# Twin-Field Quantum Key Distribution over 511 km Optical Fiber Linking two
Distant Metropolitans

Jiu-Peng Chen  The authors contributed equally to the paper. Chi Zhang  The
authors contributed equally to the paper. Yang Liu  Hefei National Laboratory
for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei 230026, China  Shanghai
Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Shanghai 201315, China  Jinan Institute of Quantum Technology, Jinan,
Shandong 250101, China  Cong Jiang  Jinan Institute of Quantum Technology,
Jinan, Shandong 250101, China  State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing 100084, China
Wei-Jun Zhang  State Key Laboratory of Functional Materials for Informatics,
Shanghai Institute of Microsystem and Information Technology, Chinese Academy
of Sciences, Shanghai 200050, China  Zhi-Yong Han  Shi-Zhao Ma  Jinan
Institute of Quantum Technology, Jinan, Shandong 250101, China  Xiao-Long Hu
State Key Laboratory of Low Dimensional Quantum Physics, Department of
Physics, Tsinghua University, Beijing 100084, China  Yu-Huai Li  Hui Liu
Hefei National Laboratory for Physical Sciences at Microscale and Department
of Modern Physics, University of Science and Technology of China, Hefei
230026, China  Shanghai Branch, CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics, University of
Science and Technology of China, Shanghai 201315, China  Fei Zhou  Jinan
Institute of Quantum Technology, Jinan, Shandong 250101, China  Hai-Feng Jiang
Teng-Yun Chen  Hefei National Laboratory for Physical Sciences at Microscale
and Department of Modern Physics, University of Science and Technology of
China, Hefei 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China  Hao Li
Li-Xing You  Zhen Wang  State Key Laboratory of Functional Materials for
Informatics, Shanghai Institute of Microsystem and Information Technology,
Chinese Academy of Sciences, Shanghai 200050, China  Xiang-Bin Wang  Shanghai
Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Shanghai 201315, China  Jinan Institute of Quantum Technology, Jinan,
Shandong 250101, China  State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing 100084, China
Qiang Zhang  Jian-Wei Pan  Hefei National Laboratory for Physical Sciences at
Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei 230026, China  Shanghai Branch, CAS Center for
Excellence and Synergetic Innovation Center in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China

###### Abstract

The basic principle of quantum mechanics Wootters and Zurek (1982) guarantee
the unconditional security of quantum key distribution (QKD) Bennett (1984);
Mayers (1996); Lo and Chau (1999); Shor and Preskill (2000); Scarani et al.
(2009) at the cost of inability of amplification of quantum state. As a
result, despite remarkable progress in worldwide metropolitan QKD networks
Gisin et al. (2002); Xu et al. (2020a) over the past decades, long haul fiber
QKD network without trustful relay has not been achieved yet. Here, through
sending-or-not-sending (SNS) protocol Wang et al. (2018), we complete a twin
field QKD (TF-QKD) Lucamarini et al. (2018) and distribute secure keys without
any trusted repeater over a 511 km long haul fiber trunk linking two distant
metropolitans. Our secure key rate is around 3 orders of magnitudes greater
than what is expected if the previous QKD field test system over the same
length were applied. The efficient quantum-state transmission and stable
single-photon interference over such a long distance deployed fiber paves the
way to large-scale fiber quantum networks.

## Introduction

Quantum non-cloning theorem Wootters and Zurek (1982) forbids perfect cloning
of unknown quantum states or non-orthogonal states simultaneously. Based on
the theorem, the first QKD protocol, BB84 Bennett (1984) has been proposed, in
which non-orthogonal states are exploited to encode random key and distributed
between two authorized users. Any probe from an unauthorized party intending
to steal the key can be seen as a kind of clone, which will inevitably bring
additional bit error according to the non-cloning theorem. This bit error can
be found during the authorized users’ post processing and thus the information
theoretical security is achieved Mayers (1996); Lo and Chau (1999); Shor and
Preskill (2000).

After that, QKD has been extensively studied Bennett (1984); Gisin et al.
(2002); Scarani et al. (2009); Xu et al. (2020a) and grown to a matured
technology in a real-world application today. Quite a few metropolitan fiber
network has been built all over the world Peev et al. (2009); Stucki et al.
(2011); Chen et al. (2009); Dynes et al. (2019); Chen et al. (2010); Wang et
al. (2010). Similar as the optical fiber communication (OFC), the quantum
state will also be attenuated exponentially with the transmission distance.
For OFC, optical amplifier is used to rely the optical signal every 80 km
Singal (2017) to build up a long haul fiber network. The amplifier for single
unknown quantum state, however doesn’t exist because the amplification of
quantum signal can also be seen as a type of clone and any cloning of unknown
quantum states will bring fatal errors Wootters and Zurek (1982). Therefore, a
simple optical amplifier can’t be used for long haul fiber QKD. So far the
longest reported field test of QKD is around 90 km Chen et al. (2021).

Trustful relay is utilized to build up the long haul trunk QKD Chen et al.
(2021) but the many relay stations must be well isolated and trustful.
Meanwhile, quantum repeater Duan et al. (2001); Yu et al. (2020) has been
invented to replace optical amplifier. But the current technology can only
reach 50 km van Leent et al. (2020); Li et al. (2020). Recently, it has been
shown that there exists an upper bound for both repeater-less QKD or OFC
Pirandola et al. (2017). Therefore, the current main challenge towards large
scale fiber quantum communication network is to beat the repeater-less bound
and demonstrate QKD over real long haul trunk line.

Among all protocols, TF QKD Lucamarini et al. (2018) promises high key rates
over long distances to beat the repeater-less bound, which enhances the key
rate to a square-root scaling ($\sqrt{\eta}$) to the channel transmittance.
Significantly, the experimental demonstrations of TF-QKD Minder et al. (2019);
Liu et al. (2019); Wang et al. (2019); Zhong et al. (2019); Fang et al.
(2020); Chen et al. (2020) achieved a record-long distribution distance of
more than 500 km Fang et al. (2020); Chen et al. (2020). However, all these
experiments were implemented in a laboratory, leaving the question that
whether this fancy protocol is feasible in a practical scenario.

In general, comparing to lab experiments, field tests in OFC network often
introduce more noise due to complex environmental fluctuation and crosstalk
from adjacent classical fiber communication. The condition becomes even more
challenging for TF-QKD, which requires optical phase stability for single
photon level interference. For example, in the lab, a vibration due to human
voice or walk will influence the optical phase and decrease the interference
visibility Chen et al. (2020). While, in the field, the human voice or
activity can’t be forbidden like in the lab. Considering these challenges,
putting the experiment to a field test is not only necessary to prove its
feasibility, but also an important step to explore the possibilities of the
future global QKD networks.

Here, we have precisely controlled the wavelength of two 500 km away
independent laser sources and fast compensated any small phase fluctuation in
the channel. Then, we present a field test of a TF-QKD experiment through a
total channel length of 511 km ultra-low-loss fiber (including 430 km long
haul fiber and 81 km fiber spool) with a total loss of 89.1 dB connecting
Qingdao and Jinan in Shandong province, China. As far as we know, this is the
longest point to point field test of QKD. The secure key rate ($3.45\times
10^{-8}$ per pulse) at 511 km is higher than the absolute PLOB bound, after
collecting 4.7 hours data for finite size and fluctuation analysis.

