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Abstract
In recent years, the feasibility of quantum key distribution (QKD) in a water channel has been verified by theory and experiment. Here, we present an experimental investigation of QKD and decoy-state QKD based on the BB84 protocol. The experiment was carried out in a 10 m water tank. The attenuation coefficient of tap water is 0.08/m, which is close to Jerlov Type II seawater. We measured the probability-of-detection matrix of polarization states, and the average fidelity of the four polarization states is up to 98.39%. For the 10 m underwater QKD experiment, 20 MHz optical pulses are generated by modulating the laser diode (LD) and attenuated to an average of 0.1 photons per pulse. The security key rate can reach 563.41 kbits/s and the quantum bit error rate (QBER) is 0.36%. Two decoy states (one of which is the vacuum state) was used in the 10 m underwater decoy-state QKD experiment, and the average QBER of signal state is 0.95%, the security key rate reaches 711.29 kbits/s. According to the parameters of the decoy-state experiment, the maximum secure transmission distance of the underwater decoy-state QKD is predicted to be 19.2 m, while it can be increased to 237.1 m in Jerlov Type I seawater with a lower dark count single photon detector (SPD).
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The theoretical unconditional security of quantum key distribution (QKD) can meet the communication security need. The first QKD protocol was proposed by Bennett and Brassard in 1984 (BB84) [1], which attracted a lot of theoretical and experimental research interest. Theoretical research focuses on reducing protocol complexity and improving system security. B92 protocol [2] and six states protocol [3] are the simplified and improved version of the BB84 protocol. Decoy-state protocol [4–6] is the perfection of BB84 protocol, which can resist the photon-number-splitting (PNS) attack. So far, the experimental demonstration system of QKD has also been developed, achieving thousands of kilometers remote QKD in free-space air channel [7,8] and hundreds of kilometers in optical fiber channel [9,10].
Seawater is also an important quantum communication channel. In recent years, due to the demands of marine exploration and submarine military activities, underwater wireless communication technology has been greatly developed, and underwater QKD technology can be used to enhance the security of information transmission. Due to the complex composition and special optical properties of seawater, the feasibility of underwater QKD has been investigated by researchers. In 2015, the vector radiative transfer theory and Monte Carlo method were used to study the absorption and scattering properties of photons in water channel [11]. For the first time, the feasibility was verified by experiments in 2017, and experimental results show that polarization states and entangled states can be well preserved in a 3.3 m seawater channel [12]. The transmission of high-dimensional twisted photons in the 3 m underwater channel has been experimentally demonstrated, and the effect of turbulence on bit error rates has also been studied [13]. After that, researchers conducted experiments on the photonic polarization states transmission and high-dimensional twisted photons transmission over a 55 m underwater channel, proving the feasibility of QKD in long-distance and high-loss underwater channel [14–16]. In 2019, a proof-of-principle experiment of polarization encoding BB84 protocol QKD over a 2.37 m water channel was reported. By measuring the Mueller matrix of underwater channel, it is proved that the seawater channel has a high degree of polarization preservation. When the attenuation coefficient of the water channel is 0.11/m, the quantum bit error rate (QBER) is 1.65%, and the security key rate can reach 337.2 bits/s [17]. In order to promote the practicality of underwater QKD, the decoy-state protocol can be considered to improve the communication security, as well as the transmission distance and security key rate.
In this paper, we performed a proof-of-principle demonstration of polarization encoding QKD and decoy-state QKD over a longer water channel. The attenuation coefficient of a 10 m water tank channel is close to Jerlov Type II seawater, which might simulate the deep-ocean communication environment. The probability-of-detection matrix and fidelity of photon polarization state were measured to verify the polarization-preserving characteristics of water channel. By reducing the bias current of the LD, placing the transceiver in a black box, and placing a band-pass filter (BPF) in front of each SPD, the interference of background light is minimized and the QBER is reduced. In addition, the security key rate can be improved by increasing the pulse repetition rate. The experimental results of underwater QKD show that the average QBER is less than 0.36%, and the security key rate reaches 563.41 kbits/s. In the decoy-state QKD experiment where two decoy states (a decoy state and a vacuum state) was used, the average QBER of signal state is 0.95%, and security key rate reaches 711.29 kbits/s.
