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Abstract
We demonstrate the underwater quantum key distribution (UWQKD) over a 10.4-meter Jerlov type III seawater channel by building a complete UWQKD system with all-optical transmission of quantum signals, a synchronization signal and a classical communication signal. The wavelength division multiplexing and the space-time-wavelength filtering technology are applied to ensure that the optical signals do not interfere with each other. The system is controlled by FPGA and can be easily integrated into watertight cabins to perform the field experiment. By using the decoy-state BB84 protocol with polarization encoding, we obtain a bit rate of secure keys of 1.82 Kbps and an error rate of 1.55% at the attenuation of 13.26 dB. We prove that the system can tolerate the channel loss up to 23.7 dB and therefore may be used in the 300-meter-long Jerlov type I clean seawater channel.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD), a way of generating and distributing secret keys based on quantum physics, is considered unconditionally secure. Following the first BB84 protocol [1], QKD has been implemented in optical fiber [2–4] and atmosphere [5,6], leaving underwater quantum key distribution (UWQKD) a last barrier to conquer. In recent years, UWQKD has been studied theoretically [7–9] and experimentally [10–17]. Most of the experimental studies remain on the level of feasibility research of UWQKD. The polarization of photons was proved experimentally to maintain high fidelity through a 3-meter-long water channel and be feasible for UWQKD [13]. Twisted photons were also proved feasible for short distance UWQKD [14], even in flowing water [15]. In subsequent works [11,12], quantum tomography was performed, showing that polarized photons and twisted photons can survive well through a 55-meter-long underwater channel. Till now, two of the experimental works demonstrated the complete QKD process [10,16]. BB84 protocol, as the most widely used protocol, was completely implemented with polarized photons over a 2.3-meter-long water channel [16]. Most recently, decoy-state method was employed in UWQKD [10,17], where the demonstrated distances are 30 meters in Jerlov types III seawater [10] (equivalent to 345-meter-long clean seawater) and 10 meters in Jerlov Type II seawater [17], respectively.
On the road to the practical application of UWQKD, many works, both scientific and technical, should be performed. One challenge is the requirement of classical optical signal in UWQKD. The classical optical signal is widely used to send classical information [3], realize time synchronization [3,5,6], and finish the task of pointing and tracking in free space QKD [5,6]. For fiber-based or free space QKD, these tasks can also be finished with the electrical signal or radio wave. However, because of the nature of seawater, electric cable or radio wave cannot be used to finish these tasks underwater, making the wireless optical communication the only choice for the underwater channel. Moreover, underwater optical communication is even more difficult due to the complexity of the underwater channel and the interactions among the quantum, classical and synchronization signals. Another challenge brought by the underwater channel is that the underwater instrument should be designed in small size and with high integration, and therefore should be controlled with elaborately designed field-programmable gate array (FPGA) system.
In this work, we successfully implement a complete decoy-state BB84 UWQKD system with polarization encoding and all-optical transmission, in which the optical setup, FPGA boards, and software work together. Compared to our previous work [16], we increase the repetition frequency of the quantum signals and adopt the decoy-state method here, therefore the bit rate of secure keys increases by roughly two orders of magnitude. Compared to the recent UWQKD work with the decoy-state method [10,17], the optical synchronization and optical classical communication are integrated into our system, so the two ends of communication do not need any other kinds of interactions based on local network. In order to meet the integration requirement in the underwater environment, the lasers and detectors are controlled by one FPGA board at each end but not the electrical equipment like arbitrary wave generator or oscilloscope. We successfully distribute secret keys through a 10.4-meter-long Jerlov type III seawater channel (The channel loss is 13.26 dB, equivalent to 170-meter-long Jerlov type I clear seawater channel). The average error rate of the sifted keys is 1.55%, and the average secure key rate is 1.82 Kbps. The system can tolerate the channel loss 23.7 dB, which indicates that our system may be used in the 300-meter-long Jerlov type I clean seawater channel.
