribution with imperfect modulation,” Phys. Rev. A 97(5), 052328 (2018).
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A. Marie and R. Alléaume, “Self-coherent phase reference sharing for continuous-variable quantum key distribution,” Phys. Rev. A 95(1), 012316 (2017).
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Y. Guo, Q. Liao, Y. Wang, D. Huang, P. Huang, and G. Zeng, “Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction,” Phys. Rev. A 95(3), 032304 (2017).
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H. Qin, R. Kumar, and R. Alléaume, “Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution,” Phys. Rev. A 94(1), 012325 (2016).
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Z. Li, Y. C. Zhang, F. Xu, X. Peng, and H. Guo, “Continuous-variable measurement-device-independent quantum key distribution,” Phys. Rev. A 89(5), 052301 (2014).
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P. Huang, J. Huang, Z. Zhang, and G. Zeng, “Quantum key distribution using basis encoding of gaussian-modulated coherent states,” Phys. Rev. A 97(4), 042311 (2018).
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T. Wang, P. Huang, Y. Zhou, W. Liu, and G. Zeng, “Pilot-multiplexed continuous-variable quantum key distribution with a real local oscillator,” Phys. Rev. A 97(1), 012310 (2018).
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F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88(5), 057902 (2002).
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C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum cryptography without switching,” Phys. Rev. Lett. 93(17), 170504 (2004).
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R. Renner and J. I. Cirac, “de Finetti representation theorem for infinite-dimensional quantum systems and applications to quantum cryptography,” Phys. Rev. Lett. 102(11), 110504 (2009).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012).
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Phys. Rev. X (2)

B. Qi, P. Lougovski, R. Pooser, W. Grice, and M. Bobrek, “Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection,” Phys. Rev. X 5(4), 041009 (2015).
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Quantum Sci. Technol. (1)

Z. Zhang, C. Chen, Q. Zhuang, F. N. Wong, and J. H. Shapiro, “Experimental quantum key distribution at 1.3 gigabit-per-second secret-key rate over a 10 dB loss channel,” Quantum Sci. Technol. 3(2), 025007 (2018).
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Rev. Mod. Phys. (2)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
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Sci. Rep. (1)

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
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Theor. Comput. Sci. (1)

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Theor. Comput. Sci. 560, 7–11 (2014).
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M. Rückmann and C. G. Schaeffer, “1 GBaud heterodyne continuous variable quantum key distribution over 26 km fiber,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2019), p. FTh4A.4.

W. Jiang, K. Okamoto, F. M. Soares, F. Olsson, S. Lourdudoss, and S. J. B. Yoo, “5 GHz channel spacing InP-based 32-channel Arrayed-Waveguide Grating,” in 2009 Conference on Optical Fiber Communication-incudes post deadline papers (2009), pp. 1–3.

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Figures (8)

Fig. 1.
Fig. 1. The OFC-based multichannel parallel CVQKD system. Alice splits an OFC with center frequency of $f_0^s$ and repetition rate of $f_r^s$ , generating $N$ sub-channels followed by independent Gaussian modulation. Bob detects the received comb by a locally generated OFC with the same parameters as Alice. The outermost two sub-channels are selected as pilot channel for phase compensation.
Fig. 2.
Fig. 2. Schematic presentation of the phase compensation process. $f^s_{n_{min}}$ and $f^s_{n_{max}}$ are the outermost two comb lines of the signal comb that sent by Alice, $f^L_{n_{min}}$ and $f^L_{n_{max}}$ are the outermost two comb lines of the LO comb that generated by Bob.
Fig. 3.
Fig. 3. Temporal decorrelation between quantum and reference signals.
Fig. 4.
Fig. 4. (a) The the excess noise of each sub-channel in terms of the ratio $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ when $N=7$ . $k$ is the channel index. (b) The excess noise in each sub-channel when the extinction ratio $Re=\{40,50,60\}$ dB. The number of sub-channel is set as $N=15$ .
Fig. 5.
Fig. 5. (a) The secret key rate of the OFC-based CVQKD system against collective attacks in the asymptotic limit with the variable parameter ${\Delta }\nu =\{10^4,10^5,10^6,10^7\}$ Hz. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate and the repetition rate of the comb is set as $f_r^s=10$ GHz. (b) The secret key rate of the OFC-based CVQKD system against collective attacks in the asymptotic limit with the variable parameter $f_r^s=\{10,20,30,40\}$ GHz. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate and the linewidth is given by $\Delta \nu =10^5$ Hz.
Fig. 6.
Fig. 6. (a) The secret key bit rate of each sub-channel when $N=15$ . (b) The secret key bit rate for the first and last sub-channels when $N={7,15,35}$ . The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate. Other parameters are $f^s_r=10$ GHz, $\Delta \nu =100$ kHz, and $Re=60$ dB.
Fig. 7.
Fig. 7. (a) Secret key bit rate $R_{tot}$ as a function of modulation variance $V_a$ and transmission distance $L$ . The number of sub-channels are $N=7$ , $N=15$ , and $N=35$ , respectively. (b) The top view of the surface in Fig. 7(a) when $N=35$ . The black solid line represents the bound between positive and negative secret key rate. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is set as 500. Other parameters are $f^s_r=10$ GHz, $\Delta \nu =100$ kHz, and $Re=60$ dB.
Fig. 8.
Fig. 8. (a) The secret key bit rate as a function of the transmission distance. Curves from top to bottom are $N=35$ , $N=15$ , $N=7$ , and the single channel case, respectively. (b) The multichannel gain of the system. Curves from top to bottom are $N=35$ , $N=15$ , and $7$ , respectively. The value of $\langle \hat {N}^{Alice}_{n_{min}}\rangle /V_a$ is chosen to optimize the key rate. Other parameters are $f^s_r=10$ GHz, $\Delta \nu =100$ kHz, and $Re=60$ dB.