2. Experimental setup and details
As shown in Fig. 1, our experiment was performed in a water tank of 10 m $\times$ 0.75 m $\times$ 0.6 m. The attenuation coefficient of tap water used in the experiment is measured to be 0.08/m (at wavelength 520 nm). The transmitter and receiver are placed in a black box respectively. There is a small hole with a diameter of 6 mm on each box for alignment, and it also plays a role in minimizing the background light. The components of underwater QKD and decoy-state QKD experimental system are shown in Fig. 2.
At the transmitter, optical pulses are generated by a laser diode (LD) which is modulated by an arbitrary waveform generator (AWG), and attenuated to less than one photon per pulse by an adjustable attenuator (ATT) to prepare a single photon source. The peak wavelength of the LD (THORLABS, LP520-SF15) is 520 nm, and the maximum output optical power is 20 mW. After that, the polarizer (P) is adjusted so that the polarization of output state is horizontal (H). The final output state of photons is then encoded as one of $|H\rangle$, $|V\rangle$, $|D\rangle$ or $|A\rangle$ by half-wave plate (HWP), where H and V represent horizontal and vertical polarizations, D and A represent $45^{\circ }$ and $-45^{\circ }$ polarizations.
At the receiver, a 50:50 beam splitter (BS), a HWP and two polarization beam splitters (PBS) are used for polarization measurement. Four single photon detectors (SPD) are used to detect photons, and their outputs are collected by an oscilloscope for off-line signal processing. The SPDs are multi-pixel photon counter (MPPC) modules (HAMAMATSU, C13366-1350GD). Each module contains 667 pixels, with a photosensitive area of 1.3 mm $\times$ 1.3 mm, and the photon detection efficiency is about 37% at 520 nm. The dark count of each MPPC module is about 2.5 kHz, which is much higher than a single pixel, but the larger photosensitive area can greatly increase the probability of photon detection and reduce the difficulty of optical path alignment. In addition, a BPF with a bandwidth of 10 nm is placed in front of each SPD to reduce the interference of background light.
Since there is no ideal single photon source device in practice, the weak coherent laser source was used as a quasi single photon source in the experiment. We use AWG to generate 20 MHz electric pulse signal and inject it into LD through a Bias-T (THORLABS, LDM9LP). The width of electric pulses is less than 10 ns, and the peak-to-peak voltage is about 4 V. The bias current of LD is close to zero (0.13 mA), which can eliminate the potential inter-pulse phase correlation and reduce the background photons [18]. In our experiment, the electric pulses are AC-coupled directly to the LD through a Bias-T network, and the input impedance is 50 $\Omega$, so the driving current is large enough, which can produce the optical pulse signal with narrow linewidth and radiation beam.
3. Results and discussion
This section first focuses on the attenuation and polarization preservation properties of water channel. Then it presents underwater QKD experimental results and underwater decoy-state QKD experimental results. Finally it predicts the maximum secure transmission distance under decoy-state QKD.
3.1 Measurement and analysis of water channel properties
Photons are mainly affected by absorption and scattering when they propagate in water channel. Macroscopically, the relationship between the power of received light $I$ and the power of transmitted light $I_0$ can be described by Beer-Lambert law [19] as
where $c(\lambda )$ is the attenuation coefficient related to the light wavelength, and $z$ is the propagation distance. $c(\lambda )$ can be expressed as the sum of absorption coefficient $a(\lambda )$ and scattering coefficient $b(\lambda )$ as For pure seawater, the absorption coefficient is mainly affected by the wavelength of light. In the blue-green optical window (430-570 nm [20,21]), photons experience less loss and therefore can propagate further. The laser wavelength used in our experiment is 520 nm. In addition, the scattering coefficient is mainly related to the density of phytoplankton and detritus [20].Then we measured the attenuation coefficient of water channel in a 10 m water tank. Considering that the light beam passes through the glass twice during the measurement, assuming the transmittance coefficient of glass is $K$, Eq. (1) can be modified to
Then we used the model (3) to fit the measured data, where $K$ was measured to be 0.917, and the result is shown in Fig. 3. The attenuation coefficient $c(\lambda )$ was estimated to be 0.08/m, and subsequent experiments are based on this water quality.For the BB84 protocol QKD based on polarization encoding, the polarization preservation properties of water channel have a great influence on the communication performance. We measured the probability-of-detection matrix of polarization states [22]. As shown in Fig. 4, the transmitter sends a linear polarization state, and the receiver records its projection probability on different orthogonal measurement bases, namely {$|H\rangle$, $|V\rangle$} or {$|D\rangle$, $|A\rangle$}. The resultant error rate is less than 1%. In addition, the fidelity of four polarization states ($|H\rangle$, $|V\rangle$, $|D\rangle$ and $|A\rangle$) through 10 m water channel was measured, and results are shown in Fig. 5. The average fidelity of four polarization states is as high as 98.39%, which means that the water channel has a high degree of polarization preservation. The slight difference of the fidelity of four polarization states is caused by imperfect polarization optics and experimental errors, such as optical path alignment.