2. System setup and algorithm
In our system shown in Fig. 1, at the Alice end (transmitter), we use eight blue lasers ($450$ nm) to implement the three-intensity decoy-state polarization-encoding BB84 protocol because of the lack of effective polarization modulators in the blue-green region. The system factors are shown in Appendix A (Table 4). We use one green laser ($520$ nm) to generate the synchronization signal, another laser ($488$ nm) to transmit the classical communication signal, and one photoelectric detector (PD) to detect the classical communication signal.
Fig. 1. System setup for the transmitter end (Alice) and the receiver end (Bob). HWP: half-wave plate, QWP: quarter-wave plate, BS: beam splitter, PBS: polarizing beam splitter, PD: photoelectric detector, SPD: single-photon detector made of photomultiplier tube, DM: dichroic mirror. Each FPGA board is connected to one personal computer by a USB wire, respectively. Telescopes are used to collimate or collect the quantum and synchronization signals at both ends.
At the Bob end (receiver), the quantum signals are detected with four single-photon detectors (SPD) made of photomultiplier tubes. One laser ($488$ nm) is used to transmit classical communication signal, and two PDs are used to detect the synchronization signal and the classical communication signal.
The dichroic mirrors are employed to combine the quantum and the synchronization signals at the Alice end and to split them at the Bob end. Two water pipes are used as the underwater channels. One is used for the propagation of quantum signals and synchronization signal, and the other is for the classical communication signal. All lasers, PDs, and SPDs are controlled by FPGA circuit boards at both ends. The FPGA circuit boards also perform key-sifting and transmit the sifted keys to the personal computers (PCs). The software in the PCs completes error-correction, error checking, and privacy amplification. With this UWQKD system, real-time secret keys can be generated.
2.1 Quantum signal source and polarization encoding
The central wavelength of quantum signal lasers is $449.5$ nm with the deviation less than $0.5$ nm, and the power stability is less than $1$% within 50 minutes as shown in Fig. 2(a). These eight lasers are organized into four groups, each group corresponding to one of the four polarizations (horizontal (H), vertical (V), $45{^\circ }$ (P), $135{^\circ }$ (M)), and the two lasers in one group generating the pulses of the signal state and the decoy state through different attenuations required by the decoy-state method. The beam splitter (BS) and reflector are used to combine the signal state and the decoy state in a group. Then the PBSs, HWP and BS are used to encode the pulses into the four polarizations and combine them together.
Fig. 2. Performance tests of the quantum lasers. (a) The power stability test in 50 minutes. The power variation is less than 0.011 mW (about 1%). (b) The temporal shape of the quantum signal pulse. SPD is used to count the photons in 50 ns, and the counts in each time slots (250 ps) are recorded and normalized to obtain the shape. The eight quantum lasers have almost completely the same temporal shapes.
The control format for the quantum signal lasers is shown in Table 1. We use four random number generator (RNG) chips to generate the random numbers (denoted by bit3, bit2, bit1 and bit0), which are fed into the FPGA to control the operation of the eight quantum signal lasers. The probabilities of choosing the signal state, decoy state and vacuum state are 0.5, 0.25, and 0.25, respectively, similar to the works of QKD underwater [10] and in atmosphere [6].
Table 1. The control format for the quantum signal lasers.
The repetition frequency of the quantum signal lasers is 20 MHz. The high repetition frequency is helpful for achieving a high secret key rate and low error rate on condition that no symbol interference occurs. In this work we adopt 20 MHz repetition frequency by comprehensive consideration of the lasers, detectors and circuits in our system. As we know, the repetition frequency is bounded above by the symbol interference. Except for highly turbid waters, the channel time dispersion is less than 1 ns when working over moderate distances [18]. For highly turbid waters (the attenuation coefficient $2.17~m^{-1}$), the channel time dispersion can reach 3 ns [18].
