Performance analysis for OFDM-based multi-carrier continuous-variable quantum key distribution with an arbitrary modulation protocol

Heng Wang; Yan Pan; Yun Shao; Yaodi Pi; Ting Ye; Yang Li; Tao Zhang; Jinlu Liu; Jie Yang; Li Ma; Wei Huang; Wei Huang; Bingjie Xu

doi:10.1364/OE.482136

1. Introduction

Quantum key distribution (QKD) allows to distribute an information-theoretically secure key between two legal communication parties through an insecure channel based on the fundamental principles of quantum mechanics [1–4]. Compared with the intensively developed discrete-variable QKD (DV-QKD) based on single-photon detection, continuous-variable QKD (CV-QKD) based on coherent detection is compatible with commercial off-the-shelf components and has the potential advantages of high secret key rate (SKR) in metropolitan area [5–9].

Currently, single-frequency optical carrier scheme is regularly used in most existing CV-QKD setups, and it has been classified into two schemes according to the modulation method of the coherent state, i.e., Gaussian modulation coherent state (GMCS) [10–12] and discrete modulation coherent state (DMCS) [13–16]. For the single-carrier GMCS CVQKD, the SKR has achieved a sub-Mbps level over 100 km [17]. Meanwhile, the SKR of the single-carrier DMCS CVQKD has reached a sub-Gbps level within metropolitan area [18,19]. Specifically, the SKRs of these single-carrier schemes can be improved by increasing the system repetition rate and optimizing the excess noise. However, the repetition rate of the state-of-art CV-QKD system has reached 5 GHz [18], where the technical challenge lies in that it requires faster data acquisition cards with high linearity, wider bandwidth homodyne detectors with low noise and faster post-processing with high complexity [20–24]. Meanwhile, the excess noise of CV-QKD system has been controlled to a reasonably low level at the order of 0.001∼0.01 [18,25,26]. To further improve the SKR, the multi-carrier CV-QKD scheme has been proposed to distribute multiplexing independent secret keys encoded on N subcarriers within a single fiber channel [27,28]. At present, two multi-carrier CV-QKD schemes have been proposed. One needs multiple CV-QKD transmitters, which is basically the same as deploying multiple CV-QKD systems [29]. The other is based on one transmitter with N independent radio-frequency (RF) oscillators while it is difficult to realize due to the complicated electrical structure [30]. In fact, the multi-carrier CV-QKD scheme can be simply realized with one transmitter and one receiver by employing the orthogonal-frequency-division-multiplexing (OFDM) method [31–33], which is efficient and economical for practical deployment. Moreover, the SKR can be intuitively improved by increasing the carrier number N for an OFDM-based multi-carrier CV-QKD. However, extra excess noise will be introduced in the imperfect multi-carrier quantum state preparation process, which will deteriorate the system performance for larger N. So, to effectively improve the system performance, there is a trade-off between the carrier number N and the extra excess noise. The extra excess noise is mainly the modulation noise that originated from the IQ imbalance and intermodulation distortion. In [31], the IQ imbalance noise has been modeled and analyzed for the QPSK multi-carrier CV-QKD scheme. Nevertheless, there is still a lack of systematic modulation noise model for multi-carrier CV-QKD with arbitrary modulation protocol (e.g. QPSK, 256QAM or Gaussian protocol), which is seriously needed to evaluate the performance of the scheme.

In this paper, a systematic modulation noise model is proposed for the OFDM-based multi-carrier CV-QKD with arbitrary modulation protocol. Based on the proposed modulation noise model, the SKRs of the OFDM-based multi-carrier CV-QKD system with QPSK, 256QAM and Gaussian modulation protocols are quantitatively evaluated by combining the security analysis method of the single-carrier CV-QKD. Specifically, the modulation noise from IQ imbalance and third-order intermodulation distortion are quantitatively modeled and analyzed, which limits the performance of the system with N subcarriers. Under practical system parameters, the simulation results show that the OFDM-based multi-carrier CV-QKD system can achieve much higher asymptotic SKR by increasing N even with imperfect modulation compared with the single-carrier CV-QKD scheme. Moreover, a maximum total SKR for multi-carrier CV-QKD can be obtained by carefully optimizing N. Therefore, our work offers a promising way to further improve the SKR performance of quantum secure network.

2. Description of the multi-carrier CV-QKD scheme

The working process of the OFDM-based multi-carrier CV-QKD scheme can be described as follows. (i) Quantum state preparation: Alice modulates N independent random keys on a N mode coherent state as based on QPSK, 256QAM or Gaussian modulation CV-QKD protocol, and sends the prepared quantum state to Bob through the quantum channel. (ii) Quantum state measurement: Bob measures the received quantum state with coherent receiver (e.g. homodyne or heterodyne detection), and gets the measurement results $\vec{y} = \{ {y_k}|k = 1,2, \ldots ,N\} $ based on digital demodulation. Alice and Bob repeat steps (i) and (ii) for n times. (iii) Post-processing: Alice and Bob perform sifting (if only Bob uses homodyne detection), parameter estimation, reverse reconciliation and privacy amplification independently for N modes of encoded quantum states. The total asymptotic SKR between Alice and Bob with N subcarriers can be written as:

(1)$$R = \sum\limits_{k = 1}^N {{R_k}} ,\quad {R_k} = {\beta _k}I({{A_k}:{B_k}} )- \chi ({{B_k}:E} )$$

where R_k is the asymptotic SKR of k-th mode, I(A_k:B_k) is the Shannon mutual entropy between Alice and Bob for k-th mode, χ(B_k:E) is the quantum mutual entropy between Bob and Eve for k-th mode, and ${\beta _k}$ is the reconciliation efficiency for k-th mode.

