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Proof-of-principle demonstration of quantum key distribution with seawater channel: towards space-to-underwater quantum communication
Optics Communications 452 (2019) 220–226
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Proof-of-principle demonstration of quantum key distribution with seawater channel: towards space-to-underwater quantum communication Dong-Dong Li a,b ,1 , Qi Shen a,b ,1 , Wei Chen a,b , Yang Li a,b , Xuan Han a,b , Kui-Xing Yang a,b , Yu Xu a,b , Jin Lin a,b , Chao-Ze Wang a,b,c , Hai-Lin Yong a,b , Wei-Yue Liu a,b,c , Yuan Cao a,b ,∗, Juan Yin a,b , Sheng-Kai Liao a,b , Ji-Gang Ren a,b ,∗ a
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, China b CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai, 201315, China c Ningbo University, Ningbo, Zhejiang, 315211, China
ARTICLE
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Keywords: Quantum key distribution Quantum communication Free-space Seawater
ABSTRACT Quantum communication via seawater channels has attracted much attention. Here, we present the design and implementation of a prototype for space-to-underwater quantum key distribution (QKD). The influence of the fluctuating atmosphere–seawater interface is theoretically analyzed. The setup consists of a specially designed optical source working in the blue–green band and a receiving optical antenna with large field-of-view (FOV). To achieve a larger FOV, photomultipliers with 25 mm active area are adopted to measure the single photon signals. The proof-of-principle experiment of QKD was implemented in the laboratory with near-coast seawater to verify the performance of the devices. This study provides a first step towards space-to-underwater quantum communication.
1. Introduction Quantum communication provides a solution for communication that is secure against eavesdropping [1,2]. As opposed to classical communication, quantum communication embraces the unconditional security that is guaranteed by the fundamental laws of quantum mechanics [3]. Since it was first proposed in 1984 [4], quantum communication has experienced massive growth [5–9]. Fiber QKD has reached a level of maturity that is sufficient for commercial implementation [10]. Satellite-to-ground QKD has been demonstrated [11,12]. In addition, via satellite relay, the intercontinental transmission of images and video conferences has been realized with quantum key protection [13], pointing toward the formation of an ultra-long-distance global quantum network. As seawater occupies more than 70% of the Earth’s surface, more underwater exploration will occur in the future with the advancement of human technology, and the demand for communication using seawater channels will increase [14,15]. It is an interesting question whether quantum communication can be used to ensure the security of communication with seawater channels, especially when the security weakness of existing underwater communication methods is getting exposed [16]. Successful demonstration of quantum communication with
seawater channels will complete the global quantum communication network, covering any corner including underwater area on the earth. Numerous potential applications will arise in underwater exploration, rescue, mining, and scientific research. Currently, only a few studies have been performed to study quantum communication through seawater channels. Theoretical studies have shown that it is feasible to use polarized photons for quantum key distribution in seawater channels with a communication distance of approximately 100 m [17,18]. Ji et al. tested the transmission of polarized photons and entangled photons using an indoor water tank and found that these properties are maintained when passing through a seawater channel [19]. However, these studies have only focused on communication scenarios between underwater devices. Another important scenario is communication between a space platform and an underwater submersible, which is an indispensable component in the construction of a global quantum-secured network. This requires the efficient transmission of photons across the atmosphere–seawater interface, especially when this surface fluctuates, which is a huge challenge. In this study, we theoretically analyze the influence of sea surface fluctuations on the QKD error rate and the receiving efficiency for such
∗ Corresponding authors at: Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, China. E-mail addresses: [email protected] (Y. Cao), [email protected] (J.-G. Ren). 1 These two authors contributed equally to this work.
https://doi.org/10.1016/j.optcom.2019.07.037 Received 7 April 2019; Received in revised form 16 July 2019; Accepted 17 July 2019 Available online 19 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 452 (2019) 220–226
Fig. 1. The tilt angle of the sea surface at different wind speed.
Fig. 2. Schematic diagram of a photon being transmitted through the atmosphere– seawater interface.
channel characteristics. Then, we design a decoy-state QKD source, which works in the ‘‘blue–green’’ band, together with a receiving antenna with a large field-of-view (FOV), which is suitable for spaceto-underwater quantum communication. Furthermore, we verify the performance of the system with real seawater in an indoor testbed and successfully extract secure keys with a total attenuation of approximately 28 dB. Our study marks a first step toward space-to-underwater QKD.
[ ] 𝑡𝑝 = 2𝑠𝑖𝑛𝜃2 𝑐𝑜𝑠𝜃1 ∕ 𝑠𝑖𝑛(𝜃1 + 𝜃2 )𝑐𝑜𝑠(𝜃1 − 𝜃2 )
(2)
Here, 𝑡𝑠 and 𝑡𝑝 denote the amplitude transmittance of the 𝒔 component and 𝒑 component, respectively. When the linearly polarized photon |𝜙𝑖𝑛 ⟩ = 𝑐1 |𝑠⟩ + 𝑐2 |𝑝⟩ (wherein |𝑐1 |2 +|𝑐2 |2 = 1) passing through the√ interface, the output quantum state
2. Theoretical analysis of the influence of the fluctuating atmosphere–seawater interface
becomes |𝜙𝑜𝑢𝑡 ⟩ = (𝑐1 𝑡𝑠 |𝑠⟩ + 𝑐2 𝑡𝑝 |𝑝⟩)∕
|𝑐1 𝑡𝑠 |2 + |𝑐2 𝑡𝑝 |2 , with an error rate
2
2 2 2 2 of 𝑒𝑒𝑟𝑟𝑜𝑟 = | ⟨𝜙⟂ 𝑖𝑛 |𝜙𝑜𝑢𝑡 ⟩ | = |𝑐1 𝑐2 | (𝑡𝑠 −𝑡𝑝 ) ∕(|𝑐1 𝑡𝑠 | +|𝑐2 𝑡𝑝 | ). In non-normal incident conditions, the polarization of the photons will change except for 𝒔 and 𝒑. According to the BB84 protocol, the four polarization √ states of incident √ photons can be chosen as |𝑠⟩, |𝑝⟩, (|𝑠⟩ + |𝑝⟩)∕ 2, and (|𝑠⟩ − |𝑝⟩)∕ 2 (equivalent to |𝐻⟩, |𝑉 ⟩, |−45◦ ⟩ and |+45◦ ⟩). The quantum bit error rate introduced by interface refraction 𝑄𝐵𝐸𝑅𝑖𝑛𝑡 can be expressed as [1,23,24](details are shown in Appendix):
In ocean water, the sea surface always fluctuates due to the wind. Consistent with some theoretical models [20,21], limited to a small area, the fluctuating surface can be regarded as a titled plane. The titled angle of the slope is denoted 𝛽. Previous studies indicate that, at slow wind speeds (u < 14 m/s), the probability of the sea surface’s inclination 𝛽 is related to the wind speed u (m/s), as shown in Fig. 1 [22]. Fig. 1 shows the probability of the sea slope when the wind speed varies. We can see that, even if the wind speed is 0 m/s, the tilt range of the sea surface is generally within 5◦ . When the wind speed increases gradually, the sea surface fluctuations increase. When the wind speed is <5 m/s, the probability that the sea surface fluctuates within 8◦ is still relatively large (>50%). Photons are refracted into the seawater through the air–water interface and transmitted to the underwater receiving antenna. The fluctuation of the sea surface results in a dramatic change in the arrival angle of the photons (> 140 mrad). If we use a receiving system with a 100 μrad FOV [11], the same as that used in traditional free space quantum communication, the receiving efficiency can be estimated to be (100 μrad∕140 mrad)2 ≈ 63 dB. This attenuation is too high to permit quantum communication. Besides, it is necessary to take into account the beam broadening effect caused by sea surface fluctuations, which will increase the link attenuation further. Therefore, sea surface fluctuations will seriously affect the receiving efficiency. In addition, as shown in Fig. 2, sea surface fluctuations will cause changes in the output angle of the photons during the refraction process, and will change the polarization of the photon, and may increase the quantum communication error rate. Fortunately, we can treat this scenario in an equivalent way. The change in polarization caused by sea surface fluctuations is equivalent to the atmosphere–seawater interface being still while the incident angles of the photons vary. During the refraction, the expression 𝑛1 𝑠𝑖𝑛𝜃1 = 𝑛2 𝑠𝑖𝑛𝜃2 holds, where 𝑛1 and 𝑛2 donates the refractive index of atmosphere and the seawater respectively, and 𝜃1 and 𝜃2 donates the incident and output angle respectively. According to the Fresnel’s Law, we have: 𝑡𝑠 = 2𝑠𝑖𝑛𝜃2 𝑐𝑜𝑠𝜃1 ∕𝑠𝑖𝑛(𝜃1 + 𝜃2 )
𝑄𝐵𝐸𝑅𝑖𝑛𝑡 =
𝑅𝑤𝑟𝑜𝑛𝑔 𝑅𝑟𝑖𝑔ℎ𝑡 + 𝑅𝑤𝑟𝑜𝑛𝑔
=
(𝑡𝑠 − 𝑡𝑝 )2 4(𝑡2𝑠 + 𝑡2𝑝 )
(3)
Here 𝑅𝑟𝑖𝑔ℎ𝑡 and 𝑅𝑤𝑟𝑜𝑛𝑔 are the detection probability of the right bits and the wrong bits, respectively. Notice that we mainly focus on the interface diffraction processing here while we did not take into consider the link loss (such as the atmospheric turbulence and underwater transmission loss), or the detector characteristics (such as dark count rate). The refractive index of the atmosphere is approximated as 𝑛1 = 1. The refractive index of the seawater depends on the temperature, the salinity, and the concentration of suspended solids and is numerically close to 𝑛2 = 1.34 [19]. As Fig. 3 shows, we can calculate the QBER introduced by the interface refraction at different incident angles. It is clear that QBER is very small when the incident angle is less than 80◦ . In addition, the maximum error rate does not exceed 1%. Therefore, the QBER induced by the fluctuating interface is within the tolerance level for quantum communication [8,25]. The interface fluctuation imposes very strict requirements on the FOV of the receiving system. We need to reconsider the design of the receiving device to overcome the variation in the arrival angle of the photons. Fortunately, the QBER induced during refraction is within an ignorable magnitude when the sea surface fluctuates. 3. Experimental implementation 3.1. The decoy-state QKD source in the ‘‘blue–green’’ region In a typical scenario of space-to-underwater QKD, the photons leave the LEO satellite and experience an effective atmosphere of approximately 10 km and then propagate in the seawater for a distance of
(1) 221
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Fig. 3. The QBER introduced by the fluctuation of the atmosphere–seawater interface.