The measured full width at half maximum (FWHM) of the optical pulse is $9.5$ ns as shown in Fig. 2(b).
2.2 Synchronization, wavelength division multiplexing (WDM) and filtering
In our system, the repetition frequency of the synchronization signal is $5$ MHz and the FWHM is $30$ ns. We use the wavelength division multiplexing (WDM) technology to combine and split the synchronization signal and the quantum signals. Specifically, a dichroic mirror with the edge at 480 nm is used at the Alice end to combine the signals by letting the quantum signals (450 nm) pass through and reflecting the synchronization signal (520 nm), and another dichroic mirror is used to split them at the Bob end. Our test shows that the ratio of the 520 nm laser passing through the dichroic mirror is less than 0.1%.
At the Bob end, the PD converts the optical synchronization signal into the electrical signal. In order to match the quantum signal, the electrical signal is delayed by a high-precision programmable delay chip, whose delay precision is $250$ ps with a maximal delay $64$ ns, and then sent to the FPGA. The FPGA converts the signal into the gate signal of repetition frequency $20$ MHz, and the gate time is set to 10 ns which is larger than the FWHM of the temporal shape of quantum signal in Fig. 2(b). We also design a self-adapting algorithm to precisely match the quantum signal. The gate signal is not applied to SPDs but used by the FPGA program. In the following processing, the FPGA program records the clicks of the SPDs to generate keys when the gate is open, and omits the clicks of the SPDs to reduce the background noise when the gate is close.
To reduce the influence of the synchronization signal and background noise, we perform filtering in space, time and wavelength domains. In time domain, we use the FPGA program to omit clicks of the SPDs out of the time gates. The clicks coming from the synchronization signal will also be omitted by turning off the gate when the synchronization pulses arrive. In wavelength domain, we use two filters behind the dichroic mirror at the Bob end. The spectral width of the filters is $20$ nm, and the central wavelength is $445$ nm. Our test shows that these two filters perform well in suppressing the background noise. In space domain, the field of view (FOV) is limited mainly by the fiber coupler and the aperture is limited by the telescope. The FOV of the fiber coupler is about 0.14 degrees and the diameter of the telescope is 5 cm. When we use SPDs directly without the fiber coupler and fiber, the noise caused by the classical signal and the background light makes the error rate extremely high, and the UWQKD experiment cannot be performed successfully.
A big challenge in our early experiment is that the count rate of the SPDs was always high when the optical synchronization signal was open, even when many 445 nm filters were used. We speculate that the synchronization laser can also send photons in $450$ nm. To protect the SPDs, we use a $520$ nm filter at the Alice end just behind the synchronization laser and solve the problem perfectly.
2.3 Optical classical communication
To build a system with all-optical transmission, the classical communication in the UWQKD system is realized by optical signal with the open-off-keying (OOK) modulation [19,20]. The wavelength of the classical communication signal is $488$ nm, and the baud rate is $20$ MHz. Horizontal and vertical polarizations are employed at two ends, respectively, so as to use the same channel to realize the duplex optical classical communication via the two PBSs.
2.4 FPGA control
FPGA is responsible for the control of the decoy-state UWQKD system, including polarization encoding and detection, key-sifting, classical information between Alice and Bob and communication with the PCs.
As shown in Fig. 3, at the Alice end, the "laser control" module reads random numbers from the RNG chips and converts them into quantum and synchronization laser control pulses to generate keys, and then sends the key information (serial number, intensity, polarization of each key) to the "key-sifting" module. The "laser control" module will send three quantum signals after one synchronization signal, and decide the polarization and intensity of the quantum signal as the format in Table 1 by lighting a specific laser.
Fig. 3. FPGA diagrams at the Alice and Bob ends. The dotted lines represent the optical links through seawater.
