Satellite-based measurement-device-independent quantum key distribution

A satellite-based measurement-device-independent quantum key distribution model is proposed and its performance is analyzed through numerical simulations. The simulation result shows that the key rate increases almost linearly as transmittance increases and the higher the key rate is, the lower its probability may be. More significantly, as the transmission distance increases, the optimal intensities of the signal states decrease while those of the decoy states increase. The optimal values in the daytime are slightly larger than those at night. This work may provide some important parameters for the relevant experiment on satellite-based quantum communication network.

In fiber-based networks, the high loss of fiber has become the main reason for limiting the transmission distance. Fortunately, schemes utilizing satellite links have been proposed to achieve longer distance quantum communication in recent years. For example, Oi et al used the micro satellite SOCRATES to demonstrate a QKD experiment with an average quantum error rate of less than 5% [18] and Liao et al used the quantum satellite Micius to complete a number of experiments on satellite-ground quantum communication [19][20][21][22]. Since fiber-based networks have been well developed, in order to establish worldwide quantum communication networks at lower cost, satellite-based quantum channels and fiber-based networks should be effectively integrated. Hence, the performance of satellite-based QKD protocols combined with fiber-based networks needs to be studied. Satellite-based MDI-QKD proposed in this paper can easily solve detector vulnerabilities caused by imperfect devices in real life and prolong the transmission distance. In previous demonstration, the longest transmission distance of MDI-QKD in optical fiber is only 404 km [16], while satellite-based MDI-QKD combined with fiber-based networks can easily reach more than 500 km. Satellite-based MDI-QKD can make full use of fiber-based networks and satellite channels, thus promoting the establishment of global quantum communication networks.
This work is organized as follows. In section 2, the theory of the satellite-based MDI-QKD is briefly introduced. In section 3, the main simulation and optimization results are shown and discussed. Finally, the conclusion is given in section 4.  Table 1. List of practical parameters for numerical simulations. e d is the total misalignment error; e 0 is the error probability of vacuum pulses; P d is the dark count of each detector; f e is the error correction inefficiency; W 0 is the beam spot radius at the transmitter; a is the radius of the receiving aperture; λ is the wavelength of the signal light in satellite links; η det is the detector efficiency; η otu is the OTU efficiency; β is the parameter in extinction factor; φ is the pointing error.

Theory
In satellite-based MDI-QKD, Alice and Bob randomly prepare their quantum states and send them to Charles, who will perform the Bell state measurement. Inside the measurement device, the photons from Alice and Bob interfere at a 50:50 beam splitter and then are projected into either horizontal or vertical polarization states through a polarizing beam splitter and finally are being detected by the signal-photon detector. Note that the photons from Alice are at 1550 nm while the photons from Bob are in the visible range, so the optical transponder unit (OTU) is needed to convert the wavelength of Bob's photons to the same as Alice's before measurement [23,24]. After the Bell state measurement, the detection results are announced in the public channel. Based on the announcement, Alice and Bob can decide whether to flip their bits or not and then distill the secure key. Unlike the original MDI-QKD protocol, Alice and Charles are on the ground, while Bob is in the outer space, as seen in figure 1. Hence, Alice uses the fiber channels in which transmittance ratio can be described as: t = 10 −αL/10 (α and L denotes the channel loss coefficient and the transmission distance respectively) to send her states to Charles, while Bob utilizes the satellite links whose transmittance can be modeled by the probability distribution of the transmittance (PDT) [25]. In real life application, the case that Alice inside the city and Charles outside the city is better because it can make full use of fiber-based networks and avoid extra light pollution at the receiver. The final key rate in satellite-based MDI-QKD is averaged over the PDT and is given by WhereR is the average (final) key rate, R(η) is the key rate at the specific value of the transmittance, P(η) is the PDT. The integral average is approximated dividing the range [0, 1] in N bins and taking the weighted sum of the key rate. When using weak coherent pulses source with decoy states, R(η) can be calculated by [11] R(η) P 11 Where P 11 Z is the probability that both Alice and Bob send single-photon in the rectilinear (Z) basis; Y 11 Z and e 11 X is the single-photon yield in the Z basis and the single-photon error rate in the diagonal (X) basis respectively; H 2 (x) is the binary entropy function given by H 2 (x) = −xlog 2 (x) − (1 − x)log 2 (1 − x); Q Z and E Z is the gain and quantum bit error rate (QBER) in the Z basis respectively; f e 1 is the error correction  inefficiency function. Generally, Q Z and E Z can be measured directly, while Y 11 Z and e 11 X need to be estimated.

Simulation results
In this section, the distance between Bob and Charles is fixed at 500 km because the case of Low-Earth-Orbit satellites such as the Chinese satellite Micius is focused on. Numerical simulation results of satellite-based MDI-QKD using the vacuum + weak decoy states scheme in the asymptotic case are given. The experimental parameters used in simulations are listed in table 1 [25][26][27].
Firstly, the intensities of Alice's (Bob's) signal states μ a (μ b ) and decoy states ν a (ν b ) are set to 0.3 (0.5) and 0.003 (0.005) respectively. When the distance between Bob and Charles is fixed at 500 km, the transmittance of down-link ranges from 0 to 0.12 [25]. When the distance between Alice and Charles (L A ) is fixed at 0 km, the key rate increases almost linearly with the increase of transmittance, as shown in figure 2.
The key rate equals to zero at the beginning because the transmittance is too low to transmit information. Next, the probability distribution of the key rate (PDR) is obtained through the key rate and PDT, as illustrated in figure 3. The simulation result shows that the higher the key rate is, the lower its probability may be.  With PDR, the final key rate of different L A can be easily calculated (only need to take the weighted sum of the key rate according to its probability) and is shown in figure 4. The maximum value of the key rate is 1.5988 × 10 −5 per pulse and the longest transmission distance is 120 km. Note that the results are not optimal since the intensities of Alice's (Bob's) signal and decoy states are set to the fixed values. In addition, a further discussion about PDR of different L A is given. As can be seen from figure 5, with the increase of L A , the key rate decreases on the whole while the shape of PDR remains the same due to the unchanged PDT.
Since the above results are not optimal, parameter optimization should be taken into account to improve the performance of the protocol. As shown in figure 6, the key rate is higher and the transmission distance is longer after using the optimal intensities of the signal and decoy states. Compared with the key rate using μ a = 0.3, ν a = 0.003, μ b = 0.5, ν b = 0.005, the maximum key rate using the optimal intensities is 2.3098 × 10 −5 per pulse, increasing by 44.47% and the longest transmission distance is 145 km, increasing by 20.83%. The optimal intensities of different L A under different atmospheric conditions (see [25] for more details) are shown in figure 7. As can be seen, with the increase of the transmission distance, the optimal intensities of the signal states decrease while those of the decoy states increase no matter what atmospheric condition is. The optimal values during the day are slightly larger than those at night.

Conclusion
In this paper, the performance of satellite-based MDI-QKD combined with fiber-based networks is analyzed. When the distance between Bob and Charles is fixed at 500 km and the distance between Alice and Charles is fixed at 0 km, the key rate increases almost linearly as transmittance increases and the higher the key rate is, the lower its probability may be. With PDR, the final key rate of different L A has been calculated and displayed. After parameter optimization, the key rate and the transmission distance increased by 44.47% and 20.83% respectively. It is also found that with the increase of L A , the key rate decreases as a whole while the shape of PDR remains unchanged. More significantly, the optimal intensities of different L A under different atmospheric conditions are given. These simulation results may provide some important parameters for the relevant experiment on satellite-based quantum communication network.