Table 1 The loss induced by absorption. 𝜆 (nm) 𝛼𝑎𝑡𝑚 (km−1 ) 𝐿𝑎𝑡𝑚 of 10 km (dB) 𝛼𝑠𝑒𝑎 (m−1 ) 𝐿𝑠𝑒𝑎 of 50 m 𝐿𝑡𝑜𝑡𝑎𝑙 (dB)
1550 0.0149 0.65 960 [26] ≈∞ ≈∞
850 0.0390 1.69 4.06 [27] 881.62 883,31
532 0.0825 3.58 0.046 [28] 9.99 13.57
480 0.0973 4.22 0.019 [28] 4.13 8.35
405 0.1277 5.54 0.022 [28] 4.78 10.32
approximately 50 m. The transmitting windows of the atmosphere and the seawater are very different, approximately 850 nm or 1550 nm for the former and in the ‘‘blue–green’’ region for the later [26]. Table 1 shows the loss induced by absorption where and represent the absorption coefficients of the atmosphere and seawater, respectively. As Table 1 illustrates, photons of 1550 nm or 850 nm are not suitable for quantum communication in the space-to- underwater scenario due to their significant losses. The loss of photons in the visible range is relatively low. In this experiment, considering the maturity of current technology, 532 nm is selected as the working wavelength to provide better performance and stability at lower cost. Fig. 4 shows a schematic of our source. To extract the security key, it is necessary to improve the signal-to-noise ratio (SNR). Narrow spectral filtering is vital and is commonly adopted. The SNR increases as the filtering linewidth decreases. However, the filtering linewidth is limited by that of the optical source. For example, it is required that the source linewidth be within 0.001 nm when an atomic filter is used [29]. In this study, the laser is aligned to operate in continuous-wave mode to reduce the spectral linewidth and improve the power stability. Highspeed pulses are acquired via external modulations. We adapted a commercial solid-state laser with a central wavelength of 532.044 nm and a narrow linewidth of approximately 5 × 10−6 nm. Compared to the 0.1-nm linewidth of the Micius QKD optical source [11,13] and the 0.036-nm linewidth of fiber QKD devices [30], the linewidth of our source is reduced by approximately 5 orders and 4 orders of magnitude, respectively. Narrow spectral linewidth is important for space-to-underwater QKD because it allows us to sharply decrease the filtering bandwidth to get rid of larger amounts of background noise. The output photons pass through an attenuator to control the intensity and then pass through a polarizer to purify the polarization so that the polarization of the output laser is vertical (V). The polarizers of Pol4, Pol3, and Pol2 are set to vertical, vertical, and horizontal (H), respectively. The rotation angles of the Pockels cells, POC4 and POC3, are both set to 45◦ . When a high voltage, which is generated via homemade electronics [31], is not applied to POC4, the polarization remains V and the beam can pass through Pol2. When a high voltage is applied, the polarization changes to H and the beam will be cut off by Pol2. When a high voltage is applied to POC3, the polarization changes to H and the laser beam can pass through Pol1. When a high voltage is not applied, the polarization remains V and the beam will be cut off
Fig. 4. (a) A schematic of the optical source. An adjustable attenuator is used to modulate the intensity to the single photon level. A polarizer (Pol4) is used to purify the polarization state of the laser, which ensures that the polarization of the output state is vertical (V). M indicates the mirrors. Three polarizers (Pol3, Pol2, and Pol1) and two Pockels cells (POC4 and POC3) are used to modulate the pulsed CW laser. A polarizer (Pol0) and one Pockels cell (POC2) are used to modulate the intensity of the pulse according to the decoy-state scheme. Two Pockels cells (POC1 and POC0) are used to prepare the output state to be one of the four polarization states: |𝐻⟩, |𝑉 ⟩, |+45◦ ⟩ and |−45◦ ⟩. (b) A picture of the optical source.
by Pol1. Therefore, we can obtain a pulsed laser by applying a high voltage to POC3 and POC4 with a controllable time delay, where the voltage signal of POC3/POC4 serves as a start/stop signal. In the experiment, we set the time delay to be 30 ns, which means that the high voltage of POC3 increases 30 ns earlier than that of POC4. Fig. 5 shows the obtained pulsed signal. The full-width-at-halfmaximum (FWHM) of the pulse is 14 ns with a Gaussian fitting. The clock frequency is as high as 1 MHz. The extinction of the pulses exceeds 20 dB. In addition to the wavelength, it is necessary to find a trustworthy coding scheme with which the photons can be transferred with high fidelity. As with the theoretical analysis presented above, we adopted a polarization coding scheme, which is the primary coding scheme in free-space quantum communication. We use POC1 and POC0 to randomly generate one of the four polarization states, |𝐻⟩, |𝑉 ⟩, |+45◦ ⟩ or |−45◦ ⟩. Here, |𝐻⟩∕|𝑉 ⟩√ represents the horizontal/vertical √ polarization, |+45◦ ⟩ = (|𝐻⟩ + |𝑉 ⟩)∕ 2, and |−45◦ ⟩ = (|𝐻⟩ − |𝑉 ⟩)∕ 2. The overall polarization fidelity is over 99.1% for all the four polarization quantum states. We adopted the decoy state method [32–34] to overcome multiphoton loopholes and avoid the photon-number-splitting attack [35]. Here, we consider the ‘‘vacuum + decoy + signal’’ state scheme. An additional Pockels cell (POC2) is used to modulate the different intensity states. The rotation angle of POC2 is set to 15◦ . When a high voltage is not applied to POC2, the output state is the signal state. When a high voltage is applied, the output state is the decoy state. We obtain the vacuum state when there is no high voltage applied to POC4 or POC3. The intensity of the signal state is three times higher than the intensity of the decoy state. The mean photon numbers of the signal state, the decoy state, and the vacuum state are 0.6, 0.2, and 0, respectively. The 222
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Fig. 6. (a) A schematic of the receiving antenna. (b) A photo of the receiving antenna. (c) The test of the receiving antenna under 10 m of water.
Fig. 5. (a) The driving voltage for POC4 (the blue line) and POC3 (the purple line). (b) The observed pulse of the optical source . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. (a) The setup of the proof-of-principle experiment. OS indicates the 532-nm optical source, CA is the controllable attenuation, WT is a 0.5-m-long water tank made of glass, D stands for the detector, and TDC indicates the time-tagged signal measurement. (b) Photos of the indoor experiment.
probability of emitting the signal state, the decoy state, and the vacuum state are 50%, 25%, and 25%, respectively. The five Pockels cells are controlled by 4-bit random numbers, which are generated by a thermal noise device for each run to determine the polarization of the output photon and the intensity. Note that only one laser is used in our optical source, which avoids the inconsistencies of physical modes (such as time, spectrum, and space mode) in the multi-laser scheme and reduces the threat of side channels to the QKD security [36,37].
To improve the receiving efficiency, we use four lenses to receive the photons. At the focal plane of each lens, we align one PMT to detect the photons. Prior to each PMT, a polarizer is inserted for a polarization measurement. The directions of the four polarizers are set to 0◦ , 90◦ , 45◦ , and 135◦ to measure the polarization of |𝐻⟩, |𝑉 ⟩, |+45◦ ⟩ or |−45◦ ⟩ respectively. For the equipment to work normally underwater, we use an integrated seal design for the optical part of the receiver, including both the optical elements and the detectors. The optical part is put in a watersealed box. The front of the optical seal box is made of 10-mm-thick optical window glass, sealed with a 5-mm-thick rubber gasket with a sealant in the gaps. The cover of the optical box is also sealed with a 5mm-thick rubber gasket. We tested the sealability of the box at different water depths in the East China Sea (30◦ 43′ 17.09′′ N 122◦ 47′ 01.68′′ E). The equipment works normally even when placed at a water depth of 10 m. However, the electronic part of the receiver is not water-sealed, including the control devices, the Time-to-Digital Converter (TDC) and the power supply. The signal cables are connected through a 15-meter green plastic pipe, which can be recognized in part (c) of Fig. 6.