At the Alice end, the "key-sifting" module temporarily stores the key information and completes the key-sifting with the aid of classical communication with Bob. The keys with the right basis will attend the error estimation (20%) or be regarded as sifted keys (80%). All of the key information (serial number, intensity, basis of each key) of keys with the right basis are sent to Bob and Alice’s PC. The polarization of the keys that attend the error estimation (20%) is sent to Bob and the polarization of sifted keys (80%) is sent to the PC through the "UART and control" module.
The "UART and control" module takes the responsibilities of sending the sifted keys from the "key-sifting" module and the post-processing information from the "classic send" module to the PC. The post-processing information generated in the PC can also be sent through the "UART and control" module to the "classic send" module.
The "classic send" module at the Alice end can send the information from the "key-sifting" module and the "UART and control" module to control the laser. When the key-sifting information and post-processing information is received by the PD, the "classic receive" module at the Alice end reads the information from the PD and sends it to the corresponding modules.
At the Bob end, the "SPD read and synchronization" module controls the delay chip to match the synchronization and the quantum signals, and completes the single-photon detection to generate the keys (serial number, polarization of each key), and sends the keys to Bob’s "key-sifting" module. Every synchronization signal is converted into three pulses inside this module so as to obtain a periodic pulse train with the period 50 ns, and these pulses are used as the gate signal. Only the photons inside the gate are used to generate the keys. The gate time is set to 10 ns to fit the FWHM of the quantum signals. Different from the "laser control" module at the Alice end, the serial numbers of the keys are generated by the counts of the gate signal.
The "key-sifting" module at the Bob end can send the key information (only serial number and basis of each key) through the "classic send" module to Alice. At the same time, this module waits for the key information of Alice from the "classic receive" module. The keys with the right basis will be sent to the PC through the "UART and control" module. The key rates and error rates of the signal state, decoy state and vacuum state are also reported to the PC.
For Bob, other modules are similar to the modules at the Alice end.
2.5 Post-processing
In QKD process, post-processing is required to generate the secure keys, including key-sifting, error correction, error checking, and privacy amplification [21,22]. For our system, the key-sifting is finished by the FPGA and the others are finished by the software in the PCs in our UWQKD experiment.
Firstly, in the key-sifting, Alice’s and Bob’s FPGAs exchange information to estimate the error rate and generate the sifted keys as we mentioned in the subsection 2.4. The percentage of total keys used for error estimation needs a best compromise between the key loss caused by error estimation and error checking. On the one hand, the more keys attend error estimation, the fewer keys can be used to produce the secret keys. On the other hand, in our experiment, if the error rate is estimated not so precisely and the real error rate is beyond the ability of LDPC, errors will occur when the whole group of keys attend the privacy amplification and the keys will be discarded in the process of error checking. In our experiment, when Alice uses 20% of the sifted keys to estimate the error rate, the LDPC correction works well as the estimated error rate is below 1.5%, far lower than the error correction ability of LDPC (more than 4% for our case). Meanwhile, the choice (20%) is appropriate here, because the error probability for the estimation is very close to zero, when the allowable deviation between the real error rate and the estimated error rate is 2.5% and the length of sifted keys is more than 8000 bits per group (8640 for our work) [23]. It is clear that more sifted keys are needed to estimate the errors if the error rate is higher.
Secondly, the rest of the sifted keys participate in the error correction, which is realized by low-density parity-check code (LDPC). In our system, the code length of LDPC code is 9216 and the code rate is 3/4B, following the criteria of IEEE802.16e [24]. Thirdly, error checking [25] and privacy amplification are performed to make sure the final keys are the same and safe. The total number of the secure keys is calculated and the sifted keys are compressed into the secure keys. The Toeplitz [26] matrix is used to complete the hash function in error checking and privacy amplification. To reduce the impact of the finite-size effect, we conduct the privacy amplification for every 256 groups of LDPC decoding data, which means that roughly 1.7 Mb of keys participate in privacy amplification once. Due to the use of Fast Fourier transform (FFT), our system takes only 2 seconds to finish each error checking and privacy amplification.
