1. Introduction

Quantum key distribution (QKD) offers an approach of sharing unconditional secure keys between two remote parties based on the principle of quantum mechanics. From the first concrete QKD protocol (BB84) established by Bennet and Brassard in 1984 [1] to the simplest entanglement based schemes (BBM92) developed by Bennett et al. in 1992 [2], QKD has become the most active research field for practicability in quantum information science. Typically, there are two main types of quantum channels, fiber links and free-space links. Due to absorptive channel losses and detector dark counts, current transmission distances are limited to the order of hundreds of kilometers in a fiber [3–7]. Fortunately, considering the photon losses and photon decoherence are almost negligible in outer space, QKD based on satellites is treated as one of the most promising solutions to realize a global quantum communication network. In order to test the feasibility of QKD based on satellites, many excellent experimental works have been performed on the ground in the past decade [8–15]. Recently, China launched the world’s first quantum science satellite (QSS) dedicated to testing the fundamentals of quantum communication in space [16]. This exciting development has made it urgent to design an efficient error correction algorithm suitable for satellite-ground QKD.

In general, the postprocessing of QKD mainly includes basis sifting, error correction and privacy amplification [17]. Previous error correction algorithms for QKD are mostly based on the Cascade protocol [18, 19], which needs many times of data exchange. However, the frequent data exchange between satellite and ground is very difficult and challenging. First, data exchange can be carried out only when a satellite passes through a ground monitoring station. The lasting time is typically several minutes for a low earth orbit satellite. Second, a few ground monitoring stations are shared by many satellites, which greatly reduces the opportunities of data exchange for QSS. Also, comparing with ground fiber communication systems, satellite-ground data interaction is slower and less stable. Thus, the postprocessing of satellite-ground QKD should be performed with as few data exchanges as possible. Here, considering its excellent performance with one-time data exchange, we improve Turbo Code to make it suitable for QKD. Furthermore, based on Turbo Code, we implement a 17-km free-space experiment to simulate satellite-ground QKD with a engineering model of an optical transmitter of QSS in Delingha, Qinghai Province.

2. Experimental implementation

We perform a 17-km free-space QKD experiment in Delingha, Qinghai province. As shown in Fig. 1, the transmitter is located in a removable container which is 17 km away from the receiver. At the transmitter, the core equipment is an engineering model of QSS. We place it on a turntable to simulate the satellite movement in orbit. The receiver terminal is at the Qinghai station of Purple Mountain Observatory, which is one of five ground stations for QSS. The quantum optical signal is transmitted by the engineering model with a telescope diameter of 200 mm. To collect the incident light efficiently, we use an optical telescope with a diameter of 1.2 m to receive quantum signals at the receiver station.

 

Fig. 1 The transmitter (a) and receiver (b) telescope. The transmitter telescope is a engineering model of QSS. We place it on a turntable to simulate the satellite movement in orbit. The receiver telescope is a Cassegrain telescope which used to receive the quantum signals at receiver station.

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To establish a stable free-space optical link, we employ an acquisition, pointing, and tracking (APT) system. The two-stage APT system, coarse and fine stage, is designed in the transmitter and the receiver. For the transmitter, the coarse pointing stage consists of a two-dimensional rotatable telescope with a turning range of ±90° in azimuth and −30° ~ 70° in elevation and a camera with a field-of-view of 2.3° × 2.3° and frame rate of 40 Hz. The fine stage is realized via a fast steering mirror (FSM) driven by piezo ceramics. Similar coarse and fine APT systems are also equipped in the ground station (receiver). When we start the QKD experiment, the receiver points the transmitter by its 671 nm beacon laser with a divergence angle of ~1 mrad. The coarse camera in the transmitter detects the 671 nm beacon laser to measure the tracking error. With the feedback control of the two-dimensional rotatable telescope and the camera, the coarse tracking error is less than 300 µrad, which is much smaller than the camera’s field of view. As a subsequent step, the fine close-loop worked via utilizing the FSM and the camera. Simultaneously, the transmitter points a beacon laser with wavelength of 532 nm and divergence angle of 1.25 mrad to the receiver. With a similar cascaded two-stage APT technique, the receiver corrects its pointing direction further. Finally, we establish a stable 17-km free-space optical link with a pointing error of 3 ~ 4 µrad, as shown in Fig. 2

 

Fig. 2 Performance of APT on the X and Y axis measured in the transmitter.