## Protocol

We adopt the sending-or-not-sending (SNS) protocol Wang et al. (2018) with
actively odd parity pairing (AOPP) Xu et al. (2020b); Jiang et al. (2019)
method for post data processing in field experiment of TF-QKD. In this
protocol, Alice and Bob use three weak coherent state (WCS) sources in the $X$
basis, $\mu_{1}$, $\mu_{2}$, 0, which are used for the decoy-state analysis.
And in the $Z$ basis, Alice and Bob randomly decide sending the WCS pulses
$\mu_{z}$ or not sending, where those pulses are used to extract the final
keys. To improve the key rate, we take the odd parity error rejection method
through AOPP in the post data processing, which can efficiently deduce the
bit-error rate of the raw keys and thus largely improves the key rate. We use
the zig-zag approach proposed in Ref. Jiang et al. (2020) to analyses the
finite key effects of AOPP, which assure the high performance of SNS-TF-QKD
protocol with AOPP in the finite key size.

In this protocol, Alice and Bob randomly choose the $X$ basis or $Z$ basis
with probabilities $1-p_{z}$ and $p_{z}$. For the $X$ basis, with
probabilities $1-p_{1}-p_{2}$, $p_{1}$ and $p_{2}$, Alice and Bob randomly
prepare and send out pulses with intensities $0$, $\mu_{1}$ or $\mu_{2}$. For
the $Z$ basis, Alice and Bob randomly decides sending the WCS pulses with
intensity $\mu_{z}$ or not sending with probabilities $\epsilon$ and
$1-\epsilon$. For the received pulse pair, if Charlie announces there is only
one detector clicks, we call it as a one-detector heralded events. The
corresponding bits of one-detector heralded events in $Z$ basis are used to
extract the final keys, and the one-detector heralded events in $X$ basis and
mismatching basis are used to estimate the yield of single-photon pairs before
AOPP. Besides, if the phase of a pulse pair in $X$ basis satisfies the phase
post-selection criteria, its corresponding one-detector heralded events can be
used to estimate the phase-flip error rate before AOPP. The detail calculation
of AOPP is presented in the supplement.

## Experiment

We realize TF-QKD in field deployed fiber between two non-adjacent
metropolises, Qingdao (Alice, $N36^{\circ}6^{\prime}13^{\prime\prime}$,
$E120^{\circ}24^{\prime}32^{\prime\prime}$) and Jinan (Bob,
$N36^{\circ}36^{\prime}50^{\prime\prime}$,
$E117^{\circ}6^{\prime}22^{\prime\prime}$) in Shandong province of China,
which are connected by a 430 km long haul fiber network, as shown in Fig 1.
The measurement station is placed in Mazhan (Charlie,
$N36^{\circ}0^{\prime}19^{\prime\prime}$,
$E118^{\circ}42^{\prime}35^{\prime\prime}$) in the middle of the fiber link.
The long haul network contains 12 fibers in the cable and mainly used for
classical fiber communication test. All the 12 fibers are G654.E ultra-low
loss (ULL) fiber with a nominal loss of 0.158 dB/km.

Figure 1: (a) Satellite image indicating the SNS-TF-QKD experiment with field-
deployed fiber between Qingdao (Alice,
$N36^{\circ}6^{\prime}13^{\prime\prime}$,
$E120^{\circ}24^{\prime}32^{\prime\prime}$) and Jinan (Bob,
$N36^{\circ}36^{\prime}50^{\prime\prime}$,
$E117^{\circ}6^{\prime}22^{\prime\prime}$). Independent lasers are used as
light source for encoding; SNSPDs are used for detection in Mazhan (Charlie,
$N36^{\circ}0^{\prime}19^{\prime\prime}$,
$E118^{\circ}42^{\prime}35^{\prime\prime}$) in the middle of the link. (b)/(c)
Experimental setup of Alice/Bob. In Alice’s/Bob’s station, a commercial kilo-
Hz continuous-wave fiber laser is locked to an ultra-low-expansion (ULE) glass
cavity as the light source. AOM: acoustic optical modulator, BS: beam
splitter, PM: phase modulator, HWP: half wave plate, PBS: polarization beam
splitter, QWP: quarter wave plate, PD: photoelectric detector. Then the
frequency locked stable laser is applied to accomplish encoding by two phase
modulators (PMs) and three intensity modulators (IMs). Finally the encoded
light pulses are attenuated into single photon level with passive attenuator
(ATT) and sent to the measurement station. PC: polarization controller. (d)
Experimental setup of Charlie. In Charlie’s station, two electric polarization
controllers (EPCs) are used to real-time feedback the polarization of the
pulses from Alice and Bob, Dense Wavelength Division Multiplexers (DWDMs) to
filter the leakage of classical communication in other fibers and the
nonlinear scattering from strong reference pulses, and circulators (CIRs) to
block the reflection of SNSPDs. (e) Sectional view of a bundle of optical
fibers in the field-deployed optical cable. Three of a bundle of 12 fibers are
used in this experiment: one for single-photon level signal transmission (QKD
link), one for clock synchronization and “start” signal (Synchronization), and
one for wavelength calibration to lock the optical frequency between Alice’s
and Bob’s independent lasers (Wavelength calibration). Besides our experiment,
classical communications are running in the other 8 (Classical communication)
and one is idle (Idle link) of these 12 fibers.

Among the 12 fibers, we have rented 3 for our experiment and the other 9 are
still for classical fiber communication of China Unicom Ltd. Our three fibers
are utilized to transfer quantum signal, optical synchronizing signal and
laser frequency locking signal, respectively. The time frequency dissemination
signal is used to lock the two independent lasers’ frequency in Jinan and
Qingdao. And for the synchronization channel and wavelength locking channel,
erbium-doped fiber amplifiers (EDFAs) are used about per 70 km to amplify the
signal. While the quantum single doesn’t have any relay node.

Similar to our previous TF-QKD experiment in the lab Chen et al. (2020); Liu
et al. (2019), Alice and Bob use independent lasers: the commercial kilo-Hz
continuous-wave fiber laser are locked to ultra-low-expansion (ULE) glass
cavity in both sides as the light source. The linewidth of both Alice’s and
Bob’s laser is less than 1 Hz. The central wavelength of Alice’s laser is set
to 1550.12460 nm. We then split the light into two, one part is to transfer
quantum signal and the other one is used for wavelength locking. Instead of
locking two lasers in a same room, the field experiment needs to lock the two
lasers 400 km away. Alice adopts the time frequency dissemination technology
Predehl et al. (2012), and sends the frequency locking laser through the the
84.1 dB (430 km) fiber channel with 6 EDFAs in between. The light interferes
with Bob’s laser at a photodiode. Based on the beating result, Bob set his
laser wavelength with an acoustic-optic modulator (AOM). The relative
frequency drift of the two sources is measured to be approximately 0.1 Hz/s,
giving an accumulated phase difference of about $\pi/60$ per hour. Therefore,
instead of continuing calibrating the wavelength, we only need to calibrate
the wavelength difference per hour or two, which is good enough for the
experiment. This can eliminate the crosstalk noise from the wavelength locking
channel.

After locking the wavelength, the fiber fluctuation becomes the main source of
phase noise. The ambient temperature change during the day-night cycle affects
the effective length of the deployed fiber, and thus the phase and the arrival
time of the signal. We monitor the arrival time of the signal by statistic the
arriving time of the reference pulse at the PBS’s idler port from Alice and
Bob. As shown in Fig 2(a), without any feedback the arrival time is strongly
related to the ambient temperature, which changes up to 20 ns in one day. The
arrival time drift may also deteriorate the interference of the signal, by
affecting the overlapping between the two pulses.