3.2. The receiving antenna with large FOV Fig. 6 shows a schematic of the receiving antenna. The FOV depends on the active area of the detector and the focal length of the receiving system. To enlarge the FOV, we need to increase the active area of the detector and reduce the focal length of the receiving system. In the experiment, we adopted a photomultiplier (PMT) (Hamamatsu CH326) with a large photosensitive surface for quantum signal detection. The diameter of the photosensitive area is 25 mm (currently the highest), and the detection efficiency is approximately 8% at 532 nm. The lens we used is 75 mm in diameter with a focal length of 175 mm. The photons are directly focused on the target surface of the detector without fiber coupling. Therefore, we obtain a FOV as high as 143 mrad (8◦ ), which covers the arrival angle of the photons with a probability of more than 50% up to a wind speed of 5 m/s. This FOV is approximately 3 orders of magnitude higher than the 100 μrad commonly used in free-space quantum communications [11]. This scale of FOV can help reduce the requirement for alignment accuracy and establish optical links between space platforms and underwater submersibles. For example, an accuracy of 0.5◦ is possible for satellite orbital predictions [11]. Because the receiving FOV is much larger than the accuracy, the receiving antenna can easily point to the satellite according to the predicted orbital data to establish a communication link without a complicated and expensive Acquiring– Pointing–Tracking (APT) system. In addition, the receiving aperture is equivalent to the caliber of the current classic underwater optical communication system [14,15], which ensures a high receiving efficiency.
4. Proof-of-principle experimental realization We conducted an indoor quantum key distribution experiment to verify the functionality of the system. We custom-made a 350 mm × 350 mm × 500 mm sink to house the seawater and built an indoor link. We expanded the spot from 3 mm to 150 mm to simulate beam broadening during satellite-submarine quantum communication and ensured that the spot covered the receiving lenses. We achieved a proof-of-principle experimental realization of underwater quantum key distribution using a water tank. The entire setup is as follows (see Fig. 7). The photons are attenuated by a controllable attenuator then passed through a water tank, which was 0.5 m long and full of seawater. We 223
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Table 2 The polarization fidelity of the four quantum states. The average fidelity is about 98.35%. |𝐻⟩ 98.57%
Polarization Fidelity
|𝑉 ⟩ 97.28%
|+45◦ ⟩ 98.70%
5. Discussion There is still a lot of work to be done to increase the secure key rate, even with the current setup. First of all, by adopting the efficient QKD scheme [39] instead of the unbiased situation in the proof-of-principle experiment, the sifting efficiency can be increased from nearly 50% to approaching 100%. Thus, the secure key rate will be greatly increased. Secondly, the secure key rate is decreased more than 40% due to the finite key size effect in our experiment. By increasing the experimental time to get a larger data block, we can greatly reduce the finite key size effect and increase the secure key rate. In addition, the secure key rate and the quantum communication distance can be extended by optimizing the parameters and the data processing [40]. There is much room for the system updating to improve the system performance. First, our system’s operating frequency is 1 MHz, which is much lower than the 100 MHz of the Micius satellite [11] and the 1 GHz of fiber devices [30]. Increasing the frequency can effectively increase the bit rate. Second, reducing the pulse width can effectively reduce the coincident time window. In combination with ultra-narrow spectral filtering techniques, the signal-to-noise ratio can be effectively improved. Compared with the filter components with the bandwidth of several nanometers in many free-space QKD experiments [11,41], we can adopt more high-precision filtering techniques, (such as atomic filtering [42]), to further reduce the filter spectral width by 2 ∼ 3 orders of magnitude to strictly control the background noise. We would like to mention that the optical source used in our experiment is not phase randomized. Phase randomization is one of the important assumptions in decoy-state QKD [39]. Without phase randomization, the security would be compromised [43]. Here we generate the optical pulse with external modulators from a CW laser, thus the relative phase between pulses is usually regarded as correlated. For a proof-of-principle experimental test, this can be treated just as a minor flaw and does not affect the final conclusion. In the updated system, this problem should be settled by active phase randomization [44]. Recently, measurement-device-independent quantum key distribution based on an air–water channel has been proposed and theoretically analyzed [45]. As the measurement-device-independent (MDI) protocol can remove all detector side channels [46,47], MDI-QKD over the seawater channel may attract more attention both theoretically and experimentally in the future.
|−45◦ ⟩ 98.86%
Table 3 The measured and derived parameters of the QKD experiment. Parameters
Value
Parameters
Value
𝑄𝜇 𝑄𝜈 𝑌0 𝐸𝜇 𝐸𝜈 𝑓 𝜖𝑠𝑒𝑐
5.06 × 10−4 2.01 × 10−4 9.90 × 10−6 3.74% 5.64% 1.10 10−9
Raw key size Acquisition time Raw key rate Secure key size without finite-size effect Secure key rate without finite-size effect Secure key size with finite-size effect Secure key rate with finite-size effect
86 632 bits 705 s 122.88 bps 8289 bits 11.76 bps 4805 bits 6.82 bps
adopted four photomultiplier tubes (PMT), from Hamamatsu, to detect the receiving photons. The diameter of the active area is approximately 25.4 cm. The detection efficiency is approximately 8% at 532 nm. With this setup, the entire attenuation of the link is approximately 28 dB. The 0.5-m long seawater tank contributes 5 dB of attenuation. The detector efficiency contributes 11 dB. The remaining 12 dB is due to geometric attenuation and the transmittance of the receiving antenna. Notice that we have found out that the secure key cannot be extracted when the attenuation exceeds 30 dB with our devices. Thus we choose to perform QKD at the attenuation of 28 dB, which is near to the maximum tolerable limit. The average polarization fidelity of the entire link is more than 98%, as illustrated in Table 2. In our QKD experiment, we extracted the final secure key from the raw data following the standard decoy BB84 post-processing procedure. In our algorithm, only the signal state is used for secure key generation, while the decoy state is used for parameter estimation. The lower bound of the final key rate per pulse is given by 𝑅𝑝𝑢𝑙𝑠𝑒 ≥ 𝑞𝑝𝜇 {−𝑄𝜇 𝑓 (𝐸𝜇 )𝐻2 (𝐸𝜇 ) + 𝑄1 [1 − 𝐻2 (𝑒1 )]}
(4)
where 𝑞 = 1∕2 is the basis reconciliation factor. 𝑝𝜇 is the probability of the emitting signal states. 𝑄𝜇 and 𝐸𝜇 are the gain and the error rate of the signal states, respectively. 𝑓 is the error correction efficiency. 𝐻2 (𝑥) = −𝑥𝑙𝑜𝑔2 (𝑥) − (1 − 𝑥)𝑙𝑜𝑔2 (1 − 𝑥) is the binary Shannon entropy function. 𝑄1 and 𝑒1 are the gain and the phase error rate, respectively, when the source generates single-photon states. The low-density paritycheck (LDPC) code was used for error correction. The check matrix is selected according to the bit error rate, which is estimated by sampling the sifted key. The errors are corrected with Log-likelihood Ratio Belief Propagation algorithm. The process iterates until the CRC check passed or reaching the maximum number of cycles. We ran our QKD system with real seawater samples collected near the coast of the Gouqi Island located in the East China Sea (30◦ 43′ 17.09′′ N 122◦ 47′ 01.68′′ E).Table 3 shows the measured and derived parameters for the experiment. 𝑄𝜇 and 𝑄𝜈 represent the gains of the signal and decoy states, respectively. 𝑌0 is the measured yield of the vacuum state. 𝐸𝜇 is the observed error rate of the signal state, while 𝐸𝜈 is the observed error rate of the decoy state. The experiment lasted for 705 s. The total accumulated raw key size was 86632 bits. The average raw key rate was approximately 123 bps with an error rate of 3.74%. In the experiment, we obtained a secure key rate of 11.76 bps in the asymmetric scenario. The secure key size is approximately 8.29 kb for an experimental time of more than 705 s. Next, we take the finite size effect into consideration. We estimated the bounds for 𝑄𝐿 1 and 𝑒𝑈 following the analysis introduced in Ref. [38] by considering 1 the statistical fluctuations for the vacuum states and the gains for the signal states and the decoy states within six standard deviations, which ensures final key rates with a secure parameter of 𝜖𝑠𝑒𝑐 = 10−9 . The secure key rate is approximately 6.82 bps with a total secure key of 4805 bits.
