Eurasian-Scale Experimental Satellite-based Quantum Key Distribution with Detector Efficiency Mismatch Analysis (2024)

I Introduction

The development of computational devices opens new possibilities to attack cryptographic systems, whose security is based on computational assumptions.In particular, powerful enough quantum computing devices have the potential to compromise the security of widely deployed cryptographic tools[1, 2], such as those based on the complexity of integer factorization[3].One of the solutions is to switch to information security tools[4, 5, 6, 7, 8],which are considered to be resistant to attacks with quantum computers.Specifically, quantum key distribution (QKD), which is the technology that allows establishing secure communications between distinct parties with the security guaranteed by the law of physics[4, 5], offers such a possibility.The idea behind QKD is to encode information in states of single photons and exchange them in a way that prevents uncontrollable eavesdropping.By following such a QKD protocol properly, two distant parties share strings of random bits — quantum-generated secret keys.Although remarkable progress in QKD has been performed[4, 5, 6],one of the most important problems is photon loss in the channel,so that the distance of peer-to-peer QKD with reasonable key rates is about thousand of kilometers[9, 10].This seems to be a limitation to applying the QKD technology at the global scale.

Overcoming this challenge of the distance is possible with the use of satellite-to-ground QKD[11, 12, 13, 14].The launch of a low-Earth-orbit Micius satellite for implementing decoy-state QKD[11],realizing intercontinental quantum-secured communications[13],and deploying an integrated space-to-ground quantum communication network of the length over 4,600 kilometres[14],have been successfully demonstrated during the last years (for a review, see Refs.[15, 16]).The overall losses for the satellite-to-ground link are small compared to ground-level transmission due to negligible attenuationabove the atmosphere and insignificant vertical turbulence in the lower atmosphere.

Several satellite-based QKD research projects demonstrating diverse technology concepts have been initiated[17, 18, 19, 20, 21, 22, 23].The various QKD models[24, 25, 26, 27, 28, 29] investigating feasible concepts and designs of quantum networks in spacehave been significantly developed since the first satellite-based quantum communication experiments.However, these models commonly use a limited dataset from the quantum experiments provided by the Micius satellite to validate these QKD models.Hence, it results in restricted validation of satellite-based QKD models.Specifically, they do not take into account several potentially important imperfections in the realization of QKD protocols.For example, the problem of detector efficiency mismatch[30, 31, 32],which has been recently investigated in the context of the decoy-state QKD protocol[33, 34],seems to be important for consideration in the security analysis.

In this paper,we report on the results of establishing a quantum-secured link between the Zvenigorod and Nanshan ground stations using the Micius satellite.We realize the decoy-state QKD protocol and obtain a sifted key length of 2491kbits and a total final key length of 310kbits.This has enabled transferring messages that are encrypted via a one-time pad with quantum-generated keys over 3800 km.Also, we present a modification of the security analysis of the realization of satellite-based QKD decoy-state protocol by taking into account the effect of the detection-efficiency mismatch.We also simulate the QKD protocol for that satellite passage, validate our semi-empirical model for a realistic receiver, and find its agreement with the experimental data.Our results pave the way to the considerations of realistic imperfection of the QKD systems,which are important in the context of their practical security.

The paper is organized as follows.In Sec.II, we describe the developed ground station and report the results of the joint QKD experiment betweenthe Micius satellite and the Zvenigorod ground station.In Sec.III, we provide a semi-empirical model of satellite-to-ground QKD and compare its outcome to experimental results.The four detector efficiency mismatch analyses of secure key generationduring satellite-to-ground quantum communication are considered in Sec.IV.Finally, in Sec.V, we provide the results of the quantum secure message transfer between Nanshan and Zvenigorod;we summarize our results and conclude in Sec.V

II Satellite-to-ground QKD experiment

In this section, we describe our ground station[36] that is equipped with a 600-mm-aperture reflector telescope and placed at the Zvenigorod observatory(Moscow region, $55$°$42^{\prime}\,$N, $36$°$45^{\prime}\,$E, altitude of 198m).The main characteristics are listed in Tables1, 2, and 3.Also, we present the results of the joint satellite-to-ground QKD experiment and provide the observable optical characteristics and noise level.A general scheme of the quantum communication procedure is given in Fig.1a.