In the following, we describe detailly the OFDM-based multi-carrier CV-QKD scheme with local local oscillator (LLO), as shown in Fig. 1. At Alice’s site, a continuous optical carrier with central frequency f_A that generated from Alice laser is split into two branches. In the upper branch, the optical carrier is modulated in an IQ modulator by OFDM signal I_s and Q_s generated from an OFDM generator. The processing routine of the OFDM generator is demonstrated in Fig. 2. Firstly, a serial of true random bit stream with high bit rate originated from quantum random number generator (QRNG) is converted to N independent parallel bit streams by a serial-to-parallel conversion (S/P). Secondly, the N parallel bit streams are mapped with arbitrary modulation distribution based on the chosen CV-QKD protocol, such as QPSK, 256QAM or Gaussian distribution. After inverse fast Fourier transform (IFFT), the processed parallel data is converted to the desired OFDM signal by a parallel-to-serial conversion (P/S), given by

(2a)$${I_s} = \sum\limits_{k = 1}^N {{I_k}\cos ({2\pi {f_k}t} )- } \sum\limits_{k = 1}^N {{Q_k}\sin ({2\pi {f_k}t} )}$$

(2b)$${Q_s} = \sum\limits_{k = 1}^N {{I_k}\sin ({2\pi {f_k}t} )+ } \sum\limits_{k = 1}^N {{Q_k}\cos ({2\pi {f_k}t} )}$$

where N is the total number of the generated subcarriers, f_k= kΔf corresponds to the frequency of k-th subcarrier and Δf denotes the frequency spacing between N subcarriers. It is noteworthy from Fig. 2 that the cyclic prefix (CP) and training sequence (TS) are appended into the OFDM signal in processing routine for avoiding mutual crosstalk between subcarriers in fiber channel and achieving the data-aided time domain equalization at Bob’s site.

Fig. 1. Schematic diagram of the OFDM-based multi-carrier CV-QKD with local local oscillator. LD: laser diode; OFDM: orthogonal frequency division multiplexing; MZM: Mach-Zehnder modulator; PBS: polarization beam splitter; VOA, variable optical attenuator; PBC: polarization beam combiner; SMF: single mode fiber; PSA: polarization synthesis analyzer; OC: optical coupler; BPD: balanced photo-detector; DSP: digital signal processing

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Fig. 2. The data processing of the OFDM generator. S/P: serial-to-parallel conversion; TS: training sequence; IFFT: inverse fast Fourier transform; P/S: parallel-to-serial conversion; CP: cyclic prefix; DAC: digital-to-analog converter.

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In the upper branch at Alice’s site in Fig. 1, the OFDM-based optical signal after IQ modulation is attenuated by a variable optical attenuator (VOA) to generate the encoded multi-carrier quantum state

(3)$$|{{X_{sig}} + j{P_{sig}}} \rangle ={\otimes} _{k = 1}^N|{{X_k} + j{P_k} + \Delta {X_k} + j\Delta {P_k}} \rangle$$

where X_k and P_k are the expected modulated quadrature components for the k-th subcarrier. ΔX_k and ΔP_k are the extra modulation noise on X_k and P_k, respectively, which mainly come from IQ imbalance and intermodulation distortion in the multi-carrier IQ modulation process. The detailed theoretical model of the modulation process is shown in next section. In the lower branch, the optical carrier is sent into a frequency shifter (e.g. MZM with carrier suppression double sideband modulation) with f_s for generating the desired pilot tone with frequency f_A-f_s [11]. The generated pilot tone is used for eliminating the fast-drift phase noise in the LLO-CV-QKD system. Significantly, the intense pilot tone and the weak quantum signal are respectively generated in two different optical paths, which is beneficial to improve the preparation accuracy of the multi-carrier quantum state in the case of finite DAC quantization bits and modulation extinction ratio (ER). Finally, the multi-carrier quantum signal and the pilot tone are co-transmitted into a single mode fiber (SMF) channel with different frequency bands and orthogonal polarization states for avoiding the crosstalk between them.

At Bob’s site, the received quantum signal and pilot tone are separated by a polarization beam splitter (PBS). In order to separate the multi-carrier quantum signal and pilot tone efficiently, a polarization synthesis analyzer (PSA) is used for correcting the polarization deterioration resulted from the fiber channel disturbance. The separated quantum signal and pilot tone are independently coupled into two balanced photo-detectors (BPDs) with their corresponding LLO signals of frequency f_B supplied by the Bob laser. The detected quantum signal and pilot tone are analog-to-digital converted for the subsequent DSP, respectively. As is shown in Fig. 3, the frequency f_s-Δf_AB is firstly efficiently estimated from the pilot tone, with which the central frequency Δf_AB of the OFDM-based quantum signal can be obtained. Secondly, the OFDM-based quantum signal y_sig and the pilot tone y_pilot are bandpass filtered, down-converted and matched filtered for recovering their quadrature components. Thirdly, the fast-drift phase noise of the OFDM-based quantum signal can be compensated based on the pilot-assisted channel equalization method, which is recently applied in optical fiber LLO-CV-QKD system [34]. After removing CP, the serial OFDM-based quantum signal is converted to N parallel signals and then transformed to N independent subcarriers by FFT. Subsequently, the phase noises of N independent subcarriers are further compensated based on data-aided time domain equalization method, respectively. Finally, a high-efficient post-processing setup for the multi-carrier CV-QKD is required to extract N independent secure keys. For distilling the final key based on the security analysis method of the single-carrier CV-QKD, the excess noise of the designed multi-carrier CV-QKD system, mainly including laser intensity noise, DAC quantization noise, modulation noise, fiber channel crosstalk noise, and phase noise, should be controlled to be a reasonably low level. Moreover, the shot noise for each subcarrier should be calibrated in real time at Bob’s site [35].