3.2 Underwater QKD experimental results
We performed the proof-of-principle experimental results of 10 m underwater QKD. The outputs of four SPDs when sending horizontal polarization signal are shown in Fig. 6. In Fig. 6(a), SPD1 and SPD2 represent $|H\rangle$ and $|V\rangle$ orthogonal basis, SPD3 and SPD4 represent $|D\rangle$ and $|A\rangle$ orthogonal basis. Obviously, the pulse count of SPD1 is the largest, the pulse count of SPD2 is close to the dark count, and the pulse counts of SPD3 and SPD4 are approximately equal and slightly less than half of SPD1, which is due to the non-ideal characteristics of the device. Since the number of received pulses per second is large enough and the pulse delay roughly obeys the Gaussian distribution, synchronization can be completed by a statistical method. For the pulses in the output of four SPDs, the time corresponding to the rising edge can be expressed as
where $t_0$ is the starting time of each group of data, $T$ is the pulse period, and $d_i$ represents the delay of the $i$-th pulse. For each group of data, the following delay values are counted and the temporal shape of signal pulses is shown in Fig. 6(b). We can determine a 20 ns time window according to the temporal shape of signal pulses, and the pulses not in the time window are regarded as noise pulse and need to be removed. In addition, if more than one SPD detects pulses in the same pulse period, it means that the photon number in the corresponding transmitter pulse is greater than 1, or a part of SPDs detect noise pulses, so all pulses in the pulse period need to be removed.After the above data processing, we can get the effective pulse outputs of SPDs. We obtain the average sifted key rate of four polarization states is 678.7 kbits/s, and the average QBER is 0.36%, as shown in Fig. 7. The post-processing was performed off-line in the computer, including error correction and privacy amplification, and finally the security key can be obtained. The lower bound of security key rate can be determined by the GLLP formula [23] as,
where $q$ is the sifting factor, which is 0.5 in our experiment. $Q_\mu$ and $E_\mu$ represent gain and QBER respectively. $Q_1$ and $e_1$ are, respectively, the gain and QBER when single-photon states are emitted by source. $H_2(e)$ is the binary Shannon entropy, i.e. $H_2(e) = -e\log _2e - (1-e)\log _2(1-e)$, where $e$ represents QBER. $f(e) = I(e) / H_2(e)$ is the error correction efficiency, where $I(e)$ is the information quantity leaked during actual error correction, and $f(e) \geq 1$, obviously. The Cascade algorithm was used for error correction, and then the error checking was performed to ensure that the keys obtained by the receiver and sender are exactly the same.In formula (6), $Q_\mu$ and $E_\mu$ can be obtained from experimental data, and $Q_1$ and $e_1$ can be estimated by the following formula,
where $p_m$ is the probability of multi-photons in the pulse emitted by source, which can be obtained from average photon number per pulse $\mu$ as The experimental parameters are shown in Table 1. According to formula (6), we can get the lower bound of the average security key rate of four polarization states is 563.41 kbits/s.
Table 1. Main parameters of the underwater QKD experiment.
3.3 Underwater decoy-state QKD experimental results
For the BB84 protocol QKD system using a weak coherent laser source, eavesdroppers can obtain the keys exactly consistent with the receiver and transmitter through PNS attack. Therefore, we performed the proof-of-principle experiment of decoy-state BB84 protocol to improve the security of a QKD system. The decoy-state BB84 protocol we use includes a signal state, a decoy state and a vacuum state, whose average photon numbers per pulse are $\mu =0.9$, $v_1=0.4$ and $v_2=0$, and the mixture ratios are 50%, 25%, and 25%, respectively.