II.1 Zvenigorod ground station

The ground station has a two-stage acquiring, pointing, and tracking (APT) system that ensures high angular stability for the optical signaland suitable operating conditions for the photon states decoder during quantum communication.The general scheme and the hardware at the Zvenigorod ground station are illustrated in Fig.1b and Fig.1c, respectively.

The coarse control of the station is achieved by a motorized mount of the primary telescope,which uses the calculated satellite location in orbit, and an auxiliary 70-mm telescope with a camera for pointing to the satellite.The algorithm on the camera determines the image center of the satellite, and the telescope orientation is adjusted.Figure1d shows a frame from the coarse camera during satellite tracking.The tracking error of the coarse control stage is typically less than 100 $\mu$rad.

The ground station features an optical signal analysis and processing system (APS) for adjusting the beam deflection,rotating the polarization reference frame[35], and spectrum filtering the photons.The main task of APS is the fine control of the beams, which includes a fast-steering mirror and a high-speed camera.As a result, the high-frequency deflections of the co-aligned beaconbeam with quantum signals are corrected to an accuracy of less than 10 $\mu$rad.Moreover, there are spectrum filters to suppress the background noise and a motorized half-wave plate to reduce decoding errors (see Sec.III).

The polarization analyzer is configured in the classical BB84 scheme with a 50:50 beam splitterand includes four types of channels: horizontal (H), vertical (V), diagonal (D), and antidiagonal (A) polarization.To distinguish orthogonal quantum states, two polarization beam splitters with additional polarizers on the reflection side are utilized.The mean polarization contrast ratio of the decoder is more than 350:1, which is measured in multiphoton mode.The quantum signals are coupled into four optical fibers with a core diameter of 105 $\mu$m and detected by the single-photon module.The time-to-digital converter with picosecond resolution is used to record the registration moment.

Synchronization pulses are also detected by the single-photon module, and detection moments are recorded by the time-to-digital converter.Such an extra light channel allows us to find a correlation between the quantum states transmitted from the satelliteand recorded on the ground station for the following sifting procedure[37, 38],as well as to suppress the background noise with temporal filtering[11].

III Experiment and model validation

For the characterization of the experimental results obtained in the joint QKD experimenton March 1, 2022, the simulation of the source of quantum states,time-dependent communication channels and optical characteristics of the receiver is completed.Here we consider a modified semi-empirical satellite-to-ground QKD model[39]implied for the Zvenigorod ground station and the Micius satellite to determine the bit generation rate after basis reconciliationand quantum bit error rate (QBER) for the sources of the signal, decoy, and vacuum states.

The temporal filtering of quantum states reduces satellite and constant background noise by around five times.As a result, the background signal $Y_{0}$ per one sent pulse is equal to $\mathcal{N}/(5f)$ (for details, see AppendixB).WE note that it already contains afterpulse signals, owing to the special experimental way the data $\mathcal{N}$ is acquired.

Figure3 depicts the obtained and the simulated sifted bit generation rates forthe signal, decoy, and vacuum states over the satellite-to-ground QKD experiment under clear weather conditions ($\varkappa=0.22$).For the QKD experiment, the model accurately depicts the mean value of the bit generation rate.Although there is some difference between the simulated and experimental data,it might be explained by additional losses induced by pointing errors between the Micius satellite and the Zvenigorod ground station.During the quantum communication session,the resulting sifted key rate changes from around 20.7 kbit/s at 600 km to 2.9 kbit/s at 1,100 km, as shown in Fig.3a.

According to Ref.[11], the average polarization contrast ratio of the Micius satellite source is 225:1,whereas the time-dependent part of the intrinsic errors $e^{\rm Rx}_{\mathrm{det,}\xi}$is caused by the rotation of the polarization states and the birefringent elements in the receiver optical path.This may be measured empirically before the quantum communication session using the motorized half-wave plates.

Before the communication session, we compute the relative rotation of the polarization reference frames between the satellite and ground station based on the predicted trajectory of the Micius satellite[40, 41].Then the APS system of the receiver (see Fig.1) uses the calculated rotation angles to manage the motorized half-wave plate.Therefore, to measure the intrinsic errors of the receiver, we simulate the rotation of each polarization state by sending classicallaser beams into the ground station and then execute dynamical compensation following the computed rotation function[35].Figure4a illustrates the intrinsic errors $e^{\rm Rx}_{\mathrm{det,}\xi}$ of the Zvenigorod ground stationafter the dynamical compensation of the polarization state rotations for the Micius satellite passage on March 1, 2022.