Fig. 3. The DSP routine of the OFDM-based multi-carrier LLO-CV-QKD scheme. CP: cyclic prefix; S/P: serial-to- parallel conversion; FFT: fast Fourier transform.

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3. Modulation noise model of the multi-carrier CV-QKD

In the OFDM-based multi-carrier modulation process, the preparation accuracy of the multi-carrier quantum state will suffer from the IQ imbalance and intermodulation distortion. Therefore, a detailed modulation noise model of the multi-carrier CV-QKD scheme should be derived to effectively evaluate the SKR.

The OFDM-based multi-carrier quantum signal after the IQ modulation can be written as:

(4)$${E_{sig}}(t )= 2{A_{sig}}{e^{j2\pi {f_A}t}}\left\{ {{G_1}\sin \left[ {\sum\limits_{k = 1}^N {{\gamma_k}\cos ({2\pi {f_k}t + {\varphi_k}} )} } \right] + j{G_2}\sin \left[ {\sum\limits_{k = 1}^N {{\gamma_k}\sin ({2\pi {f_k}t + {\varphi_k}} )} } \right]} \right\}$$

with

(5a)$${\gamma _k} = {\mu _k}\sqrt {{I_k}^2 + {Q_k}^2}$$

(5b)$$\cos {\varphi _k} = \frac{{{I_k}}}{{\sqrt {{I_k}^2 + {Q_k}^2} }}$$

where the IQ imbalance factor G₁= [1 + κ_kexp(jθ_k)]/2 and G₂= [1 + κ_kexp(-jθ_k)]/2, and κ_k and θ_k are the gain imbalance and quadrature skew of the k-th subcarrier modulation, respectively. µ_k= V_kπ/V_π/2 is the modulation index of the k-th subcarrier. V_k and V_π are the driving amplitude and the half-wave voltage of the employed I/Q modulator, respectively, and they should be chosen as a reasonably lower value for guaranteeing the small signal condition and the multi-carrier quantum state with enough signal-to-noise ratio (SNR) in practical experiment. A_sig is the amplitude of the quantum signal. Significantly, under IQ balance (κ_k= 1 and θ_k= 0) and no intermodulation distortion, Eq. (4) can be simplified as:

(6)$${E_{sig}}(t )= 2{A_{sig}}{e^{j2\pi {f_A}t}}\sum\limits_{k = 1}^N {{\gamma _k}{e^{j({2\pi {f_k}t + {\varphi_k}} )}}} = 2{A_{sig}}{e^{j2\pi {f_A}t + j2\pi {f_k}t}} \otimes _{k = 1}^N|{{I_k} + j{Q_k}} \rangle$$

Comparing Eq. (3) and Eq. (6), one can see that the expected modulated quadrature X_k and P_k of the k-th subcarrier are determined by the random variables I_k and Q_k with arbitrary modulation distribution.

In order to derive modulation noise model for ΔX_k and ΔP_k shown in Eq. (3), we apply the Jacobi-Anger expansion [36] into Eq. (4), which can be expanded with the pth-order Bessel function of the first kind J_p(·) (p = 0, ± 1, ± 2, ± 3, …) as follows:

(7)$${E_{sig}}(t )= 2{A_{sig}}{e^{j2\pi {f_A}t}}\left\{ \begin{array}{l} {G_1}\sum\limits_{{p_1} ={-} \infty }^{ + \infty } {{J_{{p_1}}}({{\gamma_1}} )\cdots \sum\limits_{{p_k} ={-} \infty }^{ + \infty } {{J_{{p_k}}}({{\gamma_k}} )} } \cdots \sum\limits_{{p_N} ={-} \infty }^{ + \infty } {({{\gamma_{_N}}} ){J_{{p_N}}}} \cdot \\ \sin \left[ \begin{array}{l} \left( {2\pi {f_1}t{p_1} + {p_1}{\varphi_1} + {p_1}\frac{\pi }{2}} \right) +\cdots{+} \left( {2\pi {f_k}t{p_k} + {p_k}{\varphi_{_k}} + {p_k}\frac{\pi }{2}} \right)\\ +\cdots{+} \left( {2\pi {f_N}t{p_N} + {p_N}{\varphi_{_N}} + {p_N}\frac{\pi }{2}} \right) \end{array} \right]\\ + j{G_2}\sum\limits_{{p_1} ={-} \infty }^{ + \infty } {{J_{{p_1}}}({{\gamma_1}} )\sum\limits_{{p_k} ={-} \infty }^{ + \infty } {{J_{{p_k}}}({{\gamma_2}} )} } \cdots \sum\limits_{{p_N} ={-} \infty }^{ + \infty } {{J_{{p_N}}}({{\gamma_{_N}}} )} \cdot \\ \sin \left[ \begin{array}{l} ({2\pi {f_1}t{p_1} + {p_1}{\varphi_1}} )+\cdots{+} ({2\pi {f_k}t{p_k} + {p_k}{\varphi_{_k}}} )\\ +\cdots{+} ({2\pi {f_N}t{p_N} + {p_N}{\varphi_{_N}}} )\end{array} \right] \end{array} \right\}$$

From Eq. (7), we can quantify the frequency component at f₀+ f_k for expressing the k-th subcarrier in consideration of the IQ imbalance and the third-order intermodulation distortion, which is given by