As shown in Fig. 8, in order to perform the synchronization more accurately, a signal frame was designed, which contains $10^4$ pulse periods, including $5000$ signal states, $2500$ decoy states, and $2500$ vacuum states. $100$ vacuum states are placed in the frame head, and the remaining signal states, decoy states and vacuum states are arranged in the frame in a random order. Since the receiving oscilloscope we use has only four channels, the synchronization signal is mixed with the SPD1 output signal after attenuation, as shown in Fig. 9. Then we can separate the synchronization signal and SPD1 output signal according to the difference in signal amplitude.
Figure 10 shows the characteristics of received signals when sending horizontal polarization signal in the decoy-state QKD experiment. As can be seen from Fig. 10(b), the temporal shapes of signal state and decoy state are normalized by dividing their maximum photon counts respectively, and the time window is also set to 20 ns. After the above processing, we find that the average sifted key rate of signal state of the four polarization states is 2.21 Mbits/s, and the average QBER is 0.95%, as shown in Fig. 11. The lower bound of security key rate of decoy-state QKD can also be estimated by formula (6), where $Q_1$ and $e_1$ can be estimated by the following formula
Fig. 10. Received signal characteristics when sending horizontal polarization signal in the decoy-state QKD experiment.
Table 2. Main parameters of the underwater decoy-state QKD experiment.
3.4 Predicted maximum secure transmission distance
According to the GLLP formula, with the increase of transmission distance, the security key rate will gradually approach 0. So there is a maximum secure transmission distance. According to the parameters of the decoy-state QKD experiment in this paper, the maximum secure transmission distance is predicted by computer simulation, and the results are shown in Fig. 12, where $c$ is the attenuation coefficient of water channel, and $Y_0$ is the yield of 0-photon state, i.e., background rate, which includes the environment background noise, laser source background noise and SPD dark count. In our experiment, $c = 0.08$/m, $Y_0 = Q_{v2} = 2.9\times 10^{-3}$ as shown in Table 2, and the maximum secure transmission distance is 19.2 m. For Jerlov Type I seawater with an attenuation coefficient of 0.03/m, the maximum secure transmission distance can reach 51 m. When the transmission distance increases, the laser source background noise is also attenuated and its influence on $Y_0$ is reduced. In addition, SPD with smaller dark count can be used. Considering all these factors, $Y_0$ can be decreased as small as $1\times 10^{-5}$. When $Y_0 = 1\times 10^{-5}$ is adopted, the maximum secure transmission distance can reach 88.9 m when $c = 0.08$/m, and can reach 237.1 m when $c = 0.03$/m. Beyond the upper bound distance, the sender and receiver cannot determine whether the line-of-sight link has eavesdropping, or whether the non-line-of-sight scattered photons will leak the key.
The extrapolation results can be used in the short-distance underwater quantum Internet of Things, or for quantum security enhancement of underwater blue-green optical wireless communication. For example, the transmission distance of an underwater optical wireless communication system has been demonstrated to reach about 100 m at data rate of 500 Mbps [24]. It is anticipated that the underwater QKD technology will provide security guarantee for an underwater optical wireless communication channel.
4. Conclusions
In summary, we built a demonstration platform of QKD and decoy-state QKD experiments based on BB84 protocol over a 10 m underwater channel, and the attenuation coefficient of water is 0.08/m at wavelength 520 nm, which is close to Jerlov Type II seawater. We measured the probability-of-detection matrix of polarization states, and the average fidelity of the four polarization states is as high as 98.39%, which indicates that the water has good polarization preserving property. Then we performed the proof-of-principle experiments of 10 m underwater QKD and decoy-state QKD. The average security key rate of the four polarization states can reach 563.41 kbits/s in the QKD experiment and 711.29 kbits/s in the decoy-state QKD experiment, and their average QBER are both less than 1%. At last, the maximum secure transmission distance is predicted based on the parameters of the decoy-state QKD experiment. According to our simulation results, the maximum secure transmission distance is 19.2 m when adopting the $Y_0$ value in the experiment, while such distance is predicted to be 237.1 m when setting $Y_0 = 1\times 10^{-5}$ in Jerlov Type I seawater.
Funding
National Key Research and Development Program of China (2013CB329201); Key Program of National Natural Science Foundation of China (61631018); Key Research Program of Frontier Science of Chinese Academy of Sciences (QYZDY-SSW-JSC003); Strategic Priority Research Program of Chinese Academy of Sciences (XDA22000000).
Acknowledgment
The authors would like to thank Information Science Laboratory Center of USTC for the hardware & software services.
Disclosures
The authors declare no conflicts of interest.
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