The observed QBER in the satellite-to-ground QKD experiment and the predicted upper bound of average QBERfor the signal and decoy states are shown in Fig.4b.In four time frames, the announced parts of the QBER for a source of signal states are 0.93%, 0.95%, 0.87%, and 0.89%,which correlate with calculated data. The experimental data on quantum errors for decoy states may be fullyand securely opened according to the QKD decoy state protocol and therefore compared with the model predictions duringthe whole communication session.As a result, the model calculation accurately reproduces the shape of the average QBER.We note that the experimental and model QBER for vacuum states are close to 50%, as expected.

Hence, the joint quantum communication experiment between the Micius satellite and the Zvenogorod ground stationgives results that are quite close to the simulation outcomes according to a semi-empirical satellite-to-ground QKD model for realistic receivers.Thus, the successfully verified model is an excellent tool for simulating complicated and realistic QKD systemsand analyzing their practical constraints and prospects.

IV Security analysis and key rates

To reduce the possible knowledge of an eavesdropper,we estimate the secret key length and use a random matrix for the shuffled key after the error correction step,with an inefficiency $f_{ec}=1.44$[42](for details of the post-processing procedure, see Refs.[43, 44, 45].The decoy-state method is used to verify the lower bound on the single-photon counts.Therefore, if we consider statistical fluctuation and utilize Chernoff bounds with a failure probabilityof $10^{-9}$[39, 46], the secure final key is 629,000 bits.

We can also consider higher-level security by taking into account the detection efficiency mismatch[30, 31, 32].The problem is caused by the different channel optical efficiencies of the receiver and the complexityof designing detectors with the same quantum efficiency.Here we assume that the optical efficiencies of the $\xi$-type channels are equaland the detection efficiency imbalances are created by detectors with differing quantum efficiencies.

In our QKD experiment, we have four unbalanced detectors with the efficiencies $\eta_{z0}$, $\eta_{z1}$, $\eta_{x0}$ and $\eta_{x1}$,where the first subscript is a basis and the second one is a bit value.We assume that $\eta_{z1}\leq\eta_{z0}$ and $\eta_{x1}\leq\eta_{x0}$.Then the detection-efficiency mismatches in each basis are defined as $\eta_{z}=\eta_{z1}/\eta_{z0}$ and $\eta_{x}=\eta_{x1}/\eta_{x0}$.We take the values $\eta_{z}=0.60$ and $\eta_{x}=0.51$.We note that we do not demand that the fractions of the efficiencies are precisely the values given above.The formulas for the secret key rate we used require that $\eta_{z1}/\eta_{z0}$ and $\eta_{x1}/\eta_{x0}$ are not smaller than 0.60 and 0.51,respectively.The ratio of the probability of signal reception in different bases is denoted as $t_{xz}\approx 1$,according to Fig.2a and Table2.We then assume that the beam splitter is set perfectly with the basis selection probability of $1/2$.

To perform a privacy amplification of the key, we use the calculations and results of Ref.[30, 32],where the various methods of accounting for detection mismatch are addressed.Here we use an approach that allows us to obtain the maximum final key rate.

This equation can be simply explained by physical considerations.First, we divide our data into two parts: one key is generated from the positions where Alice and Bob chose the $Z$ basis and another one –where they chose the $X$ basis, and after that, these two keys are merged into one.Then, according to the Devetak and Winter theorem[47],for each basis, the secret key rate is proportional to the difference between Eve’s ignorance about Alice’s stringand Bob’s ignorance about Alice’s string with all information that legitimate users announce during communication.Bob’s ignorance can be bound from above by binary entropy from quantum bit error rate; it is the last term in each bracket of Eq.(7).The first term (Eve’s ignorance) contains some eavesdropper’s information about the state.We can get rid of this using the entropic uncertainty relations and receive the first and second parts in each bracket of Eq.(7).Also, Eve can change the initial states $\rho_{AB}$ as she wants,but we can detect interference using conditions on the observable value of received bits and errors for different bases.The derivation is described in more detail in AppendixC.This relationship is derived from general considerations about information and conditional entropy,and it is performed for our case of four detectors too.The main difference between the reasoning for two unbalanced detectors[30, 31]and our case is that both bases are used for key generation.