3. Experiments and results
3.1 Characterization of the underwater channel and the optical system
In the experiment, the seawater we used is collected by Dongfanghong Research Ship of Ocean University of China. The length of the water channel is 10.4 m and the overall attenuation is $13.26$ dB (450 nm). Thus the attenuation coefficient is $0.293~m^{-1}$ (Jerlov type III seawater) according to Beer-Lambert formula. The salinity of the seawater, which is the gram-equivalent number of soluble materials in one kilogram water, is 32‰. We tested five points of the water channel and the average temperature is $13^{\circ }C$. The fluctuation of the temperature is not observed under the measurement accuracy of $0.1^{\circ }C$, and the turbulence is also not observed in the experiment. The total photon collecting efficiency of Bob is 54.9%, and the quantum efficiency of the SPD is 20%. Therefore the overall attenuation of Bob is $9.59$ dB.
3.2 Quantum tomography
We perform quantum tomography over the 10.4 m underwater channel at the single-photon level before the QKD experiment. In the experiment, Alice’s FPGA controls the quantum lasers to send four polarization states, and Bob’s FPGA counts the photons detected by the four single-photon detectors to calculate the outcoming density matrices. The ideal density matrices of the four polarization states in this work are given by
Fig. 4. The results of quantum tomography of the polarization states. (a) The measured density matrices of the four polarization states, with Re and Im being their real parts and imaginary parts, respectively. (b) The fidelities of the four polarization states.
3.3 Decoy-state UWQKD
We demonstrate complete key distribution with our decoy-state BB84 QKD system through the 10.4 m seawater channel. The experimental setups are shown in Appendix B (Fig. 7). In the QKD experiment, the average photon numbers per pulse of the signal state and the decoy state are 0.8 and 0.1, respectively [6]. A higher secure key rate can be obtained if we optimize the average photon numbers, because the average photon numbers can have an effect on the gain of the single photons $Q_1$ [27]. However, our choice is enough for the demonstration of this system.
After the PCs at the both ends send the start order to their FPGAs, the FPGAs execute the decoy-state protocol and report the bit value of the sifted keys, the key rates and error rates to the PCs. The PCs calculate the bit rate of secure keys and generate the secure keys. The results of a $1077$ second experiment are shown in Fig. 5.
Fig. 5. The experimental results of the UWQKD experiment through the 10.4 m seawater channel. (a) The real time gain and error rate for the signal state. (b) The real time gain and error rate for the decoy state.
Figure 5(a) and (b) show the gains and error rates of the signal state and decoy state. It can be seen that the average gain of the signal state is more than $4\times 10^{-3}$ and the average error rate of the signal state is 1.55%. After the key-sifting and error estimation, we obtain a bit rate of sifted keys of more than 14 Kbps. The average error rate is low enough for the LDPC code to carry out the error correction successfully. In fact, Alice and Bob can obtain completely the same final keys. We can also see that the gain of the decoy state is about $6\times 10^{-4}$. The average error rate of the decoy state is only $2.35$% despite the large fluctuation caused by its smaller gain, because there are a considerable number of zero points in the error rates of decoy states.
The real-time bit rate of secure keys is not shown in Fig. 5, for the reason that the privacy amplification is conducted for every $1.7$ Mb sifted keys, which means that the system needs $2$ minutes to accumulate the sifted keys for one privacy amplification. Instead, we show the gains and error rates in Fig. 5 and in Table 2.
Table 2. Average gains and error rates.
As shown in Table 2, thanks to the adoption of the time gates, the average gain of the vacuum state $Q_{0}$ is $5.6\times 10^{-6}$, much lower than the background noise in pre-experiment without time gates, which means our temporal filtering works well. The gain $Q_{1}$ and error rate $e_{1}$ of single photons can be estimated by the gain $Q_{u}$ and the error rate $E_{u}$ of signal state, the gain $Q_{v}$ and the error rate $E_{v}$ of decoy state, the gain $Q_{0}$ and the error rate $E_{0}$ of vacuum state according to the decoy state method. The results are that the gain of the single photons is $2.4\times 10^{-3}$ and the error rate of the single photons is 2.25%.