(8)$$\begin{array}{l} {E_{sig(k)}}(t )= 2{A_{sig}}{e^{j\pi {f_A}t}} \cdot \\ \left\{ \begin{array}{l} {J_1}({{\gamma_k}} )[{({{G_1} + {G_2}} ){e^{j({2\pi {f_k}t + {\varphi_k}} )}} + ({{G_1} - {G_2}} ){e^{ - j({2\pi {f_k}t + {\varphi_k}} )}}} ]\\ - \sum\limits_{,n = 1\atop { \ne n \ne k\atop \scriptstyle2m + n = k}}^N {{J_2}({{\gamma_m}} ){J_1}({{\gamma_n}} )[{({{G_1} + {G_2}} ){e^{ - j({2\pi {f_k}t + 2{\varphi_m} + {\varphi_n}} )}} + ({{G_1} - {G_2}} ){e^{j({2\pi {f_k}t + 2{\varphi_m} + {\varphi_n}} )}}} ]} \\ - \sum\limits_{,s = 1\atop { \ne n \ne k\atop \scriptstyle2m - n = k}}^N {{J_2}({{\gamma_m}} ){J_1}({{\gamma_n}} )[{({{G_1} + {G_2}} ){e^{j({2\pi {f_k}t + 2{\varphi_m} - {\varphi_n}} )}} + ({{G_1} - {G_2}} ){e^{ - j({2\pi {f_k}t + 2{\varphi_m} - {\varphi_n}} )}}} ]} \\ - \sum\limits_{,s,l = 1\atop { \ne n \ne l \ne k\atop + n + l = k}}^N {{J_1}({{\gamma_m}} ){J_1}({{\gamma_n}} ){J_1}({{\gamma_l}} )[{({{G_1} + {G_2}} ){e^{ - j({2\pi {f_k}t + {\varphi_m} + {\varphi_n} + {\varphi_l}} )}} + ({{G_1} - {G_2}} ){e^{j({2\pi {f_k}t + {\varphi_m} + {\varphi_n} + {\varphi_l}} )}}} ]} \\ - \sum\limits_{,n,l = 1\atop { \ne n \ne l \ne k\atop + n - l = k}}^N {{J_1}({{\gamma_m}} ){J_1}({{\gamma_n}} ){J_1}({{\gamma_l}} )[{({{G_1} + {G_2}} ){e^{j({2\pi {f_k}t + {\varphi_m} + {\varphi_n} - {\varphi_l}} )}} + ({{G_1} - {G_2}} ){e^{ - j({2\pi {f_k}t + {\varphi_m} + {\varphi_n} - {\varphi_l}} )}}} ]} \\ - \sum\limits_{,n,l = 1\atop { \ne n \ne l \ne k\atop - n - l = k}}^N {{J_1}({{\gamma_m}} ){J_1}({{\gamma_n}} ){J_1}({{\gamma_l}} )[{({{G_1} + {G_2}} ){e^{ - j({2\pi {f_k}t + {\varphi_m} - {\varphi_n} - {\varphi_l}} )}} + ({{G_1} - {G_2}} ){e^{j({2\pi {f_k}t + {\varphi_m} - {\varphi_n} - {\varphi_l}} )}}} ]} \end{array} \right\} \end{array}$$

Under small signal condition of J₁(x)≈x/2 and J₂(x)≈x²/4, the quadrature X_sig_(k) of the k-th subcarrier can be obtained from Eq. (8) and expressed as:

(9)$$\begin{array}{l} {X_{sig(k )}} = {X_k} + \Delta {X_k} = 2{A_{sig}}\textrm{real} ({{\gamma_k}{e^{j{\varphi_k}}}} )\\ + 2{A_{sig}}real \left\{ \begin{array}{l} \frac{{{\gamma_k}}}{2}[{({{G_1} + {G_2} - 2} ){e^{j{\varphi_k}}} + ({{G_1} - {G_2}} ){e^{ - j{\varphi_k}}}} ]\\ - \frac{{{M_1}({N,k} )\gamma_m^2{\gamma_n}}}{8}[{({{G_1} + {G_2}} ){e^{ - j({2{\varphi_m} + {\varphi_n}} )}} + ({{G_1} - {G_2}} ){e^{j({2{\varphi_m} + {\varphi_n}} )}}} ]\\ - \frac{{{M_2}({N,k} )\gamma_m^2{\gamma_n}}}{8}[{({{G_1} + {G_2}} ){e^{j({2{\varphi_m} - {\varphi_n}} )}} + ({{G_1} - {G_2}} ){e^{ - j({2{\varphi_m} - {\varphi_n}} )}}} ]\\ - \frac{{{W_1}({N,k} ){\gamma_m}{\gamma_n}{\gamma_l}}}{8}[{({{G_1} + {G_2}} ){e^{ - j({{\varphi_m} + {\varphi_n} + {\varphi_l}} )}} + ({{G_1} - {G_2}} ){e^{j({{\varphi_m} + {\varphi_n} + {\varphi_l}} )}}} ]\\ - \frac{{{W_2}({N,k} ){\gamma_m}{\gamma_n}{\gamma_l}}}{8}[{({{G_1} + {G_2}} ){e^{j({{\varphi_m} + {\varphi_n} - {\varphi_l}} )}} + ({{G_1} - {G_2}} ){e^{ - j({{\varphi_m} + {\varphi_n} - {\varphi_l}} )}}} ]\\ - \frac{{{W_3}({N,k} ){\gamma_m}{\gamma_n}{\gamma_l}}}{8}[{({{G_1} + {G_2}} ){e^{ - j({{\varphi_m} - {\varphi_n} - {\varphi_l}} )}} + ({{G_1} - {G_2}} ){e^{j({{\varphi_m} - {\varphi_n} - {\varphi_l}} )}}} ]\end{array} \right\} \end{array}$$

where M₁(N, k) and M₂(N, k) are the combinatorial numbers of the condition 2m + n = k and 2m-n = k, respectively. W₁(N, k), W₂(N, k) and W₃(N, k) are the combinatorial numbers of the condition m + n + l = k, m + n-l = k and m-n-l = k, respectively. In Eq. (9), X_k is the k-th subcarrier component under the condition of IQ balance (κ_k= 1 and θ_k= 0) and no intermodulation distortion [M₁(N, k)=M₂(N, k)=W₁(N, k)=W₂(N, k)=W₃(N, k) = 0], which can be expressed as