We utilize two stage procedure to fix it. First, we synchronize Alice and
Bob’s laser pulses with Charlie’s clock Rubenok et al. (2013); Cao et al.
(2020). Charlie sends two 250 MHz synchronized laser pulse trains to Alice and
Bob through the synchronization fiber channel, respectively. The two laser
trains are used to lock the local clock of the signal generators, which then
generate signal pulses in both sides with a pulse duration of 280 ps. The
measured arrival times in Charlie side are used for feedback signal to adjust
the relative delay between Alice’s and Bob’s synchronization laser trains.
With such an arrangement, the fluctuation of the arrival time decrease to 10
ps, negligible comparing to the signal pulse duration of 280 ps.

Figure 2: (a) The signal arrival time with (orange curve) and without (blue
curve) feedback, compared with the temperature change during the measurement
(November 24, 2020, red dot). Insert: Magnified view of the signal arrival
time with feedback. (b)The phase drift rate is measured as 0.03 rad/us through
the field-deployed fiber.

After fixing the time drifting , the phase fluctuation is the target for the
second stage, which is also larger than the fluctuations in lab, due to
environment vibration. We measured a phase rate of 0.03 $rad/\mu s$ through
field-deployed fiber, or equivalent to 17 degree in 10 $\mu$s, as shown in Fig
2(b). The phase reference pulse and the corresponding phase estimation method
Chen et al. (2020); Liu et al. (2019) are used to compensate for this phase
drift in post-processing. We time-multiplex the quantum signal together with
strong phase reference pulses using intensity modulator. The superconducting
nanowire single photon detectors (SNSPD) used in our experiment has a 60 dB
dynamic range, which can measure both the quantum signal and strong phase
reference signal. According to the measurement result of phase reference
signals, we can compensate the phase difference in postprocessing, and then
the optical phase sensitive interference is achieved. (See supplementary for
details of the experimental setup)

Besides the time drifting and phase fluctuation, the polarization drift in the
deployed fiber also ruins the interference. To solve this problem, electric
polarization controllers (EPCs) and polarization beam splitters (PBSs) are
inserted before the beam splitter in Charlie. The signals from the idler port
of the PBS are directed to a SNSPD to monitor the polarization, for Alice and
Bob respectively. The target counting rate is set to 175 kHz for feedback.
Without the feedback, the intensity changes 30% in 4 hours; which is almost
stable when the feedback is on. (See Supplementary Material for details).

After fixing all these fluctuations, we also need to filter out the crosstalk
noise from adjacent fiber bundles, which is also new comparing to lab
experiment. The crosstalk noise count rate from synchronization pulses and
classical communication in the fiber bundle is around 486 count per second
(cps) and 616 cps for Alice and Bob respectively. We set our QKD wavelength
out of the noise spectrum and add a 100 GHz (0.8 nm) Dense Wavelength Division
Multiplexing (DWDM) to filter out the noise. After the filter, these crosstalk
noises drops to less than 10 cps for both sides.

Besides crosstalk noise counts, there are also noise counts from re-Railey
scattering noise Chen et al. (2020), which is caused by the backscattering of
the Railey backscattered phase reference light and detector dark counts. We
estimate this re-Railey scattering noise to about 8 cps on each single photon
detector, based on the detected reference counts. The dark counts for SNSPDs
are 4 cps and 6 cps, with detection efficiency of 77% and 91% respectively. In
total, the noise counts for each detector is about 24 cps. We then further use
a 350 ps time filtering to reduce the noise probability per gate, achieving a
noise count rate of $7\times 10^{-9}$.

With all the above improvements, we realized the field test of SNS-TF-QKD with
in total 511 km ultra low loss fiber deployed between Jinan and Qingdao. The
experimental parameters and results are shown in Tab. 1, and can be found in
detail in Supplementary Materials. With an effective system frequency of 100
MHz, we sent $1.679\times 10^{12}$ pulses in approximately 4.7 hours. A total
of $4.987\times 10^{6}$ valid detections are recorded, where one and only one
detector clicks. The raw bit-flip error rate before AOPP is 24.39%, which
drops to 0.431% after AOPP. The survived bits after AOPP decreases to 576130.
These values are in agreement with the theoretically expected values very
well, which are 0.426% and 576295 respectively.

The secure key rate considering data fluctuation and the finite data size
effect is calculated following the theory Jiang et al. (2019):

$\begin{split}R=&\frac{1}{N_{t}}\\{n_{1}^{\prime}[1-H(e_{1}^{ph^{\prime}})]-fn_{t}^{\prime}H(E_{Z})\\}\\\
&-2\log_{2}\frac{2}{\varepsilon_{cor}}-2\log_{2}\frac{1}{\sqrt{2}\varepsilon_{PA}\hat{\varepsilon}}\\},\\\
\end{split}$ (1)

where $R$ is the final key rate, $n_{1}^{\prime}$, $e_{1}^{ph}$,
$n_{t}^{\prime}$ and $E_{Z}$ are the number of untagged-bits, the phase-flip
error rate, the number of survived bits, and the bit-flip error rate of
untagged-bits after AOPP, $f$ the error correction efficiency where we set to
$f=1.1$, $N_{t}$ the total number of signal pulses, $\varepsilon_{cor}=1\times
10^{-10}$ the failure probability of error correction process,
$\varepsilon_{PA}=1\times 10^{-10}$ the failure probability of privacy
amplification process, $\hat{\varepsilon}=1\times 10^{-10}$ the coefficient of
the chain rules of smooth min- and max- entropy.

Table 1: Experimental parameters and results. The parameters $\mu_{1}$, $\mu_{2}$ and $\mu_{z}$ are the intensities of the sent pulses of weak decoy state, strong decoy state and signal state, $p_{1}$ and $p_{2}$ are the probabilities of weak decoy state and strong decoy in $X$ basis, $p_{z}$ is the probability of $Z$ basis, $\epsilon$ is the sending probability in $Z$ basis. $\mu_{1}$ | 0.100
---|---
$\mu_{2}$ | 0.298
$\mu_{z}$ | 0.422
$p_{1}$ | 0.846
$p_{2}$ | 0.076
$p_{z}$ | 0.735
$\epsilon$ | 0.269
$n_{t}^{\prime}$ | 576130
$n_{1}^{\prime}$ | 219136
$e_{1}^{ph}$ | 16.06%
$E_{Z}$ | 0.43%
$N_{t}$ | $1.679\times 10^{12}$
Valid detections | $4.987\times 10^{6}$
Key rate | $3.45\times 10^{-8}$

The secure key rate is calculated as $R=3.45\times 10^{-8}$, which is about
3.45 bps considering our system frequency. We plot this value and the
theoretical simulation in Fig. 3, where we also plot recent TF-QKD
experimental results as a comparison. With the improvement in both theory and
experiment, our secure key rate is more than 10 times higher than the absolute
PLOB bound. The secure key rate is also 3 orders higher than previous field
test MDI-QKD experiment Tang et al. (2016), if considering linear decreasing
and assuming no detector dark count. This result demonstrates the capability
of the SNS-TF-QKD protocol with AOPP in practical environment with field
deployed fibers.