In the experiment, $1963762$ bits secure keys are generated in 1076 seconds, and the bit rate of secure keys is 1823.4 bps. The system does not need to stop single-photon distribution to do the privacy amplification. The privacy amplification is conducted just when the sifted keys reach $1.7$ Mb. Secure keys are generated in situ and in real-time.
To transmit quantum and synchronization signals in one channel and classical signals in another channel, we use two water pipes filled with the same seawater. The classical signal in another water pipe has little influence on the quantum signal as the error rates shown in Fig. 5. In the system, the spectral filtering (two 450 nm filters) and spatial filtering (fiber couplers) are both important for reducing the influence of the classical signal. In addition, time filtering with a gate time of 10 ns also effectively blocks the photons of the classical signal.
In addition to the high attenuation, we observe significant scattering of Jerlov III seawater in the experiment, which reduces the collection efficiency of light. We adjust the telescope to reduce the diameter of the beam spot at the Bob end as much as possible to obtain high collection efficiency. High scattering may lead to a degradation of polarization, but this phenomenon is not observed in our experiment.
There are still some issues worthy to be considered for the practical application of UWQKD. For example, the acquisition pointing tracking (APT) in the underwater turbulence and strong attenuation environment. The addition of the APT system requires the introduction of additional classical optical signals, and our experiments have proved the feasibility of adding classical signals in the quantum key distribution system.
4. Numerical simulation of the system performance
As shown above, our system works well under the condition of $10.4$ m seawater and $13.26$ dB attenuation. In fact, our system can tolerate a much higher level of attenuation, and we now show this with numerical simulation. According to Eq. (5) in [27] and Beer-Lambert law, the overall transmittance can be written as
where $c$ is the attenuation coefficient of the seawater, and $l$ is the distance. $\eta _{opt}$ is the transmittance at the Bob end (including the internal transmittance of optical components and detector efficiency) and $\eta _{opt}$ is 9.59 dB as shown in Table 3. Then according to Eq. (10) and Eq. (11) in [27], the gain $Q_u$ and the error rate $E_u$ of signal state, and the gain $Q_v$ and the error rate $E_v$ of decoy state are given by where $u$ and $v$ are the mean photon numbers per pulse for the signal state and the decoy state, and $Q_0$ and $e_0$ are the gain and error rate of the vacuum state, respectively. $e_{det}$ is the error rate caused by the degradation of the polarization, including the influence of the channel and the optical elements, and $e_{det}$ is 1.5% in simulation without loss of generality.Table 3. System parameters of numerical simulation
According to Eq. (34), Eq. (35) and Eq. (37) in [27], the gain of the single photons $Q_1$ and error rate of the single photons $e_1$ are given by
and the secure key rate is given by Eq. (1) in [27] where $H_2(x)$ is the binary Shannon information function, and $q$ depends on the sifting rate (0.5 for BB84 protocol in our simulation). $f(E_u)$ is the bidirectional error correction efficiency.According to the method in [7], when the LDPC error correction is employed, $f(E_u)H_2 (E_u)$ can be calculated as follows
where $C$ is the code rate of LDPC. Therefore $f(E_u)H_2 (E_u)$ should be 1/4 for the reason that $C$ is 3/4 in our experiment and simulation. Here, we replace $1-C$ with $R$, thus Eq. (9) can be simplified to [7,10,27] In order to further enhance the security, we assume that all the check code information (1/3 of the sifted keys) sent through the classical channel is eavesdropped, and $R$ is 1/3 instead of 1/4. This may lead to a lower secure key rate but the keys are more secure.The random numbers for generating the Toeplitz matrix are produced by the same seeds in the PCs of Alice and Bob. This process does not require classical message interaction, while the information revealed by error checking is very few (8 bits for one error checking and privacy amplification in our system). Thus the information revealed by error checking and privacy amplification is not included into Eq. (11).