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2.1. Experiment setup

The experiment setup is shown in Fig. 3. In the transmission system, the quantum optical source is composed of eight laser diodes (LDs) which have four polarization states (H, V, +, −) with two intensity states (signal state, decoy state) of the decoy-state BB84 protocol [20–22]. These LDs are modulated by a 100 MHz pulse train based on random numbers. The random numbers are generated from a random physical noise device. This module also simultaneously generates 10 KHz clock synchronous signals to drive a synchronous LD. Finally, according to the experimental requirements, the transmitter can send four types of polarization states of single-photon pulses and synchronous optical signal to implement quantum communication and auxiliary time calibration respectively. In the transmission payload, the quantum optical signal and synchronous light are coupled into the same optical path and transmitted by a telescope system. Through a beam expander, emitted light with a narrow divergence angle can be transmitted farther with a lower loss of quantum signals. The deviation of beam directivity between synchronous light and the quantum optical signal is smaller than the divergence angle of 1/10, which is sufficient to ensure covering of the receiving telescope simultaneously.

 

Fig. 3 A schematic diagram of the QKD setup. In transmission system, the quantum optical source is composed of eight laser diodes (LDs), these LDs are modulated by the 100 MHz pulse train based on random numbers and can send four kinds of polarization states of single photon pulses and synchronous optical signal to implement quantum communication and auxiliary time calibration. In the receiving system, we use telescope to collect the photons flying from the transmitter. After the optical encoding process and detected by SPCMs, the raw key is sent to the upper computer for postprocessing of error correction and privacy amplification. HWP: half-wave plate; BS: beam splitter; PBS: polarizing beam splitter, reflecting vertically polarized photons; DM: dichroic mirror; FSM: fast steering mirror.

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In the receiving system, we use a 1.2-m-diameter telescope to collect the photons transmitted from the transmitter. An integrated optical measurement system consisting of a beam splitter (BS), two polarization beam splitters (PBS), and an half wave plate (HWP) is assembled at the telescope. The polarization compensation against the rotational movement is implemented by a HWP in the receiver, as shown in Fig. 3. Passing through a narrow-band interference filter (full-width at half-maximum bandwidth Δ_FWHM = 10 nm) used to reduce background noise, the photons are coupled in multimode fibers and then detected by single-photon counting modules (SPCMs) with low dark counts (≤ 100/s). After detection by SPCMs, the signal is recorded by a high-time-resolution time-to-digital converter (TDC). Finally, the raw key is sent to the upper computer for post-processing of error correction and privacy amplification.

2.2. Error correction based on turbo code

Error correction in QKD is the process by which Alice and Bob correct discrepancies between two strings, X and Y, via an authenticated classical channel. In any practical implementation of a QKD protocol, X and Y suffer from discrepancies mainly due to losses and fidelity of the channel. Therefore, all QKD protocols must include a classical post-processing to extract the final secret key from X and Y. In a QKD protocol, the information is encoded in binary variables. Errors are usually uncorrelated. For this reason, X and Y can be seen as the input and output of a binary symmetric channel (BSC). Typically, the crossover probability p of the BSC is supposed to be known.

There is one main measure of performance of an error correction scheme: reconciliation efficiency. Reconciliation efficiency f(p) is given by

In the following, we will use this expression to compare the efficiencies of different error correction schemes.

2.2.1. Turbo code

Turbo code can perform very close to the Shannon limit, even with a suboptimal but fast iterative decoding scheme. For reconciliation of binary strings, the Turbo decoding algorithm, specifically the maximum posterior probability (MAP) algorithm, needs to be modified for BSC.

The MAP soft decisions are defined as the log-likelihood ratio:

In QKD, Alice encodes a classical bit onto the polarization or phase of a photon and sends this photon to Bob. After repeating this step k times, Alice and Bob share two k-bit strings, and the crossover probability p of BSC is supposed as known. In public discussion, parity pair y_k is transmitted via an authenticated classical channel. Therefore, we can obtain the following expression:

We can now express the transition probability γ_k (ś, s) as

The structure of a Turbo information reconciliation is shown in Fig. 4. As a Turbo encoder, two identical convolutional encoders are used. The order of the information bits undergoes a pseudo random permutation P prior to been fed into the second convolutional encoder. Many types of permutation can be used, as long as they are sufficiently random. The permutation can guarantee that the two constituent encoders are uncorrelated. Each convolutional encoder is based on a recursive systematic convolutional code. As a turbo decoder, a pair of decoders are used. The decoders are based on the Log-MAP algorithm and output soft decision information learned from the noisy parity bits.