Figure 3: Secure key rates of the SNS-TF-QKD experiment. The red star
indicates the experimental result over 511 km field-deployed fibers, with the
secure key rate of $R=3.45\times 10^{-8}$. The yellow circle and the green
square experimental result of ref.[24] and ref.[27], respectively, with
attenuators simulating the channel loss. The four triangles indicate the
experimental results of ref.[25], ref.[26], ref.[28] and ref.[29], with in lab
fiber spools. The cyan curve is the simulation result with the experimental
parameters. The blue dotted line and brown dashed line shows the relative and
absolute PLOB bound Pirandola et al. (2017).

## Conclusion

In conclusion, we experimentally demonstrated SNS-TF-QKD with two independent
laser sources, in 511 km long haul field-deployed fibers between the city of
Jinan and Qingdao. The generated secure key rate is an order higher than the
absolute key rate limit of the repeater-less QKD protocol and 3 orders of
magnitudes greater than what is expected if the previous QKD field test system
over the same length were applied. Our work proves the feasibility of TF-QKD
in a practical environment and paves the way towards long haul fiber quantum
network. The techniques developed in the work will find immediate applications
in more general applications like in quantum repeater Duan et al. (2001), and
phase-based architecture for the quantum internet Wehner et al. (2018).

### Acknowledgments

This work was supported by the National Key R&D Program of China (Grants No.
2017YFA0303900, 2017YFA0304000, 2020YFA0309800), the National Natural Science
Foundation of China, the Chinese Academy of Sciences (CAS), Shanghai Municipal
Science and Technology Major Project (Grant No.2019SHZDZX01), Key R&D Plan of
Shandong Province (Grant No. 2019JZZY010205, 2020CXGC010105), the Taishan
Scholar Program of Shandong Province, and Anhui Initiative in Quantum
Information Technologies.

## The SNS-TF-QKD protocol with odd-parity error rejection

The 4-intensity SNS-TF-QKD protocol is used to perform the experiment, and the
theory of actively odd-parity paring (AOPP) with finite key effects is used to
get high key rate.

In the whole protocol, Alice and Bob send $N$ pulse pairs to Charlie, and
Charlie is supposed to announce an one-detector heralded event if his received
pulse pair cause only one detector clicks. At each time, Alice (Bob) randomly
chooses the $X$ basis with probability $1-p_{z}$, or a $Z$ basis with
probability $p_{z}$. If the $Z$ basis is chosen, with probability $\epsilon$,
Alice (Bob) randomly prepares a phase-randomized weak coherent state (WCS)
pulse with intensity $\mu_{z}$, and denote it as bit $1$ ($0$); with
probability $1-\epsilon$, Alice (Bob) prepares a vacuum pulse, and denote it
as bit $0$ ($1$). If the $X$ basis is chosen, Alice (Bob) randomly prepares a
vacuum pulse or coherent state pulse $\ket{e^{i\theta_{A1}}\sqrt{\mu_{1}}}$ or
$\ket{e^{i\theta_{A2}}\sqrt{\mu_{2}}}$ ($\ket{e^{i\theta_{B1}}\sqrt{\mu_{1}}}$
or $\ket{e^{i\theta_{B2}}\sqrt{\mu_{2}}}$) with probabilities
$p_{0}=1-p_{1}-p_{2}$, $p_{1}$ and $p_{2}$, respectively, where
$\theta_{A1},\theta_{A2},\theta_{B1}$ and $\theta_{B2}$ are different in
different windows, and are random in $[0,2\pi)$.

A time window that both Alice and Bob choose the $Z$ basis is called a $Z$
window. One-detector-heralded events in $Z$ windows are called effective
events. Alice and Bob get two $n_{t}$-bit strings formed by the corresponding
bits of effective events of the $Z$ windows. For a time window that both the
intensities of Alice’s and Bob’s WCS pulse is $\mu_{1}$, and if the phases of
the WCS pulses satisfy

$1-|\cos(\theta_{A1}-\theta_{B1}-\psi_{AB})|\leq\lambda,$ (2)

it is called an $X$ window. Here, $\psi_{AB}$ can take an arbitrary value,
which can be different from time to time as Alice and Bob like, so as to
obtain a satisfactory key rate for the protocol Liu et al. (2019). $\lambda$
is a positive value close to $0$, and would be optimized to obtain the highest
key rate. One-detector-heralded events in $X$ windows are called effective
events.

The data of one-detector heralded events are used to decoy method analysis. To
clearly show the calculation method, we denote the vacuum source, the WCS
source with intensity $\mu_{1},\mu_{2}$, and $\mu_{z}$ of Alice (Bob) by
$o,x,y$ and $z$ ($o^{\prime},x^{\prime},y^{\prime}$, and $z^{\prime}$). We
denote the number of pulse pairs of source
$\kappa\zeta(\kappa=o,x,y,z;\zeta=o^{\prime},x^{\prime},y^{\prime},z^{\prime})$
sent out in the whole protocol by $N_{\kappa\zeta}$, and the total number of
one-detector heralded events of source $\kappa\zeta$ by $n_{\kappa\zeta}$. We
define the counting rate of source $\kappa\zeta$ by
$S_{\kappa\zeta}=n_{\kappa\zeta}/N_{\kappa\zeta}$, and the corresponding
expected value by $\langle S_{\kappa\zeta}\rangle$.

Then we can use the decoy-state method to calculate the lower bounds of the
expected values of the counting rate of single-photon states
$|01\rangle\langle 01|$ and $|10\rangle\langle 10|$, which are Hu et al.
(2019)

$\displaystyle\langle\underline{s_{01}}\rangle$
$\displaystyle=\frac{\mu_{2}^{2}e^{\mu_{1}}\langle
S_{ox^{\prime}}\rangle-\mu_{1}^{2}e^{\mu_{2}}\langle
S_{oy^{\prime}}\rangle-(\mu_{2}^{2}-\mu_{1}^{2})\langle
S_{oo^{\prime}}\rangle}{\mu_{2}\mu_{1}(\mu_{2}-\mu_{1})},$ (3)
$\displaystyle\langle\underline{s_{10}}\rangle$
$\displaystyle=\frac{\mu_{2}^{2}e^{\mu_{1}}\langle
S_{xo^{\prime}}\rangle-\mu_{1}^{2}e^{\mu_{2}}\langle
S_{yo^{\prime}}\rangle-(\mu_{2}^{2}-\mu_{1}^{2})\langle
S_{oo^{\prime}}\rangle}{\mu_{2}\mu_{1}(\mu_{2}-\mu_{1})}.$ (4)

Then we can get the lower bound of the expected value of the counting rate of
untagged photons

$\langle\underline{s_{1}}\rangle=\frac{1}{2}(\langle\underline{s_{10}}\rangle+\langle\underline{s_{01}}\rangle),$
(5)

and the lower bound of the expected value of the untagged bits $1$,
$\langle\underline{n_{10}}\rangle$, and untagged bits $0$,
$\langle\underline{n_{01}}\rangle$

$\displaystyle\langle\underline{n_{10}}\rangle=Np_{z}^{2}\epsilon(1-\epsilon)\mu_{z}e^{-\mu_{z}}\langle\underline{s_{10}}\rangle,$
(6)
$\displaystyle\langle\underline{n_{01}}\rangle=Np_{z}^{2}\epsilon(1-\epsilon)\mu_{z}e^{-\mu_{z}}\langle\underline{s_{01}}\rangle.$
(7)

We denote the number of total pulses sent out in the $X$ windows by $N_{X}$,
and the number of corresponding error events by $m_{X}$, then we have the
error counting rate of $X$ windows

$T_{X}=\frac{m_{X}}{N_{X}}.$ (8)

Then we have

$\langle\overline{e_{1}^{ph}}\rangle=\frac{\langle
T_{X}\rangle-e^{-{2\mu_{1}}\langle
S_{oo^{\prime}}\rangle/2}}{2e^{-2\mu_{1}}\mu_{1}\langle\underline{s_{1}}\rangle},$
(9)

where $\langle T_{X}\rangle$ is the expected value of $T_{X}$.