Next we show the simulation results of the performance of our system under different attenuations. The simulation parameters are taken from our experiments, as shown in Table 3. $F$ is the repetition frequency of the quantum signals.
For different types of water, the results of the secure key rates (per pulse) as a function of the distances are shown in Fig. 6.
Fig. 6. Secure key rates (per pulse) of our system in different type of water as a function of distance. $c$ is the attenuation coefficients of the seawater.
Figure 6 shows that our system can still produce secure keys in $300$ m Jerlov type I seawater, where the attenuation is 23.7 dB. In this case, the gain of the signal state is $4.0\times 10^{-4}$ and the error rate of the signal state $E_{u}$ is 2.15%, so the secure key rate $R_{SKR}$ (per pulse) is $9.1\times 10^{-6}$. As the method previously mentioned, our system will get a bit rate of sifted keys of 1200.1 bps and a bit rate of secure keys of 27.4 bps under this environment. In real post-processing and in simulation here, we adopt the LDPC error correction scheme instead of using the ideal error correction efficiency (1.16) directly. If we use the latter in Eq. (11), the bit rate of secure keys can reach 219.2 bps. The LDPC error correction scheme we adopted consumes more secure keys than the ideal case, but it represents the real situation. In the case of $300$ m Jerlov type I seawater, taking the attenuation of Bob’s optical setup $\eta _{opt}$ (9.59 dB) into account, our system can tolerate the overall attenuation $33.3$ dB.
An interesting thing is that the experimental result yields higher secure key rate per pulse than the numerical result in the same Jerlov type III seawater. The main reason is that the ratio of the gains of the signal state to the decoy state detected experimentally at the Bob end is not 8:1 although the mean photon numbers are 0.8 and 0.1 at the Alice end. The actual ratio of the gain of the signal state to the decoy state is close to 7:1 as shown in Fig. 5. According to Eq. (11), the secure key rate is higher when the ratio of the gains of the signal state to the decoy state is lower than the ratio of mean photon numbers, thus the secure key rate in the experiment is higher than the numerical result in the same seawater.
It is worth noting that the bit rate of secure keys $R_{BSKR}$ depends on the secure key rates (per pulse) $R_{SKR}$ in the experiment. For our system, the bit rate of secure keys is given by
where $F$ is the repetition frequency of the quantum signal, and $k_1$ is 0.8 because we use 20% of the sifted keys to do error estimation. $k_2$ is 0.5 because the ratio of the three states is 2:1:1 and only signal state is used to generate sifted keys in our system. $k_3$ is 0.75 because Alice will not send the quantum signal when sending the synchronization pulses in our system.5. Conclusion
We develop an underwater decoy-state quantum key distribution system, and perform the quantum key distribution experiment through a 10.4 m Jerlov type III seawater channel. The average error rate of the sifted keys is 1.55%, and the average bit rate of secure keys is 1.82 Kbps. Instead of using radio waves or electrical cables which are not fit for the underwater channel, the two ends of our system completely use optical signals to communicate, which makes the system able to work in the underwater environment. The system is controlled by FPGA, and can be easily integrated into watertight cabins to perform the field experiment. Furthermore, we prove that our system is able to tolerate the channel attenuation up to 23.7 dB, and therefore can be used in the 300-meter-long Jerlov type I clean seawater channel.
Appendix A: system factors in the experiment
Table 4. System factors in the Experiment.
Appendix B: experimental setups of the UWQKD system
Funding
National Natural Science Foundation of China (61575180, 61701464, 12005212); Fundamental Research Funds for the Central Universities (201861012, 202165008).
Acknowledgments
The authors thank Long-wen Zhou, Zhi-min Wang and Ya Xiao for their help in preparing the paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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