(10)$${X_k} = 2{A_{sig}}{\mu _k}{I_k}$$

In Eq. (9), ΔX_k is the extra modulation noise introduced from the IQ imbalance and third-order intermodulation distortion, which includes three parts

(11)$$\Delta {X_k} = \Delta {X_{k1}} + \Delta {X_{k2}} + \Delta {X_{k3}}$$

with

(12a)$$\Delta {X_{k1}} = {A_{sig}}{\mu _k}[{({{\kappa_k}\cos {\theta_k} - 1} ){I_k} + ({{\kappa_k}\sin {\theta_k}} ){Q_k}} ]$$

(12b)$$\Delta {X_{k2}} = \frac{1}{4}{A_{sig}}\mu _m^2{\mu _n}\left\{ \begin{array}{l} ({1 + {\kappa_k}\cos {\theta_k}} )\left\{ \begin{array}{l} [{{M_1}({N,k} )+ {M_2}({N,k} )} ]Q_m^2{I_n}\\ + 2[{{M_1}({N,k} )- {M_2}({N,k} )} ]{I_m}{Q_m}{Q_n}\\ - [{{M_1}({N,k} )+ {M_2}({N,k} )} ]I_m^2{I_n} \end{array} \right\}\\ + ({{\kappa_k}\sin {\theta_k}} )\left[ \begin{array}{l} [{{M_1}({N,k} )+ {M_2}({N,k} )} ]I_m^2{Q_n}\\ + 2[{{M_1}({N,k} )- {M_2}({N,k} )} ]{I_m}{Q_m}{I_n}\\ - [{{M_1}({N,k} )+ {M_2}({N,k} )} ]Q_m^2{Q_n} \end{array} \right] \end{array} \right\}$$

(12c)$$\Delta {X_{k3}} = \frac{1}{4}{A_{sig}}{\mu _m}{\mu _n}{\mu _l}\left\{ \begin{array}{l} ({1 + {\kappa_k}\cos {\theta_k}} )\left\{ \begin{array}{l} [{{W_1}({N,k} )- {W_2}({N,k} )+ {W_3}({N,k} )} ]{I_m}{Q_n}{Q_l}\\ + [{{W_1}({N,k} )- {W_2}({N,k} )- {W_3}({N,k} )} ]{Q_m}{I_n}{Q_l}\\ + [{{W_1}({N,k} )+ {W_2}({N,k} )- {W_3}({N,k} )} ]{Q_m}{Q_n}{I_l}\\ - [{{W_1}({N,k} )+ {W_2}({N,k} )+ {W_3}({N,k} )} ]{I_m}{I_n}{I_l} \end{array} \right\}\\ + ({{\kappa_k}\sin {\theta_k}} )\left\{ \begin{array}{l} [{{W_1}({N,k} )+ {W_2}({N,k} )- {W_3}({N,k} )} ]{I_m}{I_n}{Q_l}\\ + [{{W_1}({N,k} )- {W_2}({N,k} )- {W_3}({N,k} )} ]{I_m}{Q_n}{I_l}\\ + [{{W_1}({N,k} )- {W_2}({N,k} )+ {W_3}({N,k} )} ]{Q_m}{I_n}{I_l}\\ - [{{W_1}({N,k} )+ {W_2}({N,k} )+ {W_3}({N,k} )} ]{Q_m}{Q_n}{Q_l} \end{array} \right\} \end{array} \right\}$$

where ΔX_k₁ is modulation noise resulted from the IQ imbalance. The modulation noise ΔX_k₂ and ΔX_k₃ are originated from the third-order intermodulation with the frequency relationships (2f_m-f_n= f_k and 2f_m+ f_n= f_k) and (f_m+ f_n+ f_l= f_k, f_m+ f_n-f_l= f_k and f_m-f_n-f_l= f_k), respectively.

It is obvious that the quadrature I_g and Q_g (g = k, m, n, l) of the g-th subcarrier are mutually random and independent. One can define the variance 〈I_g²〉=〈Q_g²〉=σ₁² and 〈I_g⁴〉=〈Q_g⁴〉=σ₂². According to Eq. (11), the variances of ΔX_kcan be expressed as

(13)$$\left\langle {\Delta X_k^2} \right\rangle = \left\langle {\Delta X_{k1}^2} \right\rangle + \left\langle {\Delta X_{k2}^2} \right\rangle + \left\langle {\Delta X_{k3}^2} \right\rangle$$

with

(14a)$$\left\langle {\Delta X_{k1}^2} \right\rangle = A_{sig}^2\mu _k^2\sigma _1^2({\kappa_k^2 + 1 - 2{\kappa_k}\cos {\theta_k}} )$$

(14b)$$\left\langle {\Delta X_{k2}^2} \right\rangle = \frac{{A_{sig}^2\mu _m^4\mu _n^2}}{8}({1 + 2{\kappa_k}\cos {\theta_k} + {\kappa_k}^2} )\left\{ \begin{array}{l} {[{{M_1}({N,k} )+ {M_2}({N,k} )} ]^2}\sigma_2^2\sigma_1^2\\ + 2{[{{M_1}({N,k} )- {M_2}({N,k} )} ]^2}\sigma_1^6 \end{array} \right\}$$

(14c)$$\left\langle {\Delta X_{k3}^2} \right\rangle = \frac{{A_{sig}^2\mu _m^2\mu _n^2\mu _l^2}}{4}({1 + 2{\kappa_k}\cos {\theta_k} + \kappa_k^2} )[{W_1^2({N,k} )+ W_2^2({N,k} )+ W_3^2({N,k} )} ]\sigma _1^6$$