6. Conclusions We discussed the requirements of space-to-underwater quantum communication, theoretically analyzed the fluctuating interface effects, and found that the interface has a large influence on the receiving efficiency but only a small influence on the communication error rate. We developed a ‘‘blue–green’’ band decoy-state optical source for space-to-underwater quantum communication with the repetition rate of up to 1 MHz and the pulse width of 14 ns. We designed a receiving antenna with a large FOV of up to 143 mrad. We verified the system performance through 0.5-meter seawater in the laboratory and successfully implemented quantum key distribution based on near-coast seawater channels. The accumulated secure key size is approximately 4.8 kb. The secure key rate is approximately 6.82 bps with an error rate of 3.74%. Our work provides a first step towards space-to-underwater QKD. Acknowledgments We thank many of our colleagues for experimental assistance and delightful discussions, especially Wen-Zhuo Zhang, Fei Zhou and FengZhi Li. 224
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Funding
Similarly, for the input state
National Natural Science Foundation of China(NSFC) (Grant No. 11705191, U1738142, 11654005); National Key Research and Development (R&D) Plan of China (Grant No. 2017YFA0303900); Anhui Provincial Natural Science Foundation, China (Grant No. 1808085QF180); Natural Science Foundation of Shanghai, China (Grant No. 18ZR1443600); Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F050004); Shanghai Sailing Program, China (Grant No. 18YF1425100). S.-K. Liao and Y. Cao were supported by the Youth Innovation Promotion Association of CAS, China (2017500; 2018492); CAS Key Technology Talent Program, China.
𝑅𝑟𝑖𝑔ℎ𝑡 (
𝑄𝐵𝐸𝑅𝑖𝑛𝑡 =
𝑁𝑟𝑖𝑔ℎ𝑡 + 𝑁𝑤𝑟𝑜𝑛𝑔
=
𝑅𝑟𝑖𝑔ℎ𝑡 + 𝑅𝑤𝑟𝑜𝑛𝑔
=
(A.10)
(𝑡𝑠 − 𝑡𝑝 )2 2(𝑡𝑠 2 + 𝑡𝑝 2 )
Then, the quantum bit error rate introduced by interface refraction can be expressed as:
(A.1)
Here 𝑁𝑟𝑖𝑔ℎ𝑡 and 𝑁𝑤𝑟𝑜𝑛𝑔 are the detection counts of the right bits and the wrong bits, respectively. 𝑅𝑟𝑖𝑔ℎ𝑡 and 𝑅𝑤𝑟𝑜𝑛𝑔 are the detection probability of the right bits and the wrong bits, respectively. Obviously, we have
𝑄𝐵𝐸𝑅𝑖𝑛𝑡 =
𝑅𝑤𝑟𝑜𝑛𝑔 = 𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑠⟩) + 𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑝⟩)+ |𝑠⟩ + |𝑝⟩ |𝑠⟩ − |𝑝⟩ 𝑅𝑤𝑟𝑜𝑛𝑔 ( √ ) + 𝑅𝑤𝑟𝑜𝑛𝑔 ( √ ) 2 2
=
𝑅𝑟𝑖𝑔ℎ𝑡 = 𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑠⟩) + 𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑝⟩) |𝑠⟩ + |𝑝⟩ |𝑠⟩ − |𝑝⟩ + 𝑅𝑟𝑖𝑔ℎ𝑡 ( √ ) + 𝑅𝑟𝑖𝑔ℎ𝑡 ( √ ) 2 2
(𝑡𝑠 + 𝑡𝑝 )2
2(𝑡𝑠 2 + 𝑡𝑝 2 ) |𝑠⟩ − |𝑝⟩ |𝑠⟩ − |𝑝⟩ ⟂ |𝑠⟩ + |𝑝⟩ 2 𝑅𝑤𝑟𝑜𝑛𝑔 ( √ ) = |⟨( √ ) | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| 2 2 2 𝑡𝑠 |𝑠⟩ − 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ + |𝑝⟩ = |⟨ √ |⋅| √ ⟩| 2 𝑡𝑠 2 + 𝑡𝑝 2
Here we show the details to derive Eq. (3). The quantum bit error rate introduced by interface refraction 𝑄𝐵𝐸𝑅𝑖𝑛𝑡 is defined as the proportion of the wrong bits to the total number of the received photons [1,23,24]: 𝑅𝑤𝑟𝑜𝑛𝑔
we have
|𝑠⟩ − |𝑝⟩ |𝑠⟩ − |𝑝⟩ |𝑠⟩ − |𝑝⟩ 2 ) = |⟨ √ | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| √ 2 2 2 𝑡𝑠 |𝑠⟩ − 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ − |𝑝⟩ = |⟨ √ |⋅| √ ⟩| 2 𝑡𝑠 2 + 𝑡𝑝 2 =
Appendix
𝑁𝑤𝑟𝑜𝑛𝑔
|𝑠⟩−|𝑝⟩ √ , 2
𝑅𝑤𝑟𝑜𝑛𝑔 𝑅𝑟𝑖𝑔ℎ𝑡 + 𝑅𝑤𝑟𝑜𝑛𝑔 (
0+0+
(A.2)
(
1+0+1+0+
= (A.3)
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 𝑡2𝑠 +𝑡2𝑝
( )2 𝑡𝑠 − 𝑡𝑝 ( ) 4 𝑡2𝑠 + 𝑡2𝑝
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(
+
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 𝑡2𝑠 +𝑡2𝑝
(
+
)2 𝑡 +𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(
+
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(
+
)2 𝑡 +𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(A.11)
√
( )2 𝑛 𝑠𝑖𝑛𝜃 1 − 1 𝑛 1 − 𝑛1 𝑠𝑖𝑛2 𝜃1 ∕𝑛2 )2 2 = √ 𝑛1 𝑠𝑖𝑛𝜃1 2 4[1 + (𝑐𝑜𝑠𝜃1 1 − ( 𝑛 ) + 𝑛1 𝑠𝑖𝑛2 𝜃1 ∕𝑛2 )2 ] (1 − 𝑐𝑜𝑠𝜃1
For the input state |𝑠⟩, the output state is |𝜙𝑜𝑢𝑡 (𝑠)⟩ = |𝑠⟩. So the right detection rate can be written as
2
𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑠⟩) = |⟨𝑠| ⋅ |𝜙𝑜𝑢𝑡 (𝑠)⟩|2 = |⟨𝑠| ⋅ |𝑠⟩|2 = 1
(A.4)
During the last step, we have combined with Eq. (1), Eq. (2) and the equation of 𝑛1 𝑠𝑖𝑛𝜃1 = 𝑛2 𝑠𝑖𝑛𝜃2 .
And the wrong detection rate can be written as 2
𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑠⟩) = |⟨𝑠⟂ | ⋅ |𝜙𝑜𝑢𝑡 (𝑠)⟩| = |⟨𝑝| ⋅ |𝑠⟩|2 = 0
(A.5) References
Similarly, for the input state |𝑝⟩, we have 𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑝⟩) = |⟨𝑝| ⋅ |𝜙𝑜𝑢𝑡 (𝑝)⟩|2 = |⟨𝑝| ⋅ |𝑝⟩|2 = 1 𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑝⟩) = |⟨𝑝⟂| ⋅ |𝜙𝑜𝑢𝑡 (𝑝)⟩|2 = |⟨𝑠| ⋅ |𝑝⟩|2 = 0 For the input state |𝜙𝑜𝑢𝑡 (
|𝑠⟩+|𝑝⟩ √ , 2
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(A.6)
the output state is
𝑡𝑠 |𝑠⟩ + 𝑡𝑝 |𝑝⟩ |𝑠⟩ + |𝑝⟩ )⟩ = √ . √ 2 𝑡𝑠 2 + 𝑡𝑝 2
(A.7)
So the right detection rate can be written as 𝑅𝑟𝑖𝑔ℎ𝑡 (
|𝑠⟩ + |𝑝⟩ |𝑠⟩ + |𝑝⟩ |𝑠⟩ + |𝑝⟩ 2 ) = |⟨ √ | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| √ 2 2 2 𝑡𝑠 |𝑠⟩ + 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ + |𝑝⟩ ⟩| = |⟨ √ |⋅| √ 2 𝑡𝑠 2 + 𝑡𝑝 2 =
(A.8)
(𝑡𝑠 + 𝑡𝑝 )2 2(𝑡𝑠 2 + 𝑡𝑝 2 )
And the wrong detection rate can be written as 𝑅𝑤𝑟𝑜𝑛𝑔 (
|𝑠⟩ + |𝑝⟩ |𝑠⟩ + |𝑝⟩ ⟂ |𝑠⟩ + |𝑝⟩ 2 ) = |⟨( √ ) | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| √ 2 2 2 𝑡𝑠 |𝑠⟩ + 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ − |𝑝⟩ = |⟨ √ |⋅| √ ⟩| 2 𝑡𝑠 2 + 𝑡𝑝 2 =
(A.9)
(𝑡𝑠 − 𝑡𝑝 )2 2(𝑡𝑠 2 + 𝑡𝑝 2 ) 225
D.-D. Li, Q. Shen, W. Chen et al.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Proof-of-principle demonstration of quantum key distribution with seawater channel: towards space-to-underwater quantum communication Dong-Dong Li a,b ,1 , Qi Shen a,b ,1 , Wei Chen a,b , Yang Li a,b , Xuan Han a,b , Kui-Xing Yang a,b , Yu Xu a,b , Jin Lin a,b , Chao-Ze Wang a,b,c , Hai-Lin Yong a,b , Wei-Yue Liu a,b,c , Yuan Cao a,b ,∗, Juan Yin a,b , Sheng-Kai Liao a,b , Ji-Gang Ren a,b ,∗ a
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, China b CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai, 201315, China c Ningbo University, Ningbo, Zhejiang, 315211, China
ARTICLE
INFO
Keywords: Quantum key distribution Quantum communication Free-space Seawater
ABSTRACT Quantum communication via seawater channels has attracted much attention. Here, we present the design and implementation of a prototype for space-to-underwater quantum key distribution (QKD). The influence of the fluctuating atmosphere–seawater interface is theoretically analyzed. The setup consists of a specially designed optical source working in the blue–green band and a receiving optical antenna with large field-of-view (FOV). To achieve a larger FOV, photomultipliers with 25 mm active area are adopted to measure the single photon signals. The proof-of-principle experiment of QKD was implemented in the laboratory with near-coast seawater to verify the performance of the devices. This study provides a first step towards space-to-underwater quantum communication.