With values above, we can calculate the lower bound of untagged bits and
phase-flip error rate after AOPP, $n_{1}^{\prime}$ and $e_{1}^{\prime ph}$ by
the method proposed in Ref. Jiang et al. (2020). We have the related formulas
as follows.

$\displaystyle
u=\frac{n_{g}}{2n_{odd}},\quad\underline{n_{1}}=\varphi^{L}(\langle\underline{n_{10}}\rangle+\langle\underline{n_{01}}\rangle),$
(10a)
$\displaystyle\underline{n_{10}}=\varphi^{L}(\langle\underline{n_{10}}\rangle),\underline{n_{01}}=\varphi^{L}(\langle\underline{n_{01}}\rangle),$
(10b) $\displaystyle
n=\varphi^{L}\left(\frac{{\underline{n_{1}}}}{n_{t}}\frac{{\underline{n_{1}}}}{n_{t}}\frac{un_{t}}{2}\right),$
(10c) $\displaystyle k=u\underline{n_{1}}-2n,$ (10d) $\displaystyle
r=\frac{2n+k}{k}\ln\frac{3k^{2}}{\varepsilon(r,k)},$ (10e)
$\displaystyle\bar{M}=\varphi^{U}(2n\langle\overline{e_{1}^{ph}}\rangle),$
(10f) $\displaystyle\langle e_{\tau}\rangle=\frac{\bar{M}}{2n-r}$ (10g)
$\displaystyle\bar{M}_{s}=\varphi^{U}[(n-r)\langle e_{\tau}\rangle(1-\langle
e_{\tau}\rangle)]+r,$ (10h)

where $n_{g}$ is the number of pair if Alice and Bob perform AOPP to their raw
keys, ${n_{odd}}$ is the number of pairs with odd-parity if Bob randomly
groups all the bits in $Z_{B}$ two by two, and $n_{g}$ and $n_{odd}$ are
observed values, $\varepsilon(r,k)=10e^{-10}$ is the trace distance while
using the exponential de Finetti’s representation theorem, and
$\varphi^{U}(x),\varphi^{L}(x)$ are the upper and lower bounds while using
Chernoff bound Chernoff et al. (1952) to estimate the real values according to
the expected values.

Finally, we can get the key rate $R$ by

$\begin{split}R=&\frac{1}{N}\left\\{n_{1}^{\prime}[1-h(e_{1}^{\prime
ph})]-fn_{t}^{\prime}h(E^{\prime})-2\log_{2}{\frac{2}{\varepsilon_{cor}}}\right.\\\
&\left.-2\log_{2}{\frac{1}{\sqrt{2}\varepsilon_{PA}\hat{\varepsilon}}}\right\\}.\end{split}$
(11)

where $n_{1}^{\prime}$ is the number of the untagged bits after AOPP,
$e_{1}^{\prime ph}$ is the bit flip error rate of untagged bits after AOPP,
$h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$ is the Shannon entropy, $E^{\prime}$ is
the bit-flip error rate of the remaining bits after AOPP, $\varepsilon_{cor}$
is the failure probability of error correction, $\varepsilon_{PA}$ is the
failure probability of privacy amplification, and $\hat{\varepsilon}$ is the
coefficient while using the chain rules of smooth min- and max- entropy
Vitanov et al. (2013).

## Setup of the experiment

Since the field experimental conditions are different from that in the
laboratory where the channel can be well controlled, the real-life conditions
have to be considered now. For example, the wavelength locking of physical
separated lasers, the additional crosstalk noise from adjacent fibers, the
larger phase fluctuation and even the drifting in the signal arrival time.
Considering these challenges, we have fully upgraded our whole system in
experimental side. To implement the field SNS-TF-QKD above, we make our
experimental setup as shown in Fig. 4. Among the 12 fibers in one cable, we
have rented 3 for our experiment: one for single-photon level signal
transmission, one for clock synchronization and “start” signal distribution,
and one for wavelength calibration to lock the optical frequency between
Alice’s and Bob’s independent lasers. For about every 70 km, EDFAs are used to
maintain the optical power of these classical signals. Besides our experiment,
the other 9 are still for classical fiber communication of China Unicom Ltd.

Figure 4: chematic of our experimental setup. Alice and Bob respectively adopt
a ultra stable continuous wave (CW) laser with a linewidth less than 1 Hz by
using the technique of Pound-DreverHall (PDH) Drever et al. (1983); Pound
(1946). These light sources are then modulated by two phase modulators (PMs)
and three intensity modulators (IMs) for phase randomization, reference and
decoy intensity modulation. The pulses are then attenuated by an attenuator
(ATT) and sent out via quantum fiber channel to Charlie. At Charlie’s station,
the encoded pulses from Alice and Bob are adjusted with electric polarization
controllers (EPCs) and polarized with polarization beam splitters (PBSs),
filtered with Dense Wavelength Division Multiplexings (DWDMs) and circulators
(CIRs), and finally interfered at a beam splitter (BS). The pulses are
detected by superconducting nanowire single-photon detectors (SNSPDs), PC:
polarization controller. Besides the Quantum fiber channel, two additional
fibers are used in this experiment: one for clock synchronization and “start”
signal, and one for wavelength calibration to lock the optical frequency
between Alice’s and Bob’s independent lasers. For about every 70 km, EDFAs are
used to maintain the optical power of these classical signals.

### Stable independent light sources

Similar to our previous TF-QKD experiment in the lab Chen et al. (2020); Liu
et al. (2019), Alice and Bob respectively adopt a ultra stable continuous wave
(CW) laser with a linewidth less than 1 Hz as the light source. First, they
use a commercial continuous wave laser (NKT E15) with a linewidth of about 1
kHz at the center wavelength of 1550.12 nm as the seed laser. Then Alice locks
her seed laser to a 5-cm-long ultra-stable cavity with a finesse around
400000, and Bob locks his seed laser to a 10-cm-long ultra-stable cavity with
a finesse around 400000, using the Pound-DreverHall (PDH) technique Drever et
al. (1983); Pound (1946), to suppress their line-width from about 1 kHz to
less than 1 Hz. Since the ultra-stable cavity is machined by ultra-low-
expansion (ULE) glass, the resonant frequency of each cavity is different,
after the seed laser is respectively locked to their cavity, finally, they
additionally need to calibrate the frequency/wavelength difference of the two
stable laser sources. Compared to the experiments in the laboratory Chen et
al. (2020); Liu et al. (2019), we change the way the twin-field light sources
works to decrease the crosstalk noise. We calibrate the frequency difference
of the two independent light sources intermittently instead of continuous
frequency locking. We first calibrate the wavelength difference of the two
independent light sources before the experiment, with strong CW reference
light in the fiber; Then when the experiment is running, the strong reference
is blocked, and the wavelength between the two light sources are left to free
drifting. With stable light sources, the wavelength difference will be small
for the experiment between the wavelength calibration cycles.