At the same time, µ_k, µ_m, µ_n and µ_l are the modulation indexes of the k, m, n, l-th subcarrier modulation, respectively, which are approximately equal to each other in the multi-carrier modulation process. So, we can use µ_k to replace µ_m, µ_n and µ_l in Eqs. (14b) and (14c). Moreover, according to 〈I_k²〉=〈Q_k²〉=σ₁² and the variance of Eq. (10), the expected modulation variance of the k-th subcarrier can be expressed as

(15)$${V_A} = \left\langle {X_k^2} \right\rangle = 4A_{sig}^2\mu _k^2\left\langle {I_k^2} \right\rangle = 4A_{sig}^2\mu _k^2\sigma _1^2$$

where the modulation variance V_A is normalized in shot noise unit (SNU). From Eq. (15), it is obvious that the modulation index µ_k is directly related to the modulation variance V_A of the multi-carrier quantum state. Based on Eqs. (13), (14) and (15), we can simplify the modulation noise of the k-th subcarrier with the modulation variance V_A, given by

(16)$$\begin{array}{l} {\varepsilon _{mod}}(k) = \left\langle {\Delta X_k^2} \right\rangle = \frac{{{V_A}}}{4}({\kappa_k^2 + 1 - 2{\kappa_k}\cos {\theta_k}} )\\ + \frac{{{V_A}\mu _k^4}}{{32}}({1 + 2{\kappa_k}\cos {\theta_k} + {\kappa_k}^2} )\left\{ \begin{array}{l} {[{{M_1}({N,k} )+ {M_2}({N,k} )} ]^2}\sigma_2^2\\ + 2\left\{ \begin{array}{l} {[{{M_1}({N,k} )- {M_2}({N,k} )} ]^2}\\ \textrm{ + }W_1^2({N,k} )+ W_2^2({N,k} )+ W_3^2({N,k} )\end{array} \right\}\sigma_1^4 \end{array} \right\} \end{array}$$

One can see from Eq. (16) that the modulation noise of the k-th subcarrier in the multi-carrier CV-QKD system is related with the modulation index µ_k, the IQ imbalance factors (κ_k and θ_k) and the third-order intermodulation factors [M₁(N, k), M₂(N, k), W₁(N, k), W₂(N, k) and W₃(N, k)]. Moreover, the modulation noise is also determined by the variance σ₁² and σ₂² related with different modulation distributions (e.g. QPSK, 256QAM or Gaussian modulation) in the quantum state preparation process. One can see from Eq. (16) that the modulation noise of the single-carrier CVQKD can be also obtained, which mainly comes from the IQ imbalance noise where it has no the intermodulation distortion, i.e., [M₁(N, k)=M₂(N, k)=W₁(N, k)=W₂(N, k)=W₃(N, k) = 0].

4. Performance evaluation of multi-carrier CV-QKD

It is well known that the modulation noise is estimated as a part of excess noise at Alice’s site [37]. In this case, the extra modulation noise ε_mod(k) of the k-th subcarrier in the multi-carrier CV-QKD will cause an increment of the excess noise compared to a single-carrier CV-QKD, and the total excess noise ε_multi(k) of the k-th subcarrier can be expressed as

(17)$${\varepsilon _{multi}}(k )= {\varepsilon _{mod}}(k )+ {\varepsilon _{single}}$$

where ε_single is the excess noise of a single-carrier CV-QKD scheme without the modulation noise. Based on the asymptotic security analysis method of the single-carrier CV-QKD scheme, the asymptotic SKR performances of the multi-carrier CV-QKD with QPSK, 256 QAM and Gaussian modulation can be evaluated with modulation noise ε_mod(k) simulated from the proposed modulation noise model. It is worth noting that the asymptotic SKR that is the maximum achievable SKR can be directly reflect the performance of the designed CV-QKD system.

4.1 QPSK modulation protocol

For the QPSK modulation multi-carrier CV-QKD scheme, the variance σ₁² and σ₂² are determined to be 1 because the random variables I_g and Q_g (g = k, m, n, l) follow mutually independent binomial distribution (-1, 1). Under the total carrier number N = 10, 25, 40 and 60, the gain imbalance κ_k= 0.98 and the quadrature skew θ_k= π/50, the modulation noise ratio ${\varepsilon _{mod}}(k )/A_{sig}^2$ of the k-th subcarrier at different modulation index µ_k are calculated according to Eq. (16) and shown in Fig. 4. It is seen from Fig. 4 that ${\varepsilon _{mod}}(k )/A_{sig}^2$ of the k-th subcarrier is an increasing function of modulation index µ_k. Therefore, it is better to choose a lower modulation index µ_kto reduce the modulation noise especially for the larger total carrier number N in practical experiment, e.g. the modulation noise ratio for N = 60 is close to 0 when µ_k= 0.01.

Fig. 4. Simulated ε_mod(k)/A²_sig of the k-th subcarrier at different modulation index µ_k with N = 10, 25, 40 and 60, respectively.

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With system parameters µ_k= 0.01, κ_k= 0.98 and θ_k= π/50, the worst modulation noise ε_mod(k_worst) of the k_worst-th subcarrier in different N carriers is numerically obtained as shown in Fig. 5. According to our previous QPSK single-carrier CV-QKD experimental results [18], the single-carrier excess noise ε_single can be controlled to be about 0.007 shot noise unit (SNU) with optimized modulation variance V_A= 0.45 SNU. Suppose the k-th subcarrier share the same excess noise performance as the single-carrier CV-QKD scheme, except with the extra modulation noise induced in the multi-carrier quantum state prepration process. Then, the excess noise ε_multi(k_worst) of the k_worst-th subcarrier in the QPSK multi-carrier CV-QKD scheme can be calculated based on Eq. (16), which is shown in Fig. 5. The excess noise ε_multi(k_worst) of the k_worst-th subcarrier gradually increases with the total carrier number N, which should be set to below 60 for avoiding null key rate at 25 km transmission distance.