1. Introduction Quantum communication provides a solution for communication that is secure against eavesdropping [1,2]. As opposed to classical communication, quantum communication embraces the unconditional security that is guaranteed by the fundamental laws of quantum mechanics [3]. Since it was first proposed in 1984 [4], quantum communication has experienced massive growth [5–9]. Fiber QKD has reached a level of maturity that is sufficient for commercial implementation [10]. Satellite-to-ground QKD has been demonstrated [11,12]. In addition, via satellite relay, the intercontinental transmission of images and video conferences has been realized with quantum key protection [13], pointing toward the formation of an ultra-long-distance global quantum network. As seawater occupies more than 70% of the Earth’s surface, more underwater exploration will occur in the future with the advancement of human technology, and the demand for communication using seawater channels will increase [14,15]. It is an interesting question whether quantum communication can be used to ensure the security of communication with seawater channels, especially when the security weakness of existing underwater communication methods is getting exposed [16]. Successful demonstration of quantum communication with
seawater channels will complete the global quantum communication network, covering any corner including underwater area on the earth. Numerous potential applications will arise in underwater exploration, rescue, mining, and scientific research. Currently, only a few studies have been performed to study quantum communication through seawater channels. Theoretical studies have shown that it is feasible to use polarized photons for quantum key distribution in seawater channels with a communication distance of approximately 100 m [17,18]. Ji et al. tested the transmission of polarized photons and entangled photons using an indoor water tank and found that these properties are maintained when passing through a seawater channel [19]. However, these studies have only focused on communication scenarios between underwater devices. Another important scenario is communication between a space platform and an underwater submersible, which is an indispensable component in the construction of a global quantum-secured network. This requires the efficient transmission of photons across the atmosphere–seawater interface, especially when this surface fluctuates, which is a huge challenge. In this study, we theoretically analyze the influence of sea surface fluctuations on the QKD error rate and the receiving efficiency for such
∗ Corresponding authors at: Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, China. E-mail addresses: [email protected] (Y. Cao), [email protected] (J.-G. Ren). 1 These two authors contributed equally to this work.
https://doi.org/10.1016/j.optcom.2019.07.037 Received 7 April 2019; Received in revised form 16 July 2019; Accepted 17 July 2019 Available online 19 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
D.-D. Li, Q. Shen, W. Chen et al.
Optics Communications 452 (2019) 220–226
Fig. 1. The tilt angle of the sea surface at different wind speed.
Fig. 2. Schematic diagram of a photon being transmitted through the atmosphere– seawater interface.
channel characteristics. Then, we design a decoy-state QKD source, which works in the ‘‘blue–green’’ band, together with a receiving antenna with a large field-of-view (FOV), which is suitable for spaceto-underwater quantum communication. Furthermore, we verify the performance of the system with real seawater in an indoor testbed and successfully extract secure keys with a total attenuation of approximately 28 dB. Our study marks a first step toward space-to-underwater QKD.
[ ] 𝑡𝑝 = 2𝑠𝑖𝑛𝜃2 𝑐𝑜𝑠𝜃1 ∕ 𝑠𝑖𝑛(𝜃1 + 𝜃2 )𝑐𝑜𝑠(𝜃1 − 𝜃2 )
(2)
Here, 𝑡𝑠 and 𝑡𝑝 denote the amplitude transmittance of the 𝒔 component and 𝒑 component, respectively. When the linearly polarized photon |𝜙𝑖𝑛 ⟩ = 𝑐1 |𝑠⟩ + 𝑐2 |𝑝⟩ (wherein |𝑐1 |2 +|𝑐2 |2 = 1) passing through the√ interface, the output quantum state
2. Theoretical analysis of the influence of the fluctuating atmosphere–seawater interface
becomes |𝜙𝑜𝑢𝑡 ⟩ = (𝑐1 𝑡𝑠 |𝑠⟩ + 𝑐2 𝑡𝑝 |𝑝⟩)∕
|𝑐1 𝑡𝑠 |2 + |𝑐2 𝑡𝑝 |2 , with an error rate
2
2 2 2 2 of 𝑒𝑒𝑟𝑟𝑜𝑟 = | ⟨𝜙⟂ 𝑖𝑛 |𝜙𝑜𝑢𝑡 ⟩ | = |𝑐1 𝑐2 | (𝑡𝑠 −𝑡𝑝 ) ∕(|𝑐1 𝑡𝑠 | +|𝑐2 𝑡𝑝 | ). In non-normal incident conditions, the polarization of the photons will change except for 𝒔 and 𝒑. According to the BB84 protocol, the four polarization √ states of incident √ photons can be chosen as |𝑠⟩, |𝑝⟩, (|𝑠⟩ + |𝑝⟩)∕ 2, and (|𝑠⟩ − |𝑝⟩)∕ 2 (equivalent to |𝐻⟩, |𝑉 ⟩, |−45◦ ⟩ and |+45◦ ⟩). The quantum bit error rate introduced by interface refraction 𝑄𝐵𝐸𝑅𝑖𝑛𝑡 can be expressed as [1,23,24](details are shown in Appendix):
In ocean water, the sea surface always fluctuates due to the wind. Consistent with some theoretical models [20,21], limited to a small area, the fluctuating surface can be regarded as a titled plane. The titled angle of the slope is denoted 𝛽. Previous studies indicate that, at slow wind speeds (u < 14 m/s), the probability of the sea surface’s inclination 𝛽 is related to the wind speed u (m/s), as shown in Fig. 1 [22]. Fig. 1 shows the probability of the sea slope when the wind speed varies. We can see that, even if the wind speed is 0 m/s, the tilt range of the sea surface is generally within 5◦ . When the wind speed increases gradually, the sea surface fluctuations increase. When the wind speed is <5 m/s, the probability that the sea surface fluctuates within 8◦ is still relatively large (>50%). Photons are refracted into the seawater through the air–water interface and transmitted to the underwater receiving antenna. The fluctuation of the sea surface results in a dramatic change in the arrival angle of the photons (> 140 mrad). If we use a receiving system with a 100 μrad FOV [11], the same as that used in traditional free space quantum communication, the receiving efficiency can be estimated to be (100 μrad∕140 mrad)2 ≈ 63 dB. This attenuation is too high to permit quantum communication. Besides, it is necessary to take into account the beam broadening effect caused by sea surface fluctuations, which will increase the link attenuation further. Therefore, sea surface fluctuations will seriously affect the receiving efficiency. In addition, as shown in Fig. 2, sea surface fluctuations will cause changes in the output angle of the photons during the refraction process, and will change the polarization of the photon, and may increase the quantum communication error rate. Fortunately, we can treat this scenario in an equivalent way. The change in polarization caused by sea surface fluctuations is equivalent to the atmosphere–seawater interface being still while the incident angles of the photons vary. During the refraction, the expression 𝑛1 𝑠𝑖𝑛𝜃1 = 𝑛2 𝑠𝑖𝑛𝜃2 holds, where 𝑛1 and 𝑛2 donates the refractive index of atmosphere and the seawater respectively, and 𝜃1 and 𝜃2 donates the incident and output angle respectively. According to the Fresnel’s Law, we have: 𝑡𝑠 = 2𝑠𝑖𝑛𝜃2 𝑐𝑜𝑠𝜃1 ∕𝑠𝑖𝑛(𝜃1 + 𝜃2 )
𝑄𝐵𝐸𝑅𝑖𝑛𝑡 =
𝑅𝑤𝑟𝑜𝑛𝑔 𝑅𝑟𝑖𝑔ℎ𝑡 + 𝑅𝑤𝑟𝑜𝑛𝑔
=
(𝑡𝑠 − 𝑡𝑝 )2 4(𝑡2𝑠 + 𝑡2𝑝 )
(3)
Here 𝑅𝑟𝑖𝑔ℎ𝑡 and 𝑅𝑤𝑟𝑜𝑛𝑔 are the detection probability of the right bits and the wrong bits, respectively. Notice that we mainly focus on the interface diffraction processing here while we did not take into consider the link loss (such as the atmospheric turbulence and underwater transmission loss), or the detector characteristics (such as dark count rate). The refractive index of the atmosphere is approximated as 𝑛1 = 1. The refractive index of the seawater depends on the temperature, the salinity, and the concentration of suspended solids and is numerically close to 𝑛2 = 1.34 [19]. As Fig. 3 shows, we can calculate the QBER introduced by the interface refraction at different incident angles. It is clear that QBER is very small when the incident angle is less than 80◦ . In addition, the maximum error rate does not exceed 1%. Therefore, the QBER induced by the fluctuating interface is within the tolerance level for quantum communication [8,25]. The interface fluctuation imposes very strict requirements on the FOV of the receiving system. We need to reconsider the design of the receiving device to overcome the variation in the arrival angle of the photons. Fortunately, the QBER induced during refraction is within an ignorable magnitude when the sea surface fluctuates. 3. Experimental implementation 3.1. The decoy-state QKD source in the ‘‘blue–green’’ region In a typical scenario of space-to-underwater QKD, the photons leave the LEO satellite and experience an effective atmosphere of approximately 10 km and then propagate in the seawater for a distance of
(1) 221
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Fig. 3. The QBER introduced by the fluctuation of the atmosphere–seawater interface.