Here we give a brief introduction on the PDH method. As shown in Fig. 5, after
frequency modulated by an acoustic optical modulator (AOM), the seed laser
with a frequency of $\nu$ is phase modulated by an electro-optical modulator
(EOM) with a radio-frequency (RF) of $\frac{\Omega}{2\pi}$ coupling in the ULE
ultra-stable cavity through a polarization beam splitter (PBS) and a quarter-
wave plate (QWP), then the input of the cavity contains optical signals at
frequencies of $\nu$ \- $\frac{\Omega}{2\pi}$ , $\nu$ and $\nu$ \+
$\frac{\Omega}{2\pi}$. The signals at frequencies of $\nu$ \-
$\frac{\Omega}{2\pi}$ and $\nu$ \+ $\frac{\Omega}{2\pi}$ are sidebands of the
modulation. Then detected by a photoelectric detector (PD), the reflected
carrier wave is the beat-note signal between the carrier and sidebands. If the
carrier frequency $\nu$ of the seed laser is close to a resonant frequency of
the cavity, the reflected carrier wave can be regarded as a combination of the
backward leakage of cavity-stored signal and the reflected incident signal.
Further with the reference signal from the oscillator, the corresponding error
signal is obtained by demodulating the beat-note by a mixer. The phase shifter
is required to set the phase relationship between the demodulation and
detected signals to be in phase, leading to a maximum demodulation efficiency.
Thus based on the error signal, we use two proportional integral servo
controllers to realize the feedback. The slow feedback with a bandwidth of
about 6 kHz is accomplished by adjusting the piezoelectric (PZT) of the seed
laser, and the fast feedback with a bandwidth of about 100 kHz by adjusting
the carrier frequency of the AOM.

Figure 5: Scheme of the PDH technique.

### The encoding details

Based on these two high-performance narrow line-width coherent light sources,
Alice and Bob individually implement the encoding by applying two phase
modulators (PMs) to encode the light to 16 different phase slices and three
intensity modulators (IMs) to 5 different intensities respectively. With each
basic period of 1 $\mu$s time sequence, the first 500 ns they send 100 signal
pulses with 4 different random intensities and 16 phase values are
respectively used as the signal state pluses, strong decoy state pluses, weak
decoy state pluses, and vacuum state pluses, each with a 280 ps pulse duration
and 4.72 ns interval, the following 400 ns they send 4 strong phase reference
pulses with a same intensity and determined phase values, each with a 100 ns
pulse duration, the last 100 ns they send nothing as the recovery time for the
superconducting nanowire single-photon detectors (SNSPDs).

For the phase encoding, during the basic period of 1 $\mu$s, Alice (Bob)
randomly modulates the phase to $\theta_{A}$ ($\theta_{B}$) on 100 signal
pulses with a 5 ns duration in the first 500 ns, $\theta_{A}$,
$\theta_{B}\in\\{0,\pi/8,2\pi/8...15\pi/8\\}$, in the next 400 ns, Alice
orderly modulates the phase to 0, $\pi/2$, $\pi$, $3\pi/2$, while Bob
modulates the phase to $\pi$ on 4 reference pulses with a each 100 ns
duration, and then they both modulate the phase to $\pi$ on vacuum state
pulses in the last 100 ns. Following the phase modulation, the two users
modulate the light to 5 different intensities with three IMs. During the basic
period of 1 $\mu$s, Alice and Bob randomly modulate the pulse intensity to
$\mu_{x}$, $\mu_{x}\in\\{\mu_{z},\mu_{1},\mu_{2},0\\}$ on 100 signal pulses
with a 280 ps duration and 4.72 ns interval in the first 500 ns, $\mu_{z}$ is
the intensity of the signal state, $\mu_{1}$ is the intensity of the weak
decoy state, $\mu_{2}$ is the intensity of the strong decoy state, and $0$ is
the vacuum state, in the next 400 ns, they modulate the intensity to
$\mu_{ref}$ on 4 reference pulses with a each 100 ns duration, finally they
modulate the intensity to $0$ on vacuum state pulses in the last 100 ns.

With pre-generated quantum numbers, all the modulation signals are generated
by a high-speed AWG (Tektronix, AWG70002A) with a maximum output of 500 mV and
amplified by the electronic amplifiers, thus, in a basic period, the encoding
waveform of the phase and intensity modulators would randomly generate
different durations and amplitudes patterns, especially the full wave driving
voltage of the PM is approximate to 9 V peak-peak value, normal RF amplifiers
are difficult to maintain without distortion after amplifying the waveform
with a such high gain, however, the distortion of the driving waveform of the
PM would inevitably contribute to QBER in $X$-basis and that of the IM would
decrease the stability of the encoding, therefore by applying 5 linear voltage
amplifiers (LTC6433) with a maximum output of 5.2 V peak-peak to drive the two
PMs and three IMs, Alice and Bob can maintain low modulation error rate and
high encoding stability. Additionally, Alice and Bob insert an active bias
point feedback to the first IM which encodes the decoy state and place all the
PMs and IMs into a thick foam box to reduce the fluctuations of the ambient
temperature to improve the stability of the encoding. After that the signals
from both sides are attenuated into single photon level with passive
attenuators and sent to the measurement station (Charlie) through a total
channel length of 511 km with a loss of 89.1 dB.

### The field fiber links

In the experiment, Alice and Bob are placed in two non-adjacent metropolises,
Qingdao (Alice, $N36^{\circ}6^{\prime}13^{\prime\prime}$,
$E120^{\circ}24^{\prime}32^{\prime\prime}$) and Jinan (Bob,
$N36^{\circ}36^{\prime}50^{\prime\prime}$,
$E117^{\circ}6^{\prime}22^{\prime\prime}$) which are separated by other two
metropolises, Zibo and Weifang, while Charlie in Mazhan (Charlie,
$N36^{\circ}0^{\prime}19^{\prime\prime}$,
$E118^{\circ}42^{\prime}35^{\prime\prime}$) in the middle of the fiber link,
in Shandong province, China. We in total used three fiber links, the QKD link,
synchronization link and wavelength calibration link. They are all in a same
bundle, meanwhile there are one idle fiber link and 8 running fiber links of
classical communication in it. Based on the QKD link, Alice and Bob spatia
separately apply a high speed arbitrary waveform generator (AWG) to randomly
generate encoding waveforms, then amplify the waveforms by electronics
amplifiers to drive the phase modulators (PMs) and intensity modulators (IMs)
to modulate their stable CW laser, ultimately send the modulated pulses to
Charlie for interference.

A total channel length of 511 km ultra-low-loss fiber (G654.E ULL) of the QKD
link connecting Alice, Charlie and Bob, which consists of 430 km buried fiber
with a loss of 76 dB and 81 km ultra-low loss optical fiber spool with a loss
of 13.1 dB. The physical separation distance is more than 300 km between Alice
and Bob. The link from Alice (Bob) to Charlie consists of 207.6 km (222.7 km)
buried fiber with a loss of 35.5 dB (40.5 dB) and 55.5 km (25.5 km) ultra-low
loss fiber spool with a loss of 9 dB (4.1 dB). The total loss of the ultra-low
loss fiber link is 89.1 dB (Alice to Charlie 44.5 dB, Bob to Charlie 44.6 dB),
which is around 0.174 dB/km on average. The excess loss than the nominal loss
of G654.E fiber (0.158 dB/km) is attributed to the fiber connections at the
nodes along the link. There are two (three) nodes along the link between Alice
(Bob) and Charlie, the corresponding lengths between nodes are shown in Tab.
2.