Fig. 5. Calculated noise of the k_worst-th subcarrier at different total carrier number N in the QPSK multi-carrier CV-QKD. The blue dash dot line denotes the worst modulation noise ε_mod(k_worst), the magenta solid line corresponds to the excess noise ε_multi(k_worst) and the red dash line represents the null key rate threshold at 25 km.

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Under the modulation variance V_A= 0.45 SNU, the reconciliation efficiency β_k= 95% (k = 1, 2, …, N), and the excess noise ε_multi(k_worst) calculated by Eq. (16), the single SKR ${R_{{k_{\textrm{worst}}}}}$ of the k_worst-th subcarrier is evaluated by using the semidefinite programming (SDP) security analysis method [38,39], as shown in Fig. 6(a). In fact, the single SKR can be further improved by employing the security analysis method in [40] for QPSK CV-QKD protocol. Note that the modulation noise ε_mod(k_worst) of the k_worst-th subcarrier is the worst in N subcarriers. So, the total SKR of the N multi-carrier CV-QKD system with extra modulation noise can be obtained as $R \ge N{R_{{k_{\textrm{worst}}}}}$ according to Eq. (1), and the total SKR $N{R_{{k_{\textrm{worst}}}}}$ is shown in Fig. 6(b). From Fig. 6(b) that the total SKR of the multi-carrier CV-QKD can be greatly improved by increasing N. Moreover, the multi-carrier SKR gain is defined as $N{R_{{k_{\textrm{worst}}}}}/{R_{single}}$ and shown in Fig. 7, where R_single is the SKR of the single-carrier CV-QKD system without modulation noise. It is noteworthy that the multi-carrier SKR gain drastically drops with the increase of total carrier number N in long distance transmission due to the extra modulation noise. Besides, the optimal carrier number N can be numerically calculated based on the proposed modulation noise model as shown in Fig. 8, where the optimal N = 61, 56 and 43 for L = 5 km, 10 km and 25 km, respectively. In Fig. 8, the maximum N at different distances is obtained when SKR ${R_{{k_{\textrm{worst}}}}} = 0$, which reflects the null SKR threshold of the k_worst-th subcarrier.

Fig. 6. The SKR of the QPSK multi-carrier CV-QKD with N = 10, 25, 40 and 60, respectively. (a) Single SKR of the k_worst-th subcarrier, (b) total SKR of the QPSK multi-carrier CV-QKD.

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Fig. 7. Calculated multi-carrier SKR gains for the QPSK multi-carrier CV-QKD with N = 10, 25, 40 and 60, respectively.

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Fig. 8. Calculated QPSK multi-carrier SKR gains at different total carrier number N with transmission distance L = 5 km, 10 km, 25 km, respectively.

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4.2 256QAM modulation protocol

The SKR performance of the multi-carrier CV-QKD with 256QAM discrete modulation protocol is evaluated in a similar way based on the proposed modulation noise model with system parameters µ_k= 0.01, κ_k= 0.98 and θ_k= π/50. For 256QAM modulation, the random variables I_g and Q_g (g = k, m, n, l) obey mutually independent discrete Maxwell-Boltzmann distribution, and the corresponding probability density function is given by [19]

(18)$${\rho _{{x_i}}} = \frac{{\exp ({ - \nu x_i^2} )}}{{\sum\limits_{i = 1}^{16} {\exp ({ - \nu x_i^2} )} }}$$

where x_i is normalized within (-1, 1). ν is a positive free parameter whose optimal value is about 0.04. Therefore, the variance σ₁² of the quadrature I_g and the variance σ₂² of the quadrature I_g² can be obtained as, respectively

(18a)$$\sigma _1^2 = \sum\limits_{i = 1}^{16} {{{[{{x_i} - E({{x_i}} )} ]}^2}{\rho _{{x_i}}}}$$

(18b)$$\sigma _2^2 = \sum\limits_{i = 1}^{16} {{{[{x_i^2 - E({x_i^2} )} ]}^2}\rho _{{x_i}}^2}$$

with σ₁²= 0.37 and σ₂²= 0.11 applied in Eq. (16). So, the single SKR of the k_worst-th subcarrier in 256QAM multi-carrier CV-QKD can be evaluated by using the improved SDP method in [39], as is shown in Fig. 9(a). Moreover, the modulation variance V_A= 5 SNU, the reconciliation efficiency β_k= 95%, the quantum efficiency η=0.56, the electronic noise υ_ele= 0.15 SNU and the excess noise ε_single= 0.03 SNU are chosen in the following simulations referring to our previous 256QAM single-carrier CV-QKD experiment [19]. Note that the single-carrier excess noise ε_single= 0.03 SNU has excluded the IQ imbalance noise by DSP in the single-carrier CV-QKD experiment [19]. Furthermore, the corresponding total SKR and the multi-carrier SKR gain are respectively given in Fig. 9(b) and Fig. 10 for verifying the performance of 256QAM multi-carrier CV-QKD. Besides, the multi-carrier SKR gains with different N are calculated in Fig. 11, where the optimal N = 96, 85 and 72 for L = 25 km, 50 km and 100 km, respectively.

Fig. 9. The SKR of the 256QAM multi-carrier CV-QKD with N = 10, 40, 80 and 120, respectively. (a) Single SKR of the k_worst-th subcarrier, (b) total SKR of the 256QAM multi-carrier CV-QKD.

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Fig. 10. Calculated multi-carrier SKR gains for the 256QAM multi-carrier CV-QKD with N = 10, 40, 80 and 120, respectively.