Table 1 The loss induced by absorption. 𝜆 (nm) 𝛼𝑎𝑡𝑚 (km−1 ) 𝐿𝑎𝑡𝑚 of 10 km (dB) 𝛼𝑠𝑒𝑎 (m−1 ) 𝐿𝑠𝑒𝑎 of 50 m 𝐿𝑡𝑜𝑡𝑎𝑙 (dB)
1550 0.0149 0.65 960 [26] ≈∞ ≈∞
850 0.0390 1.69 4.06 [27] 881.62 883,31
532 0.0825 3.58 0.046 [28] 9.99 13.57
480 0.0973 4.22 0.019 [28] 4.13 8.35
405 0.1277 5.54 0.022 [28] 4.78 10.32
approximately 50 m. The transmitting windows of the atmosphere and the seawater are very different, approximately 850 nm or 1550 nm for the former and in the ‘‘blue–green’’ region for the later [26]. Table 1 shows the loss induced by absorption where and represent the absorption coefficients of the atmosphere and seawater, respectively. As Table 1 illustrates, photons of 1550 nm or 850 nm are not suitable for quantum communication in the space-to- underwater scenario due to their significant losses. The loss of photons in the visible range is relatively low. In this experiment, considering the maturity of current technology, 532 nm is selected as the working wavelength to provide better performance and stability at lower cost. Fig. 4 shows a schematic of our source. To extract the security key, it is necessary to improve the signal-to-noise ratio (SNR). Narrow spectral filtering is vital and is commonly adopted. The SNR increases as the filtering linewidth decreases. However, the filtering linewidth is limited by that of the optical source. For example, it is required that the source linewidth be within 0.001 nm when an atomic filter is used [29]. In this study, the laser is aligned to operate in continuous-wave mode to reduce the spectral linewidth and improve the power stability. Highspeed pulses are acquired via external modulations. We adapted a commercial solid-state laser with a central wavelength of 532.044 nm and a narrow linewidth of approximately 5 × 10−6 nm. Compared to the 0.1-nm linewidth of the Micius QKD optical source [11,13] and the 0.036-nm linewidth of fiber QKD devices [30], the linewidth of our source is reduced by approximately 5 orders and 4 orders of magnitude, respectively. Narrow spectral linewidth is important for space-to-underwater QKD because it allows us to sharply decrease the filtering bandwidth to get rid of larger amounts of background noise. The output photons pass through an attenuator to control the intensity and then pass through a polarizer to purify the polarization so that the polarization of the output laser is vertical (V). The polarizers of Pol4, Pol3, and Pol2 are set to vertical, vertical, and horizontal (H), respectively. The rotation angles of the Pockels cells, POC4 and POC3, are both set to 45◦ . When a high voltage, which is generated via homemade electronics [31], is not applied to POC4, the polarization remains V and the beam can pass through Pol2. When a high voltage is applied, the polarization changes to H and the beam will be cut off by Pol2. When a high voltage is applied to POC3, the polarization changes to H and the laser beam can pass through Pol1. When a high voltage is not applied, the polarization remains V and the beam will be cut off
Fig. 4. (a) A schematic of the optical source. An adjustable attenuator is used to modulate the intensity to the single photon level. A polarizer (Pol4) is used to purify the polarization state of the laser, which ensures that the polarization of the output state is vertical (V). M indicates the mirrors. Three polarizers (Pol3, Pol2, and Pol1) and two Pockels cells (POC4 and POC3) are used to modulate the pulsed CW laser. A polarizer (Pol0) and one Pockels cell (POC2) are used to modulate the intensity of the pulse according to the decoy-state scheme. Two Pockels cells (POC1 and POC0) are used to prepare the output state to be one of the four polarization states: |𝐻⟩, |𝑉 ⟩, |+45◦ ⟩ and |−45◦ ⟩. (b) A picture of the optical source.
by Pol1. Therefore, we can obtain a pulsed laser by applying a high voltage to POC3 and POC4 with a controllable time delay, where the voltage signal of POC3/POC4 serves as a start/stop signal. In the experiment, we set the time delay to be 30 ns, which means that the high voltage of POC3 increases 30 ns earlier than that of POC4. Fig. 5 shows the obtained pulsed signal. The full-width-at-halfmaximum (FWHM) of the pulse is 14 ns with a Gaussian fitting. The clock frequency is as high as 1 MHz. The extinction of the pulses exceeds 20 dB. In addition to the wavelength, it is necessary to find a trustworthy coding scheme with which the photons can be transferred with high fidelity. As with the theoretical analysis presented above, we adopted a polarization coding scheme, which is the primary coding scheme in free-space quantum communication. We use POC1 and POC0 to randomly generate one of the four polarization states, |𝐻⟩, |𝑉 ⟩, |+45◦ ⟩ or |−45◦ ⟩. Here, |𝐻⟩∕|𝑉 ⟩√ represents the horizontal/vertical √ polarization, |+45◦ ⟩ = (|𝐻⟩ + |𝑉 ⟩)∕ 2, and |−45◦ ⟩ = (|𝐻⟩ − |𝑉 ⟩)∕ 2. The overall polarization fidelity is over 99.1% for all the four polarization quantum states. We adopted the decoy state method [32–34] to overcome multiphoton loopholes and avoid the photon-number-splitting attack [35]. Here, we consider the ‘‘vacuum + decoy + signal’’ state scheme. An additional Pockels cell (POC2) is used to modulate the different intensity states. The rotation angle of POC2 is set to 15◦ . When a high voltage is not applied to POC2, the output state is the signal state. When a high voltage is applied, the output state is the decoy state. We obtain the vacuum state when there is no high voltage applied to POC4 or POC3. The intensity of the signal state is three times higher than the intensity of the decoy state. The mean photon numbers of the signal state, the decoy state, and the vacuum state are 0.6, 0.2, and 0, respectively. The 222
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Fig. 6. (a) A schematic of the receiving antenna. (b) A photo of the receiving antenna. (c) The test of the receiving antenna under 10 m of water.
Fig. 5. (a) The driving voltage for POC4 (the blue line) and POC3 (the purple line). (b) The observed pulse of the optical source . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. (a) The setup of the proof-of-principle experiment. OS indicates the 532-nm optical source, CA is the controllable attenuation, WT is a 0.5-m-long water tank made of glass, D stands for the detector, and TDC indicates the time-tagged signal measurement. (b) Photos of the indoor experiment.
probability of emitting the signal state, the decoy state, and the vacuum state are 50%, 25%, and 25%, respectively. The five Pockels cells are controlled by 4-bit random numbers, which are generated by a thermal noise device for each run to determine the polarization of the output photon and the intensity. Note that only one laser is used in our optical source, which avoids the inconsistencies of physical modes (such as time, spectrum, and space mode) in the multi-laser scheme and reduces the threat of side channels to the QKD security [36,37].
To improve the receiving efficiency, we use four lenses to receive the photons. At the focal plane of each lens, we align one PMT to detect the photons. Prior to each PMT, a polarizer is inserted for a polarization measurement. The directions of the four polarizers are set to 0◦ , 90◦ , 45◦ , and 135◦ to measure the polarization of |𝐻⟩, |𝑉 ⟩, |+45◦ ⟩ or |−45◦ ⟩ respectively. For the equipment to work normally underwater, we use an integrated seal design for the optical part of the receiver, including both the optical elements and the detectors. The optical part is put in a watersealed box. The front of the optical seal box is made of 10-mm-thick optical window glass, sealed with a 5-mm-thick rubber gasket with a sealant in the gaps. The cover of the optical box is also sealed with a 5mm-thick rubber gasket. We tested the sealability of the box at different water depths in the East China Sea (30◦ 43′ 17.09′′ N 122◦ 47′ 01.68′′ E). The equipment works normally even when placed at a water depth of 10 m. However, the electronic part of the receiver is not water-sealed, including the control devices, the Time-to-Digital Converter (TDC) and the power supply. The signal cables are connected through a 15-meter green plastic pipe, which can be recognized in part (c) of Fig. 6.