Table 2: The fiber length between nodes. Parameters | Fiber Length
---|---
Qingdao (Alice) to Huangshan | 68 km
Huangshan to Zhucheng | 68.3 km
Zhucheng to Mazhan (Charlie) | 71.3 km
Fiber spool (Alice’s side) | 55.5 km
Jinan (Bob) to Bogeyan | 71.4 km
Bogeyan to Yuanshan | 28.9 km
Yuanshan to Yiyuan | 62 km
YiYuan to Mazhan (Charlie) | 60.4 km
Fiber spool (Bob’s side) | 25.5 km

Through the synchronization link (430 km length, 82.6 dB loss), Charlie
distributes synchronous clocks and the system start signals to Alice and Bob
to synchronize and trigger their AWG, and to synchronize his own high speed
multichannel time tagger locally. At the station of Charlie, an AWG
(Tektronix, AWG5202) is used to drive a multi-channel external modulation
laser module to generate 250 MHz pulses train with a wavelength of 1548.11 nm
as the synchronous clock, and every time at the beginning of the experiment,
to originate 100 kHz pulses train with a wavelength of 1560.20 nm as system
start signal. In the following the synchronous clock and system start signal
are respectively sent to Alice and Bob, and 5 EDFAs are inserted in the link
(EDFAs are placed in about every 70 km, Bogeyan, Yuanshan, Yiyuan, Zhucheng,
Huangshan) to relay them. Accordingly, at the station of Alice (Bob), a DWDM
and two photoelectric detectors are used to extract the synchronous clock and
system start signals.

Additionally, by using the wavelength calibration link, Alice send the
wavelength reference laser to Bob for calibrating the wavelength difference of
the two independent laser sources. The total length of the wavelength
calibration link is 430 km, 6 erbium-doped fiber amplifiers (EDFAs) are used
in between to compensate a loss of 84.1 dB. At the station of Bob, when he
receives the wavelength reference laser from Alice, he divides parts of his
light with a intensity of about 500 $\mu$w interferes with the received
reference light at a photodiode. Based on the beating result, Bob set his
laser wavelength to the same as Alice’s with an acoustic-optic modulator
(AOM). After the frequency/wavelength of the two lasers is calibrated to be
same, we measured the relative phase drift of the twin-field (phase drift
angle and the phase drift rate) with field fiber links between Alice and Bob,
as is shown in Fig. 6, the corresponding phase drift rate follows a Gaussian
distribution with a standard deviation of $0.03rad\cdot\mu s^{-1}$. In the
experiment, we applied accumulating of 10 basic periods (10 $\mu$s) of the
reference detections for the relative phase estimation, the corresponding
relative phase drift is around 0.3 rad (or 17 degree) which contributes to an
error is less than $4\%$ when Alice and Bob sent the same phase.

Figure 6: The relative phase drift of field fiber links.

Figure 7: The distribution of the crosstalk from the classical communication.

Since the synchronization link, wavelength calibration link and other 8
running fiber links of classical communication in the same bundle with the QKD
link. The signals running in these links would leak into the QKD link as
noise. We measured the spectral distribution of the crosstalk from the 8
running fiber links of classical communication, which is shown in Fig. 7, the
crosstalk is mainly distributed between 1548.5 nm and 1549.5 nm. Therefore we
set a wavelength of 1550.12460 nm of the QKD sources to keep away from the
main distribution range of crosstalk from the 8 running fiber links of
classical communication, and insert a 100 GHz DWDM (1550.12 nm) in measurement
station to filter the noise. Meanwhile the leakage of the synchronization
pulses is also filtered out with the DWDM. In addition, another important
source of noise is the leaked wavelength calibration light. As the wavelength
of the calibration light is the same as the signal, we cannot filter them with
the DWDM. To reduce this noise, we adopt the time-multiplexing method. We
first calibrate the wavelength difference of the two independent light sources
before the experiment, with strong CW reference light in the fiber; Then when
the experiment is running, the strong reference is blocked, and the wavelength
between the two light sources are left to free drifting. With stable light
sources, the wavelength difference will be small for the experiment between
the wavelength calibration cycles. We usually calibrate it about every one or
two hours. Finally, as is shown in Tab. 3, we tested all the noises above and
the dark count of the SNSPDs, and estimate the noises from re-Railey
scattering noise Chen et al. (2020). Sum all the above noise sources together,
we achieve about 24 cps (count per second) noise counts for each detector. We
further use a 350 ps narrow gate, to reduce the noise probability per gate,
achieving a dark count rate of $7\times 10^{-9}$.

Table 3: The noise in the system. Noise | Counts
---|---
Classic communication leakage from Alice (direct output) | 486 cps
Classic communication leakage from Bob (direct output) | 616 cps
Classic communication leakage from Alice (with filtering) | 8.6 cps (SNSPD1) and 8.5 cps (SNSPD2)
Classic communication leakage from Bob (with filtering) | 6.7 cps (SNSPD1) and 6.2 cps (SNSPD2)
Strong reference leakage | 32 cps (SNSPD1) and 35 cps (SNSPD2)
Dark count (SNSPD1) | 4 cps
Dark count (SNSPD2) | 6 cps

### The feedback details of the system

In our experiment, besides the influence of various noises as mentioned above.
Another question needs to be considered is that the vibration and temperature
fluctuations caused by the ambient would indiscriminately change the
polarization and fiber delay, which undoubtedly deteriorate the performance of
single photon interference. Therefore, in Charlie, we optimize the noises of
the system, and introduce a real-time feedback of the polarization and arrival
time of the pulses from the two encoders.

First a 100 GHz Dense Wavelength Division Multiplexing (DWDM) at the central
wavelength of 1550.12 nm is inserted to filter out the crosstalk noise from
running fiber links of classical communication which are in the same bundle
with the QKD link, and to filter out the inelastic scattering noise caused by
strong reference pulses Chen et al. (2020). In the following a circulator
(CIR) is used to block the noise caused by the reflection of the four
superconducting nanowire single-photon detectors (SNSPDs) Chen et al. (2020),
after that the pulses are polarized and split by the electric polarization
controllers (EPCs) and polarization beam splitters (PBSs), the idle beam of
the PBS is detected to carry out the arrival time and polarization feedback,
another beam is sent into a polarization-maintaining beam splitter (BS) to
accomplish the single photon interference. All the detections are recorded by
a high speed multichannel time tagger.

Figure 8: The variation of the polarization with and without feedback.

Then, we measured the fluctuation of count rate caused by the variation of
polarization. As shown in Fig. 8, the blue curve shows the variation of the
count rate without the feedback under field conditions, the intensity changes
30% in 4 hours. Consequently, in order to improve the visibility of
interference, we set a target count rate range of 150 kHz to 200 kHz of the
idle beam of the PBS used for feedback, meanwhile adjust the EPC once a second
to make the actual count rate meet it. The red curve shows the variation of
the count rate with a feedback, which is almost stable. Additionally, the
vibration and temperature fluctuations caused by the ambient would cause non-
overlapping between the two pulses of interference. Also we measured the
arrival time of the light pulses from one of the encoder (Bob) and found it is
related to the temperature. As shown in Fig. 2(a) in main text, the blue curve
shows the variation of the arrival time, the red circle shows the temperature.
Obviously, the arrival time is positively correlated with temperature, and the
peak-to-peak value of the variation of the arrival time in a day is about 20
ns. To make the two pulses from Alice and Bob overlap each other, we set a
target value to adjust the phase of the synchronization signals from Charlie’s
AWG once 20 seconds to make the actual arrival time meet the target. The
orange curve shows the arrival time with a feedback, the according peak-to-
peak value of variation of the arrival time in 6 hours is less than 80 ps.