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Fig. 11. Calculated 256QAM multi-carrier SKR gains at different total carrier number N with transmission distance L = 25 km, 50 km and 100 km, respectively.

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4.3 Gaussian modulation protocol

We also estimate the SKR performance of the multi-carrier CV-QKD with Gaussian modulation protocol based on the proposed modulation noise model with system parameters µ_k= 0.01, κ_k= 0.98 and θ_k= π/50. The random variables I_g and Q_g (g = k, m, n, l) are mutually independent and follow the normal Gaussian distribution $N\left( {0,\,\sigma _1^2 } \right)$ with amplitude normalization (-1, 1), whose with probability density function is given by

(19)$$f(x )= \frac{1}{{\sqrt {2\pi } {\sigma _1}}}{e^{ - \frac{{{x^2}}}{{2\sigma _1^2}}}}$$

and the corresponding random variables I_g² and Q_g² follow the Gaussian distribution N(σ₁², 2σ₁⁴), which can be verified as

(20a)$$E({{x^2}} )= \int_{ - \infty }^{ + \infty } {{x^2}} f(x )dx = \sigma _1^2$$

(20b)$$\sigma _2^2\textrm{ = }E({{x^4}} )- {[{E({{x^2}} )} ]^2} = \int_{ - \infty }^{ + \infty } {{x^4}} f(x )dx - {\sigma _1}^4 = 2{\sigma _1}^4$$

Therefore, the variance σ₁²= 1/9 and σ₂²= 2/81 can be obtained when the standard normal distribution N(0, σ²= 1) is normalized to (-1, 1) from the amplitude section (-3σ, 3σ). Similarly, applying σ₁² and σ₂² in Eq. (16), the single SKR of the k_worst-th subcarrier for the Gaussian modulated multi-carrier CV-QKD can be evaluated based on the no-switch protocol security analysis method [12], as shown in Fig. 12(a). In the SKR calculation, the modulation variance V_A= 5 SNU, the quantum efficiency η=0.56, the electronic noise υ_ele= 0.1 SNU, the reconciliation efficiency β_k= 95% and the excess noise ε_single= 0.03 SNU are chosen as in [11]. Similarly, the corresponding total SKR and multi-carrier SKR gain are given in Fig. 12(b) and Fig. 13, respectively, which also verifies that the SKR can be greatly increased based on the multi-carrier CV-QKD scheme. Moreover, the multi-carrier SKR gains at different N are demonstrated Fig. 14, where the optimal N = 155, 130 and 110 for L = 50 km, 100 km and 150 km, respectively.

Fig. 12. The SKR of the Gaussian modulated multi-carrier CV-QKD with N = 10, 80, 120 and 160, respectively. (a) Single SKR of the k_worst-th subcarrier, (b) total SKR of the Gaussian modulated multi-carrier CV-QKD.

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Fig. 13. Calculated multi-carrier SKR gains for Gaussiam modulated multi-carrier CV-QKD with N = 10, 80, 120 and 160, respectively

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Fig. 14. Calculated Gaussian modulated multi-carrier SKR gains at different total carrier number N with transmission distance L = 50 km, 100 km and 150 km, respectively.

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5. Conclusion

We have demonstrated a systematic modulation noise model for OFDM-based multi-carrier CV-QKD with arbitrary modulation protocol. Specifically, the extra modulation noise including the IQ imbalance noise and the third-order intermodulation noise in the multi-carrier quantum state preparation is theoretically modeled and analyzed, which limits the SKR performance of system with total carrier number N. Under considering the extra modulation noise, the SKR performance of the OFDM-based multi-carrier LLO-CV-QKD system with QPSK, 256QAM and Gaussian modulation protocol can be quantitatively evaluated by combining the security analysis method of the single-carrier CV-QKD system. Under practical system parameters, our simulation results show that the OFDM-based multi-carrier CV-QKD scheme can greatly improve the asymptotic SKR by increasing N even with imperfect modulations. Moreover, a maximum total SKR can be obtained by carefully optimizing N. Note that the IQ imbalance noise and the third-order intermodulation noise can be further reduced by optimizing the related parameters and designing the corresponding compensation algorithm at Bob’s site for achieving a better SKR performance in the practical multi-carrier CVQKD system. Compared with the single-carrier CV-QKD scheme, our work offers a promising way to significantly improve the performance of CV-QKD. In the future, other noises from the channel and receiver in practical experiment should be further considered to evaluate the performance of multi-carrier CV-QKD system, such as the fiber channel crosstalk noise between these subcarriers.

Funding

National Key Research and Development Program of China (2020YFA0309704); Sichuan Science and Technology Program (2021YJ0313, 2022YFG0330, 2022ZYD0118); National Natural Science Foundation of China (61901425, 62101516, 62171418, 62201530, U19A2076); Technology Innovation and Development Foundation of China Cyber Security (JSCX2021JC001); Chengdu Major Science and Technology Innovation Program (2021-YF08-00040-GX); Chengdu Key Research and Development Support Program (2021-YF05-02430-GX, 2021-YF09-00116-GX); Foundation of Science and Technology on Communication Security Laboratory (61421030402012111); Major Project of the Department of Science and Technology of Sichuan (2022ZDZX0009).

Disclosures

The authors declare that there are no conflicts of interest related to the article.

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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Performance analysis for OFDM-based multi-carrier continuous-variable quantum key distribution with an arbitrary modulation protocol

Abstract

1. Introduction

2. Description of the multi-carrier CV-QKD scheme

3. Modulation noise model of the multi-carrier CV-QKD

4. Performance evaluation of multi-carrier CV-QKD

4.1 QPSK modulation protocol

4.2 256QAM modulation protocol

4.3 Gaussian modulation protocol

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (14)

Equations (29)

Optics Express