Trans-Media Continuous-Variable Quantum Key Distribution via Untrusted Entanglement Source

We propose a continuous-variable quantum key distribution (CVQKD) scheme based on the atmosphere-to-seawater channel. In particular, an untrusted entanglement source (two modes of the compressed vacuum state, TMSV) is deployed on the surface of the seawater, each mode of TMSV is sent to one of the two legitimate parts, respectively. In this way, we can establish a trans-media CVQKD link between the atmosphere and the underwater. Meanwhile, a suitable non-Gaussian operation, namely photon subtraction, is introduced to enhance the performance of the trans-media CVQKD scheme. Security analysis shows that the proposed scheme can establish a trans-media quantum communication system on the atmosphere-to-seawater channel. Moreover, the maximum transmission distance can be further extended by using proper photon subtraction operation. Our scheme provides a guidance for applying CVQKD to trans-media channels.


Introduction
Quantum key distribution (QKD) [1], [2] allows two legitimate parties, Alice, and Bob to remotely establish secure communication. Its security is guaranteed by the laws of quantum physics [3]. In general, there are two kinds of QKD technologies, i.e., discrete-variable quantum key distribution (DVQKD) [4] and continuous-variable quantum key distribution (CVQKD) [5], [6]. In CVQKD, the sender usually encodes key bits in the quadrature of the optical field, while the receiver can obtain the secret key bits by using high-speed and high-efficiency coherent detection techniques. This strategy usually has a repetition rate higher than that of single-photon detections so that CVQKD could potentially achieve higher secret key rates [7].
Based on the properties of quantum physics, it is theoretically proven that CVQKD is unconditionally secure in the fiber channel [8], [9]. At present, with people's research on communication channels, CVQKD has developed from fiber channel to free-space channel, and they have the same secret. For example, underwater quantum key distributions [10], atmosphere quantum key distributions [11], etc. However, these are all under a single media channel, and there is almost no CVQKD involving trans-media channels, so it is not suitable for the following scenarios, communication between satellite to submarine or ship to submarine-detector. Even so, the transmission performance of CVQKD is weaker than the fiber Channel [12]. This is mainly because the quantum signal is affected by absorption and scattering in free-space channels. Fortunately, non-Gaussian modulated photon subtraction technology [13] has been proven to effectively improve the performance of CVQKD by increasing the degree of entanglement between quantum.
In this paper, we propose a trans-media CVQKD scheme and use photon subtraction technology to provide performance for this scheme. Specifically, we consider that the trans-media channel is composed of two parts, the atmosphere, and seawater. First, by placing an untrusted entanglement source (Charlie) [14] on the seawater surface, we have successfully constructed a trans-media CVQKD scheme from the atmosphere to the seawater. Security analysis shows that the proposed scheme can establish a trans-media quantum communication system in the atmosphere-toseawater and generate a positive secret key rate. Among them, the effective transmission distance is 3885.15 m. Second, the performance of the scheme is further improved by Charlie using the photon subtraction operation. The effective transmission distance is increasing to 8693.73 m. This paper is organized as follows: In Section 2, we introduce an untrusted entanglement source to establish a trans-media transmission model for CVQKD between atmosphere and seawater. In Section 3, we give a performance analysis of the trans-media CVQKD scheme and prove its security. In the end, the conclusions are drawn in Section 4.

Trans-Media CVQKD Scheme
In this work, we consider a situation where the quantum channel consists of multi-media and thus propose a trans-media CVQKD scheme based on the atmosphere-to-seawater channel to establish a trans-media quantum communication system.
Unlike the traditional CVQKD scheme, the sender Alice protects the source of entanglement. We place an untrusted entanglement source on the surface of the seawater and are controlled by thirdparty Charlie [15]. Charlie's participation enables us to build a quantum communication network involving trans-media transmission of atmosphere and seawater. Fig. 1 (a) shows this application scenario where Charlie is located on the surface of seawater. Alice is in the upper layer; Charlie's transmit quantum single to Alice through the atmosphere. Bob's is in the lower layer; Charlie's transmit quantum single to Bob through the seawater. Fig. 1 (b) shows the model diagram. First, Charlie prepares the entanglement sources using a laser with a wavelength of 550 nm, half of which is transmitted to Alice through the atmospheric channel, and the other half is transmitted to Bob through the seawater channel after photon subtraction. Among them, quantum memory exists in the transmission of quantum single in the atmosphere or seawater. We regard the entire transmission process, then the signal loss can be expressed by the transmittance T air ∈ [0, 1]. In the same way, the loss of quantum single in seawater can be expressed by the transmittance T water ∈ [0, 1]. When Alice and Bob receive the quantum signal, they first perform a dynamic polarization control process on the signal and then perform homodyne detection or heterodyne detection. Finally, Alice and Bob respectively tell each other their detection results on the public channel and extract a string of secret keys by using parameter estimation, information reconciliation, and privacy amplification.