Hence, we estimate the final key rate using Eq.(7) applied to the divided data among two bases $X$ and $Z$.As a result, 310,400 bits of the secret key were finally shared between Alice and Bob.

V Results and Discussion

Once we had achieved the shared secret key between Zvenigorod ground station and the Micius satellite,our colleagues from China carried out the same procedures with Nanshan ground station near Urumqi to get the shared key across these stations.As a result, a secret key is established between China and Russia at a distance of 3800 km on Earth.Figure5 illustrates the transfer of encoded messages with two shared keys totaling 4kbytes.Each side used a basic bitwise XOR operation to encrypt and decrypt the image size of 2kbytes, providing unconditional security.

We have reported satellite-to-ground QKD between the Micius satelliteand the developed Zvenigorod ground station that enabled Eurasian-scaled secure message transfer.By simulating the parameters of the quantum communication link,we have validated our semi-empirical QKD model for the Zvenigorod ground station[39]with the results of the presented joint QKD experiment.Based on these experimental and simulated findings, we give a more secure analysis of the key rate than the general decoy state protocol,taking into account the detection efficiency mismatch.That kind of practical imperfection is common in realistic QKD devices, and we offer a possible solution to the problem in this research.

Acknowledgments

We are grateful to our colleagues from the University of Science and Technology of China for their helpful assistance and numerous recommendations throughout the joint experiment with the Micius satellite.The communications session has been completed on March 1, 2022.We thank A. Tayduganov and O. Fat’yanov for helpful discussions.We thank the Priority 2030 program at the National University of Science and Technology MISIS (Project No. K1-2022-027).

Appendix A Supplementary Data

Appendix B Noise estimation

The noise during satellite quantum communication is defined by the coefficients$T$ and$C$ in Eq.(1).The coefficient$T$ represents the mean number of sunlight photons at 850 nm reflected from the Micius satellite surface to the ground station,with the divergence angle equal to the quantum signal.Meanwhile, the background noise due to stray light and detector dark count is given by the coefficient$C$.

To determine the coefficient$C$, we point the telescope at the night sky and record the count rate, which is $290\pm 60$ clicks per second.However, the coefficient$T$ cannot be measured in advance and must be estimated directly during quantum communication.For instance, when the registered quantum signals are synchronized with the satellite clock,the noise may be found as the number of counts outside the time filtering frame and compared with the counting rate of registered vacuum states.

As a result, the coefficient$T$ is equal to $1.8\cdot 10^{6}\pm 0.1\cdot 10^{5}$ photons per second,and the atmospheric extinction coefficient $\varkappa$ is estimated at $0.25\pm 0.05$.We note that to estimate the atmospheric extinction coefficient accurately,we should utilize a sufficiently high count rate, as it is done in Sec.II.2 for the total count rate of the raw key.

Appendix C Derivation of the secret key rate

A complete analysis for the case of detection-efficiency mismatch is presented in Refs.[30, 31].Here we use these articles as a guide and present the general ideas on how to adopt this theory in the case of four detectors.The main result of this section is Eq.(7) for the secret key rate.

Denote $\rho$ is the density operator distributed by Eve (according to the equivalent entanglement-based formulation,also known as the source replacement scheme).Then, we express the process of imperfect measurement as two stages:the attenuation on detectors $\rho\mapsto G_{b}\rho G_{b}$,where $G_{b}$ is the attenuation operator corresponding to the measurement in the basis $b$, followed by a perfect measurement.The operator $G$ is not unitary but turns out to be Hermitian.$\operatorname{Tr}G_{b}\rho G_{b}=p_{\rm det}^{b}$ is the detection probability in the basis $b\in{X,Z}$ and $\widetilde{\rho}=G_{b}\rho G_{b}/p_{\rm det}^{b}$.

The upper bound on the entropy for the $X$ basis is made similar to Eq.(14).Thus, we obtain the desired Eq.(7) by substituting Eq.(14) and the corresponding one for the $X$ basis in Eq.(13).

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