### Detailed Experimental Results

Tab. 4 shows the parameters for the loss of fiber and the transmittance of
optical elements, as well as the SNSPD efficiencies. For the optical elements,
we measured the optical transmittance including the polarization controller
(PC), the dense-wavelength-division-multiplexer (DWDM), the circulator (CIR),
the polarization beam splitter (PBS) and the polarization maintaining beam
splitter (BS). Here, transmittances are given for each of the two inputs (A/B)
and outputs (ch1/ch2), as appropriate. The SNSPD efficiency measurements
included the PC efficiency.

Table 4: Experimental parameters of efficiencies of fiber and optical elements. Fiber Length | 511 km
---|---
$Loss_{FiberA}$ | 44.50dB
$Loss_{FiberB}$ | 44.60dB
PC-A | 99.1%
PC-B | 99.1%
DWDM-A | 93.1%
DWDM-B | 92.0%
CIR-A | 88.9%
CIR-B | 86.9%
PBS-A | 82.5%
PBS-B | 77.5%
BS-A-ch1 | 45.3%
BS-A-ch2 | 44.4%
BS-B-ch1 | 43.7%
BS-B-ch2 | 44.0%
SNSPD-ch1 | 77.0%
SNSPD-ch2 | 91.0%

Tab. 5 summarizes the experimental results. It shows the total sending number
of signal pulses ($N_{total}$) and the final key rate $R$ for the best
possible accepted phase difference $Ds$ (in degrees). In our experimental
implementation, we applied a digital window to select the signal in the middle
of each pulse, where the interference is expected to be better and all the
detections are filtered according to it, before the detections are announced
by Charlie. The results of untagged bits in the $Z_{1}$-windows ($n_{1}$),
phase-flip error rate $e_{1}^{ph}$ and the error rates in $Z$ basis before and
after applying the AOPP method are also given respectively. “QBER(X11)” and
“QBER(X22)” represent for the error rates of the decoy states “11” and “22”.

In the following rows, we list the numbers of pulses Alice and Bob sent in
different decoy states, labelled as “Sent-ABCD”, where “A” (“B”) is “X” or “Z”
indicating the basis which Alice (Bob) has used; “C” (“D”) is “0”, “1”, “2” or
“3”, indicating the intensity which Alice (Bob) has chosen within “vacuum”,
“$\mu_{1}$”, “$\mu_{2}$” or “$\mu_{z}$”. As with the numbers of sent pulses,
the numbers of detections are listed as “Detected-ABCD”. The numbers of valid
detections in $Z$ basis before AOPP are listed as “Detected-Valid-$Z$(Before
AOPP)” and the corresponding survived bits after AOPP are listed as “Survived-
bits(After Aopp)”. The total valid detections reported by Charlie is denoted
as “Detected-Valid-ch”, where “ch” can be “Det1” or “Det2” indicating the
responsive detector of the recorded counts. The table also gives the numbers
of detections falling within the accepted difference range $Ds$, listed as
“Detected-ABCD-Ds-Ch”, where “Ds” indicates that only the data within the
accepted range $Ds$ are counted, “Ch” indicates the detection channel. The
numbers of correct detections are listed as “Correct-ABCD-Ds-Ch”, and used to
calculate the $X$ basis error rate. “Detected-ZZError” and “Detected-
ZZCorrect” mean the numbers of error and correct detections in $Z$ basis
respectively.

Table 5: Experimental results. Fiber Length | 511 km | Fiber loss | 89.1 dB
---|---|---|---
$Ds$ | $10^{\circ}$ | $R$ | $3.45\times 10^{-8}$
$N_{total}$ | $1.679\times 10^{12}$ | Sent-ZZ | 907428400000
Sent-ZX00 | 19393400000 | Detected-ZX00 | 269
Sent-ZX01 | 204410800000 | Detected-ZX01 | 294733
Sent-ZX02 | 21217400000 | Detected-ZX02 | 88940
Sent-ZX30 | 8707200000 | Detected-ZX30 | 49610
Sent-XZ00 | 21229000000 | Detected-XZ00 | 275
Sent-XZ10 | 198592000000 | Detected-XZ10 | 279328
Sent-XZ20 | 23111200000 | Detected-XZ20 | 96684
Sent-XZ03 | 9517200000 | Detected-XZ03 | 51635
Sent-XX00 | 1394800000 | Detected-XX00 | 20
Sent-XX01 | 7057600000 | Detected-XX01 | 10150
Sent-XX02 | 0 | Detected-XX02 | 0
Sent-XX10 | 2174200000 | Detected-XX10 | 3037
Sent-XX20 | 1235800000 | Detected-XX20 | 5281
Sent-XX11 | 79038400000 | Detected-XX11 | 225307
Sent-XX22 | 1517800000 | Detected-XX22 | 12519
$n_{1}$(Before AOPP) | 1255190 |  $n_{1}$(After AOPP) | 219136
$e_{1}^{ph}$(Before AOPP) | 8.078% |  $e_{1}^{ph}$(After AOPP) | 16.067%
QBER$(Z-{Before})$ | 24.393% | QBER$(Z-{After})$ | 0.431%
Detected-Valid-$Z$(Before AOPP) | 2631682 | Survived-bits(After AOPP) | 576130
QBER(X11) | 4.9% | QBER(X22) | 5.8%
Detected-Valid-Det1 | 2088506 | Detected-Valid-Det2 | 2898644
Detected-XX11-Ds-Ch1 | 11162 | Detected-XX11-Ds-Ch2 | 15073
Correct-XX11-Ds-Ch1 | 10593 | Correct-XX11-Ds-Ch2 | 14353
Detected-ZZError | 641867 | Detected-ZZCorrect | 1989815

Different accepted phase difference ranges $Ds$ lead to different detection
counts and QBERs in $X$ basis. We list the QBERs of the $X$ basis when Alice
and Bob send decoy states $\mu_{1}$ and $\mu_{2}$ with different phase
difference range ($Ds$), which are listed in Tab. 6. It also shows the
different detection counts according to different $Ds$ and the optimized
secure key rates that we extract by searching through ranges of these
parameter values.

Table 6: QBERs and detections in $X$ basis and key rates with 511 km fiber. Results$\mid$Ds/2 | deg=2∘ | deg=8∘ | deg=10∘ | deg=12∘ | deg=15∘ | deg=30∘
---|---|---|---|---|---|---
QBER(X11) | 4.8% | 4.8% | 4.9% | 5.0% | 5.2% | 7.1%
QBER(X22) | 3.4% | 5.8% | 5.8% | 5.7% | 5.2% | 6.7%
Detections of X11 | 6210 | 21432 | 26235 | 31562 | 39046 | 76935
Detections of X22 | 348 | 1260 | 1511 | 1780 | 2206 | 4302
Key Rates | $1.85\times 10^{-8}$ | $3.37\times 10^{-8}$ | $3.45\times 10^{-8}$ | $3.43\times 10^{-8}$ | $3.36\times 10^{-8}$ | $2.23\times 10^{-8}$

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