Atmospheric Channel
In a turbulent atmosphere, the transmission of quantum information can be affected by the beam, resulting from the unstable adjustment of the radiation source and insufficient coverage area at the receiver [16]. There are currently two atmospheric channel models, one is called fluctuating loss due to beam wandering [17] as shown in Fig. 2, and the other is randomly blocked quantum channel [18]. In this study, we found that the fluctuating loss due to beam wandering can better reflect the importance of photon subtraction. In a weak turbulent atmosphere, the beam-deflection distance r determines the transmittance of the atmospheric channel, whereas, in a strong turbulent atmosphere, atmospheric transmittance becomes complicated. It can also depend on the elliptical beam profile that includes the half axis K 1 and K 2 with an angle θ .
In a turbulent atmosphere case, atmospheric transmittance can be described as T air [10], [18], as shown in Appendix B. The turbulence intensity, which affects the transmission efficiency of the CVQKD system, can be expressed by Rytov variance [19], given by where L is vertical propagation distance, N are optical wavenumber, and R 2 n is the refraction index structure parameters [20].
Without loss of generality, weak turbulence is usually regarded as a simplified model of strong turbulence. In this model, the atmospheric transmittance decreases as the beam deflection distance r and the beam spot radius k. We note that the fluctuation of the beam deflection distance r obeys Rice distribution [21]. When the distance between the Rytov variance R 2 n and the central aperture is reduced to zero, the resulting distribution can be simplified to the log-negative Weibull where T air ∈ [0, T air 0 ], and P(T air ) = 0. The mean value of the fading probability distribution can be calculated by In Fig. 2(b), shows the transmission probability distribution at different distances in spring. This demonstrates the effect of weak turbulence on the transmission distance of communication system. In Appendix B, shows characteristics of the atmospheric transmittance with numerical simulations.

Seawater Channel
Quantum state transmission of seawater channel is weaker than that of atmospheric channel because of the absorption and scattering of light in seawater. The effect of light absorption is since the light intensity weakens sharply as the transmission distance, while the effect of light scattering is that the area radiated by light increases with the transmission distance. Both effects are irreversible thermal processes [22]. In Fig. 3(a), shows a beam-spread model diagram of the surface information when Bob is located at points A (3 m), B (11 m), and C (28 m), respectively. The brighter the yellow bright spot, the greater the light intensity, that is, the weaker the absorption effect, which can be calculated as where ι is the wavelength, C represents chlorophyll concentration, and a c is the chlorophyll-specific absorption coefficient obtained through statistics, and a w is the absorption coefficient in seawater. Then, the smaller the halo radius, the more concentrated the light beam, that is, the weaker the scattering effect, which can be given by The overall effect on the transmitted states can be derived Q at (ι) = Q ab (ι) + Q sc (ι). The relevant values are given in Appendix C. Fig. 3(b) shows the relationship of the transmittance and depth in different seawaters, where A, B, and C correspond to the points in Fig. 3(a). We find that the transmittance decreases as the transmission depth increases.

Security Analysis
In this section, we analyze the performance of CVQKD under the three quantum channel combinations of atmosphere-to-atmosphere, seawater-to-seawater, and atmosphere-to-water. We obtain the relationship between transmission distance and the secret key rate. The detailed calculation process is shown in Appendix D.

Atmosphere-to-Atmosphere
We assume that Alice and Bob are both deployed in the atmospheric channel to better analyze the atmospheric channel part of the trans-media CVQKD scheme. In this case, the two media channels of the trans-media CVQKD scheme are atmospheric channels. Atmospheric information is a complex channel, and the transmission of quantum signals in the atmosphere is extremely susceptible to seasonal and strong turbulence. As shown in Fig. 4(a), the secret key rate of the trans-media CVQKD scheme is in the weak turbulent atmosphere. Spring and summer have the maximum transmission distance, which is related to the clean atmosphere, while the transmission distance in autumn and winter is short, which is related to the turbid atmosphere in these two seasons. Also, the maximum effective transmission distance is 8360 m in summer. However, when the quantum signal transmits in strong turbulence, the effective transmission distance will be further shortened. The secret key rate of the trans-media CVQKD scheme is in a strong turbulent atmosphere, as shown in Fig. 4(b). It can be seen from the figure that even in summer, the maximum effective transmission distance is only 6200 m. Through the above analysis, the CVQKD scheme can obtain the best performance only under the conditions of a clean atmosphere and weak turbulence, which means that the best season for quantum signal transmission is spring or summer.