3.2. The receiving antenna with large FOV Fig. 6 shows a schematic of the receiving antenna. The FOV depends on the active area of the detector and the focal length of the receiving system. To enlarge the FOV, we need to increase the active area of the detector and reduce the focal length of the receiving system. In the experiment, we adopted a photomultiplier (PMT) (Hamamatsu CH326) with a large photosensitive surface for quantum signal detection. The diameter of the photosensitive area is 25 mm (currently the highest), and the detection efficiency is approximately 8% at 532 nm. The lens we used is 75 mm in diameter with a focal length of 175 mm. The photons are directly focused on the target surface of the detector without fiber coupling. Therefore, we obtain a FOV as high as 143 mrad (8◦ ), which covers the arrival angle of the photons with a probability of more than 50% up to a wind speed of 5 m/s. This FOV is approximately 3 orders of magnitude higher than the 100 μrad commonly used in free-space quantum communications [11]. This scale of FOV can help reduce the requirement for alignment accuracy and establish optical links between space platforms and underwater submersibles. For example, an accuracy of 0.5◦ is possible for satellite orbital predictions [11]. Because the receiving FOV is much larger than the accuracy, the receiving antenna can easily point to the satellite according to the predicted orbital data to establish a communication link without a complicated and expensive Acquiring– Pointing–Tracking (APT) system. In addition, the receiving aperture is equivalent to the caliber of the current classic underwater optical communication system [14,15], which ensures a high receiving efficiency.
4. Proof-of-principle experimental realization We conducted an indoor quantum key distribution experiment to verify the functionality of the system. We custom-made a 350 mm × 350 mm × 500 mm sink to house the seawater and built an indoor link. We expanded the spot from 3 mm to 150 mm to simulate beam broadening during satellite-submarine quantum communication and ensured that the spot covered the receiving lenses. We achieved a proof-of-principle experimental realization of underwater quantum key distribution using a water tank. The entire setup is as follows (see Fig. 7). The photons are attenuated by a controllable attenuator then passed through a water tank, which was 0.5 m long and full of seawater. We 223
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Table 2 The polarization fidelity of the four quantum states. The average fidelity is about 98.35%. |𝐻⟩ 98.57%
Polarization Fidelity
|𝑉 ⟩ 97.28%
|+45◦ ⟩ 98.70%
5. Discussion There is still a lot of work to be done to increase the secure key rate, even with the current setup. First of all, by adopting the efficient QKD scheme [39] instead of the unbiased situation in the proof-of-principle experiment, the sifting efficiency can be increased from nearly 50% to approaching 100%. Thus, the secure key rate will be greatly increased. Secondly, the secure key rate is decreased more than 40% due to the finite key size effect in our experiment. By increasing the experimental time to get a larger data block, we can greatly reduce the finite key size effect and increase the secure key rate. In addition, the secure key rate and the quantum communication distance can be extended by optimizing the parameters and the data processing [40]. There is much room for the system updating to improve the system performance. First, our system’s operating frequency is 1 MHz, which is much lower than the 100 MHz of the Micius satellite [11] and the 1 GHz of fiber devices [30]. Increasing the frequency can effectively increase the bit rate. Second, reducing the pulse width can effectively reduce the coincident time window. In combination with ultra-narrow spectral filtering techniques, the signal-to-noise ratio can be effectively improved. Compared with the filter components with the bandwidth of several nanometers in many free-space QKD experiments [11,41], we can adopt more high-precision filtering techniques, (such as atomic filtering [42]), to further reduce the filter spectral width by 2 ∼ 3 orders of magnitude to strictly control the background noise. We would like to mention that the optical source used in our experiment is not phase randomized. Phase randomization is one of the important assumptions in decoy-state QKD [39]. Without phase randomization, the security would be compromised [43]. Here we generate the optical pulse with external modulators from a CW laser, thus the relative phase between pulses is usually regarded as correlated. For a proof-of-principle experimental test, this can be treated just as a minor flaw and does not affect the final conclusion. In the updated system, this problem should be settled by active phase randomization [44]. Recently, measurement-device-independent quantum key distribution based on an air–water channel has been proposed and theoretically analyzed [45]. As the measurement-device-independent (MDI) protocol can remove all detector side channels [46,47], MDI-QKD over the seawater channel may attract more attention both theoretically and experimentally in the future.
|−45◦ ⟩ 98.86%
Table 3 The measured and derived parameters of the QKD experiment. Parameters
Value
Parameters
Value
𝑄𝜇 𝑄𝜈 𝑌0 𝐸𝜇 𝐸𝜈 𝑓 𝜖𝑠𝑒𝑐
5.06 × 10−4 2.01 × 10−4 9.90 × 10−6 3.74% 5.64% 1.10 10−9
Raw key size Acquisition time Raw key rate Secure key size without finite-size effect Secure key rate without finite-size effect Secure key size with finite-size effect Secure key rate with finite-size effect
86 632 bits 705 s 122.88 bps 8289 bits 11.76 bps 4805 bits 6.82 bps
adopted four photomultiplier tubes (PMT), from Hamamatsu, to detect the receiving photons. The diameter of the active area is approximately 25.4 cm. The detection efficiency is approximately 8% at 532 nm. With this setup, the entire attenuation of the link is approximately 28 dB. The 0.5-m long seawater tank contributes 5 dB of attenuation. The detector efficiency contributes 11 dB. The remaining 12 dB is due to geometric attenuation and the transmittance of the receiving antenna. Notice that we have found out that the secure key cannot be extracted when the attenuation exceeds 30 dB with our devices. Thus we choose to perform QKD at the attenuation of 28 dB, which is near to the maximum tolerable limit. The average polarization fidelity of the entire link is more than 98%, as illustrated in Table 2. In our QKD experiment, we extracted the final secure key from the raw data following the standard decoy BB84 post-processing procedure. In our algorithm, only the signal state is used for secure key generation, while the decoy state is used for parameter estimation. The lower bound of the final key rate per pulse is given by 𝑅𝑝𝑢𝑙𝑠𝑒 ≥ 𝑞𝑝𝜇 {−𝑄𝜇 𝑓 (𝐸𝜇 )𝐻2 (𝐸𝜇 ) + 𝑄1 [1 − 𝐻2 (𝑒1 )]}
(4)
where 𝑞 = 1∕2 is the basis reconciliation factor. 𝑝𝜇 is the probability of the emitting signal states. 𝑄𝜇 and 𝐸𝜇 are the gain and the error rate of the signal states, respectively. 𝑓 is the error correction efficiency. 𝐻2 (𝑥) = −𝑥𝑙𝑜𝑔2 (𝑥) − (1 − 𝑥)𝑙𝑜𝑔2 (1 − 𝑥) is the binary Shannon entropy function. 𝑄1 and 𝑒1 are the gain and the phase error rate, respectively, when the source generates single-photon states. The low-density paritycheck (LDPC) code was used for error correction. The check matrix is selected according to the bit error rate, which is estimated by sampling the sifted key. The errors are corrected with Log-likelihood Ratio Belief Propagation algorithm. The process iterates until the CRC check passed or reaching the maximum number of cycles. We ran our QKD system with real seawater samples collected near the coast of the Gouqi Island located in the East China Sea (30◦ 43′ 17.09′′ N 122◦ 47′ 01.68′′ E).Table 3 shows the measured and derived parameters for the experiment. 𝑄𝜇 and 𝑄𝜈 represent the gains of the signal and decoy states, respectively. 𝑌0 is the measured yield of the vacuum state. 𝐸𝜇 is the observed error rate of the signal state, while 𝐸𝜈 is the observed error rate of the decoy state. The experiment lasted for 705 s. The total accumulated raw key size was 86632 bits. The average raw key rate was approximately 123 bps with an error rate of 3.74%. In the experiment, we obtained a secure key rate of 11.76 bps in the asymmetric scenario. The secure key size is approximately 8.29 kb for an experimental time of more than 705 s. Next, we take the finite size effect into consideration. We estimated the bounds for 𝑄𝐿 1 and 𝑒𝑈 following the analysis introduced in Ref. [38] by considering 1 the statistical fluctuations for the vacuum states and the gains for the signal states and the decoy states within six standard deviations, which ensures final key rates with a secure parameter of 𝜖𝑠𝑒𝑐 = 10−9 . The secure key rate is approximately 6.82 bps with a total secure key of 4805 bits.