Tables (1)

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Table 1. Numerical Parameters

Equations (35)

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s ^ ( t ) = n = n m i n n m a x a ^ n exp { j [ φ ( t ) + 2 π f n s t ] } ,
V k Ra ( σ ) = V k σ 2 e V k 2 2 σ 2 ,
( X r B P r B ) = ( cos θ ^ r sin θ ^ r sin θ ^ r cos θ ^ r ) ( X r A P r A ) .
θ ^ r = tan 1 ( P r B X r A X r B P r A X r B X r A + P r B P r A ) .
θ ^ k = θ ^ n m i n + k n m i n n m a x n m i n ( θ ^ n m a x θ ^ n m i n ) .
( X ^ k B P ^ k B ) = ( cos θ ^ k sin θ ^ k sin θ ^ k cos θ ^ k ) ( X k A P k A ) .
Δ θ n m i n = θ ^ n m i n θ n m i n , Δ θ n m a x = θ ^ n m a x θ n m a x .
Δ θ k e r r o r = θ ^ k θ k = k n m i n n m a x n m i n ( Δ θ n m a x Δ θ n m i n ) + Δ θ n m i n .
V k e r r o r = V n m i n e r r o r + ( k n m i n n m a x n m i n ) 2 ( V n m a x e r r o r + V n m i n e r r o r ) ,
V n m a x e r r o r = χ n m a x + 1 | α n m a x | 2 , V n m i n e r r o r = χ n m i n + 1 | α n m i n | 2 ,
χ n m a x = χ n m i n = 2 η T η T + 2 v e l η T + ε ,
τ i j = D L Δ λ i j ,
θ ^ n m a x = φ n m a x B ( t ) φ n m a x A ( t A ) , θ ^ n m i n = φ n m i n B ( t ) φ n m i n A ( t A ) ,
θ ^ n m a x = φ n m a x B ( t τ k n m a x ) φ n m a x A ( t A ) , θ ^ n m i n = φ n m i n B ( t + τ k n m i n ) φ n m i n A ( t A ) .
φ n m a x B ( t ) = φ n m a x B ( t τ k n m a x ) + 2 π f n m a x τ k n m a x + N n m a x , φ n m i n B ( t + τ k n m i n ) = φ n m i n B ( t ) + 2 π f n m i n τ k n m i n + N n m i n .
θ ^ k = φ n m i n B ( t + τ k n m i n ) φ n m i n A ( t A ) + k n m i n n m a x n m i n [ φ n m a x B ( t τ k n m a x ) φ n m i n B ( t + τ k n m i n ) ] .
V k d i s p = < ( Δ φ ( τ k n m i n ) ) 2 > + ( k n m i n n m a x n m i n ) 2 ( < ( Δ φ ( τ k n m a x ) ) 2 > + < ( Δ φ ( τ k n m i n ) ) 2 > ) ,
< ( Δ φ ( τ k n m a x ) ) 2 > = 2 τ k n m a x t n m a x c , < ( Δ φ ( τ k n m i n ) ) 2 > = 2 τ k n m i n t n m i n c ,
t n m a x c = t n m i n c = 1 π Δ ν .
ε L E = 2 N ^ n m a x A l i c e R e + 2 N ^ n m i n A l i c e R e
ε k = V a ( V k e r r o r + V k d i s p ) + ε L E .
ε k a t t a c k = V a ( V ~ k e r r o r + V k d i s p ) + ε L E + ε E v e ,
V a V ~ k e r r o r + ε E v e V a V k e r r o r .
V a r ( θ ^ k θ k a t t a c k ) < V a r ( θ ^ k θ k ) .
V a r ( θ ^ k a t t a c k θ k ) < V a r ( θ ^ k θ k ) .
R t o t = k R k ,
K k = β I A B χ B E .
G = R t o t R s ,
I A B = 1 2 log 2 V B V B | A = 1 2 log 2 V + χ t o l 1 + χ t o l ,
χ B E = S ( ρ E ) d m B p ( m B ) S ( ρ E m B ) ,
χ B E = i = 1 2 G ( λ i 1 2 ) i = 3 5 G ( λ i 1 2 ) ,
λ 1 , 2 2 = 1 2 ( A ± A 2 4 B ) ,
A = V 2 + T 2 ( V + χ l i n e ) 2 + 2 T ( 1 V 2 ) , B = T 2 ( 1 + V χ l i n e ) 2 ,
λ 3 , 4 2 = 1 2 ( C ± C 2 4 D ) ,
C = A χ h o m + V B + T ( V + χ l i n e ) T ( V + χ t o t ) , D = B V + B χ h o m T ( V + χ t o t ) ,

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