Seawater-to-Seawater
We assume that Alice and Bob are both deployed in the seawater channel to better analyze the seawater channel part of the trans-media CVQKD scheme. In this case, the two media channels of the trans-media CVQKD scheme are seawater channels.
As shown in Fig. 5, the secret key rate of the trans-media CVQKD scheme for both homodyne and heterodyne measurements in the seawater channel. The pure seawater has the best performance, and the maximum transmission distance is 19.6 m. We can see that the better the water quality, the less the dissolved organic matter contained in the transmission medium, and the more conducive to the transmission of optical signals. Compare Fig. 5(a) and Fig. 5(b), it can be found that not only does the water quality affect performance, but also the measurement method affects performance. The maximum transmission distance of heterodyne detection is only 13.94 m, while that of homodyne detection is 19.62 m. Homodyne measurement is one of the conditions to  obtain optimal performance in the CVQKD scheme. It is worth noting that although the transmission distance of the protocol in the seawater channel is less than that in the atmospheric channel, it provides a feasible method for the trans-media quantum communication between satellite and submarine.

Atmosphere-to-Seawater
Through the above security analysis, we know that the performance of the atmosphere and seawater in the trans-media CVQKD scheme is completely different. To optimize the performance of the trans-media CVQKD scheme, we define the distance L dist ance from Charlie to Alice and the distance L dept h from Charlie to Bob. As shown in Fig. 6, L dist ance and L dept h have the following relationship, orange represents the distance of L dist ance , and blue represents the distance of L dept h .The initial value of L dist ance is 1Km. When L dist ance increases by 200 m, L dept h increases by 0.3575 m. For example, when the L dist ance is 5000 m, then the L dept h is 7.15 m.
We have established the relationship between the secret key rate and transmission distance of the trans-media CVKQD scheme, as shown in Fig. 7. Fig. 7(a) is a security analysis diagram of atmospheric-to-water CVQKD under the background of weak turbulence in summer. It is not difficult to find that the effective transmission distance of the multi-media channel is not as good as that of the atmospheric channel, but better than that of the seawater channel. This is because the attenuation coefficient of the atmospheric channel is smaller than that of the seawater channel, so the quantum signal can be transmitted in the atmosphere for a longer distance. The transmission of photons in the atmosphere is mainly limited by atmospheric turbulence. Turbulent eddies whose size is large compared to the size of the beam induce beam wandering [23]. But in water, photons inevitably interact with water molecules and other particulate matter in the water, where absorption and scattering seriously weaken the transmitted light signal. In the underwater environment, substances such as chlorophyll can absorb blue and red light (Fig. C9). These substances and other colored dissolved organic matter increase the turbidity of the water, thereby reducing the distance of light travel. In addition, the concentration of colored dissolved organic matter will also change with the depth of the ocean, thereby changing the corresponding light attenuation coefficient [24]. For example, when Bob is placed in pure water, the trans-media CVQKD scheme maximum transmission distance is 4065.5 m. This transmission performance cannot meet the practicability of the trans-media CVQKD scheme. To further improve the performance of the trans-media CVQKD scheme, we will perform appropriate photon subtractions on Charlie's side. As shown in Fig. 7(b), the black line is the performance before improvement, red line and blue line are improved performance, where red line is a one-photon subtraction operation and the blue line is a two-photon subtraction operation. The simulation results show that the transmission performance of the trans-media CVQKD scheme can be effectively improved by using photon subtraction. It is worth noting that the trans-media CVQKD scheme achieves the best performance in one-photon subtraction operations. There are two main reasons for this phenomenon. One is that the success rate of one-photon subtraction is higher than that of two-photon subtraction [25]. Second, the photon reduction operation will cause the loss of quantum signals. Due to the loss of enough data, Alice and Bob cannot generate the secret key rate. The results also show that the improved scheme provides a feasible method to realize the actual trans-media transmission in the future.