 

Fig. 4 A structure of Turbo code system. In the part of turbo encoder, two identical convolutional encoders are used. In part of turbo decoder, it consists of a pair of decoders which work cooperatively in order to refine and improve the estimate of the original information bits.

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2.2.2. Simulation and analysis

In this section we discuss the simulation performances of Cascade and our Turbo Code for a block length of 10⁴. We implement Cascade described in [5] and our Turbo Code decoded with Log-MAP algorithm. The remaining bit error probability is below 10⁻³ for both Cascade and Turbo Code.

The emulated results of Cascade and Turbo Code are shown in Tab. 1. In the case of a low QBER, better efficiency can be achieved by Cascade, which is very close to the Shannon limit of 1. However, the error correction based on Cascade needs many times of data exchange and the efficiency is significantly deteriorated as the QBER increases. In contrast with Cascade, only one-time data exchange is needed in Turbo Code. Furthermore, the efficiency of error correction based on Turbo Code is good even for high QBERs. Thus, Turbo code is more suitable for satellite-ground QKD.

Table 1. EC f(p) achieved by Turbo code and Cascade. Denote number_Cas and number_Tur are the number of messages exchanged. The error correction efficiencies, f_Cas and f_Tur, represent error correction based on Turbo code and Cascade, respectively.

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3. Postprocessing and results

Due to the detector’s dark counts and stray light, the background count rate was less than 400 s⁻¹ in our experiment. A 2-ns time coincident window was obtained, with contributions of the timing jitter from the synchronous optical signals of 620 ps, avalanche photo diodes (APDs) of 350 ps and the optical pulse width of 125 ps.

The experiment lasted for nearly 2 h at nighttime. During the process, the average channel loss, taking into account the transmission and collection efficiency of the telescopes, the free-space channel loss, and the fiber coupling efficiency, was around 40 dB. Note that there was an additional loss of 1 ~ 2 dB loss introduced by the time coincidence window.

The final key rate is one of main results in an QKD experiment. In our experiment, we obtain the final key rate of 513 bps with the QBER of 3.43 %. For comparison, Cascade and Turbo Code are employed in the same raw key groups. As shown in Tab. 2, with one-time data exchange, the final key rate of experiments based on Turbo Code is even higher than that of Cascade. Our results experimentally show that Turbo Code is a good alternative to Cascade and is potentially useful for satellite-ground quantum communication.

Table 2. Experimental parameters and results. With T seconds effective key distribution, we get some raw key with QBER. We respectively adopt Turbo code and Cascade for error correction. After the process of error correction, we evaluate the R_Tur and R_Cas. In our table, f represents the error correction efficiencies. R_Tur and R_Cas are the final key generation rate based on Turbo code and Cascade.

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4. Conclusion and discussions

We implement a 17-km free-space QKD with efficient Turbo Code error correction. The results show that Turbo Code can be used to reconcile two correlated discrete random variables. In terms of final key generation, using only one-time data exchange, it offers a similar behavior with Cascade for low QBER. For a QBER over 2.5%, Turbo code performs better than Cascade. For satellite-ground QKD, the error correction will require an timely processing, with as few times of data exchanges as possible. Thus, our results experimentally demonstrate that the efficient error correction based on Turbo Code is suitable for satellite-ground QKD. Another possible efficient error correction scheme for satellite-ground QKD might be Low Density Parity Code (LDPC) [23, 24]. Comparing with LDPC, the Turbo Code developed in this work performs better in the case of small block size, which is more suitable for online QKD. Furthermore, due to atmosphere turbulence and the changes in distance, QBER might be very unstable during the process of satellite-ground QKD. Given the QBER as p, the error correction efficiency of the code can be written as $f=\frac{1}{H(p)}\cdot (\frac{1}{R}-1)$. The upper bound of the code rate is $R_{\max }=\frac{1}{1+H(p)}$. To achieve the best error correction efficiency, we need to design the error correction code according to the different QBERs. For the LDPC code, we need to construct different generator matrixes for different code rates. It is hard to construct a new generator matrix in real time when the QBER varies. Thus it is not convenient in satellite-ground QKD since the error rate is constantly changing. While for the Turbo code, it is easy to construct code simply by changing puncturing ratio.