6. Conclusions We discussed the requirements of space-to-underwater quantum communication, theoretically analyzed the fluctuating interface effects, and found that the interface has a large influence on the receiving efficiency but only a small influence on the communication error rate. We developed a ‘‘blue–green’’ band decoy-state optical source for space-to-underwater quantum communication with the repetition rate of up to 1 MHz and the pulse width of 14 ns. We designed a receiving antenna with a large FOV of up to 143 mrad. We verified the system performance through 0.5-meter seawater in the laboratory and successfully implemented quantum key distribution based on near-coast seawater channels. The accumulated secure key size is approximately 4.8 kb. The secure key rate is approximately 6.82 bps with an error rate of 3.74%. Our work provides a first step towards space-to-underwater QKD. Acknowledgments We thank many of our colleagues for experimental assistance and delightful discussions, especially Wen-Zhuo Zhang, Fei Zhou and FengZhi Li. 224
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Funding
Similarly, for the input state
National Natural Science Foundation of China(NSFC) (Grant No. 11705191, U1738142, 11654005); National Key Research and Development (R&D) Plan of China (Grant No. 2017YFA0303900); Anhui Provincial Natural Science Foundation, China (Grant No. 1808085QF180); Natural Science Foundation of Shanghai, China (Grant No. 18ZR1443600); Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F050004); Shanghai Sailing Program, China (Grant No. 18YF1425100). S.-K. Liao and Y. Cao were supported by the Youth Innovation Promotion Association of CAS, China (2017500; 2018492); CAS Key Technology Talent Program, China.
𝑅𝑟𝑖𝑔ℎ𝑡 (
𝑄𝐵𝐸𝑅𝑖𝑛𝑡 =
𝑁𝑟𝑖𝑔ℎ𝑡 + 𝑁𝑤𝑟𝑜𝑛𝑔
=
𝑅𝑟𝑖𝑔ℎ𝑡 + 𝑅𝑤𝑟𝑜𝑛𝑔
=
(A.10)
(𝑡𝑠 − 𝑡𝑝 )2 2(𝑡𝑠 2 + 𝑡𝑝 2 )
Then, the quantum bit error rate introduced by interface refraction can be expressed as:
(A.1)
Here 𝑁𝑟𝑖𝑔ℎ𝑡 and 𝑁𝑤𝑟𝑜𝑛𝑔 are the detection counts of the right bits and the wrong bits, respectively. 𝑅𝑟𝑖𝑔ℎ𝑡 and 𝑅𝑤𝑟𝑜𝑛𝑔 are the detection probability of the right bits and the wrong bits, respectively. Obviously, we have
𝑄𝐵𝐸𝑅𝑖𝑛𝑡 =
𝑅𝑤𝑟𝑜𝑛𝑔 = 𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑠⟩) + 𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑝⟩)+ |𝑠⟩ + |𝑝⟩ |𝑠⟩ − |𝑝⟩ 𝑅𝑤𝑟𝑜𝑛𝑔 ( √ ) + 𝑅𝑤𝑟𝑜𝑛𝑔 ( √ ) 2 2
=
𝑅𝑟𝑖𝑔ℎ𝑡 = 𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑠⟩) + 𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑝⟩) |𝑠⟩ + |𝑝⟩ |𝑠⟩ − |𝑝⟩ + 𝑅𝑟𝑖𝑔ℎ𝑡 ( √ ) + 𝑅𝑟𝑖𝑔ℎ𝑡 ( √ ) 2 2
(𝑡𝑠 + 𝑡𝑝 )2
2(𝑡𝑠 2 + 𝑡𝑝 2 ) |𝑠⟩ − |𝑝⟩ |𝑠⟩ − |𝑝⟩ ⟂ |𝑠⟩ + |𝑝⟩ 2 𝑅𝑤𝑟𝑜𝑛𝑔 ( √ ) = |⟨( √ ) | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| 2 2 2 𝑡𝑠 |𝑠⟩ − 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ + |𝑝⟩ = |⟨ √ |⋅| √ ⟩| 2 𝑡𝑠 2 + 𝑡𝑝 2
Here we show the details to derive Eq. (3). The quantum bit error rate introduced by interface refraction 𝑄𝐵𝐸𝑅𝑖𝑛𝑡 is defined as the proportion of the wrong bits to the total number of the received photons [1,23,24]: 𝑅𝑤𝑟𝑜𝑛𝑔
we have
|𝑠⟩ − |𝑝⟩ |𝑠⟩ − |𝑝⟩ |𝑠⟩ − |𝑝⟩ 2 ) = |⟨ √ | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| √ 2 2 2 𝑡𝑠 |𝑠⟩ − 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ − |𝑝⟩ = |⟨ √ |⋅| √ ⟩| 2 𝑡𝑠 2 + 𝑡𝑝 2 =
Appendix
𝑁𝑤𝑟𝑜𝑛𝑔
|𝑠⟩−|𝑝⟩ √ , 2
𝑅𝑤𝑟𝑜𝑛𝑔 𝑅𝑟𝑖𝑔ℎ𝑡 + 𝑅𝑤𝑟𝑜𝑛𝑔 (
0+0+
(A.2)
(
1+0+1+0+
= (A.3)
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 𝑡2𝑠 +𝑡2𝑝
( )2 𝑡𝑠 − 𝑡𝑝 ( ) 4 𝑡2𝑠 + 𝑡2𝑝
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(
+
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 𝑡2𝑠 +𝑡2𝑝
(
+
)2 𝑡 +𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(
+
)2 𝑡 −𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(
+
)2 𝑡 +𝑡 (𝑠 𝑝 ) 2 2 𝑡𝑠 +𝑡2𝑝
(A.11)
√
( )2 𝑛 𝑠𝑖𝑛𝜃 1 − 1 𝑛 1 − 𝑛1 𝑠𝑖𝑛2 𝜃1 ∕𝑛2 )2 2 = √ 𝑛1 𝑠𝑖𝑛𝜃1 2 4[1 + (𝑐𝑜𝑠𝜃1 1 − ( 𝑛 ) + 𝑛1 𝑠𝑖𝑛2 𝜃1 ∕𝑛2 )2 ] (1 − 𝑐𝑜𝑠𝜃1
For the input state |𝑠⟩, the output state is |𝜙𝑜𝑢𝑡 (𝑠)⟩ = |𝑠⟩. So the right detection rate can be written as
2
𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑠⟩) = |⟨𝑠| ⋅ |𝜙𝑜𝑢𝑡 (𝑠)⟩|2 = |⟨𝑠| ⋅ |𝑠⟩|2 = 1
(A.4)
During the last step, we have combined with Eq. (1), Eq. (2) and the equation of 𝑛1 𝑠𝑖𝑛𝜃1 = 𝑛2 𝑠𝑖𝑛𝜃2 .
And the wrong detection rate can be written as 2
𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑠⟩) = |⟨𝑠⟂ | ⋅ |𝜙𝑜𝑢𝑡 (𝑠)⟩| = |⟨𝑝| ⋅ |𝑠⟩|2 = 0
(A.5) References
Similarly, for the input state |𝑝⟩, we have 𝑅𝑟𝑖𝑔ℎ𝑡 (|𝑝⟩) = |⟨𝑝| ⋅ |𝜙𝑜𝑢𝑡 (𝑝)⟩|2 = |⟨𝑝| ⋅ |𝑝⟩|2 = 1 𝑅𝑤𝑟𝑜𝑛𝑔 (|𝑝⟩) = |⟨𝑝⟂| ⋅ |𝜙𝑜𝑢𝑡 (𝑝)⟩|2 = |⟨𝑠| ⋅ |𝑝⟩|2 = 0 For the input state |𝜙𝑜𝑢𝑡 (
|𝑠⟩+|𝑝⟩ √ , 2
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(A.6)
the output state is
𝑡𝑠 |𝑠⟩ + 𝑡𝑝 |𝑝⟩ |𝑠⟩ + |𝑝⟩ )⟩ = √ . √ 2 𝑡𝑠 2 + 𝑡𝑝 2
(A.7)
So the right detection rate can be written as 𝑅𝑟𝑖𝑔ℎ𝑡 (
|𝑠⟩ + |𝑝⟩ |𝑠⟩ + |𝑝⟩ |𝑠⟩ + |𝑝⟩ 2 ) = |⟨ √ | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| √ 2 2 2 𝑡𝑠 |𝑠⟩ + 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ + |𝑝⟩ ⟩| = |⟨ √ |⋅| √ 2 𝑡𝑠 2 + 𝑡𝑝 2 =
(A.8)
(𝑡𝑠 + 𝑡𝑝 )2 2(𝑡𝑠 2 + 𝑡𝑝 2 )
And the wrong detection rate can be written as 𝑅𝑤𝑟𝑜𝑛𝑔 (
|𝑠⟩ + |𝑝⟩ |𝑠⟩ + |𝑝⟩ ⟂ |𝑠⟩ + |𝑝⟩ 2 ) = |⟨( √ ) | ⋅ |𝜙𝑜𝑢𝑡 ( √ )⟩| √ 2 2 2 𝑡𝑠 |𝑠⟩ + 𝑡𝑝 |𝑝⟩ 2 |𝑠⟩ − |𝑝⟩ = |⟨ √ |⋅| √ ⟩| 2 𝑡𝑠 2 + 𝑡𝑝 2 =
(A.9)
(𝑡𝑠 − 𝑡𝑝 )2 2(𝑡𝑠 2 + 𝑡𝑝 2 ) 225
D.-D. Li, Q. Shen, W. Chen et al.
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