Conclusion
We have proposed a novel scheme for trans-media CVQKD via untrusted entanglement source, thereby establishing secure quantum communications between the atmosphere and underwater. We have also improved the performance of the proposed scheme by taking advantage of suitable photon subtraction operation. Numerical simulations show that the proposed scheme can still establish secure communication through trans-media transmission, even if the entanglement source is untrusted. Moreover, photon subtraction can lengthen the maximal transmission distance of our system. This work provides guidance for applying CVQKD to trans-media channels.

Appendix A Photon Subtraction
The TMSV state prepared by Charlie contains two modes A and B, which are sent to Alice and Bob, respectively. Then, Alice receive {â,â † } and Bob receive {b,b † }. A covariance matrix of the TMSV state can be written as where Z is matrix diag (1, −1), I is matrix diag (1,1), and v = (v a + v s )/2 is modulation variance. As shown in Fig. 1(b), Charlie use a splitter (BS 1 ) to split mode B into modes B 1 and B 2 to obtain the tripartite state Thereafter, B 1 is measured with a positive operator's quantity. {ˆ 0 ,ˆ 1 } is saved only when the A and B 2 elements are clicked. The resulting TMSV after the photon subtraction operation can be described as where t r x (·) is the partial trace of a multimode quantum state. The probability of element A being successfully clicked is calculated as , s is the number of photons subtracted, and C s n is the combinatorial number and its relationship with the transmittance. Therefore, state γ AB 2 is no longer a Gaussian state [26], which can be derived as

Appendix B Parameter Derivation of the Atmospheric Channel
There are two kinds of atmospheric channels, i.e., weak turbulence and strong turbulence. In a weak turbulent atmosphere case, atmospheric transmittance can be calculated as T air [10], where T air 0 is the maximum transmission coefficient given by T 2 , R is the scale parameter, and q represent the shape, and r is the beam-deflection distance. In a strong turbulent atmosphere case, the elliptical beam model can use five parameters to describe the received light spot, i.e., {x 0 , y 0 , k 1 , k 1 , θ }, where x 0 and y 0 partitions take the X-axis and Y-axis, which are used for reflecting the degree of the beam wandering, while the elliptical beam section half axis marked with k 1 and k 2 , and the angle between the half-axis k 1 and the x-axis of the elliptical beam section is θ . According to the above-mentioned parameters, we obtain the transmittance of a strong turbulence atmosphere, which can be approximated as  where K eff (θ − ϑ ) represents the effective point radius. For = θ − ϑ, the parameter K eff ( ) can be calculated as where W (·) is the Lambert function. In addition, the maximum transmission coefficient is calculated as where I 0 is the modified Bessel functions. Fig. B8 shows the relationship between transmission distance and transmittance. Table B1 shows that the R 2 n -values of spring, summer, autumn and winter measured over a long period of time in Hefei, Anhui, China [20]. As shown in Fig. B8(a), since the atmosphere motion in weak turbulence is relatively stable, the change of the transmittance is not obvious. We find that it performs best in spring and summer, leading to a transmittance of 0.088 even at 10 km. However, the transmittance is only 0.026 when it becomes in autumn and winter at 10 km. It is worth noting that, under strong turbulence, due to the unstable atmosphere movement, the transmittance will drop sharply, as shown in Fig. B8(b). For example, when taking into account the transmission distances 3.1 km, 3 km, 2.1 km and 1.85 km in the spring, summer, autumn, and winter, respectively, the transmittance will approach around 0.1. Although there was no significant slump in winter, it was due to more severe shocks.

Appendix C Parameter Derivation of the Seawater Channel
The total attenuation factor of light in seawater is Q at , which is the sum of the absorption factor Q ab and the scattering factor Q sc . Both Qab and Qsc are associated with the wavelength l of the light. Fig. C9 shows the effect of clear seawater on shutdown at wavelengths between 400-700 nm, but with the least light attenuation at wavelengths between 450-55 nm, which coincides with the presence of a blue-green optical window in seawater. As shown in Table C2, the corresponding parameters for the effect of seawater of different water quality when the wavelength of the signal light is 520 nm. As shown in Table C2, the corresponding parameters for the effect of seawater of different water quality when the signal light wavelength is 520 nm [27].
Assuming Eve uses the same configuration for Alice and Bob, then g = −g . In this case, the covariance matrix between Alice and Bob will be rewritten as For the simplifying calculation of the model, we take the performance analysis with direct reconciliation, resulting in the secret key rate given by where G(x ) = (x + 1) log 2 (x + 1) − x log 2 x, and the symplectic can be calculated as