Proof-of-principle experimental demonstration of twin-field type quantum key distribution

TL;DR

This paper demonstrates a practical twin-field quantum key distribution experiment that surpasses the fundamental rate-loss limit using phase stabilization and decoy states, showing promise for secure long-distance quantum communication.

Contribution

It provides the first proof-of-principle experimental demonstration of TF-QKD with phase stabilization, achieving secret key rates above the PLOB bound over long distances.

Findings

Secret key rates beat the PLOB bound at over 40 dB loss

Phase stabilization via Sagnac loop is effective for long-distance TF-QKD

Decoy states enable secure key generation in the experiment

Abstract

The twin-field (TF) quantum key distribution (QKD) protocol and its variants are highly attractive because they can beat the well-known rate-loss limit (i.e., the PLOB bound) for QKD protocols without quantum repeaters. In this paper, we perform a proof-of-principle experimental demonstration of TF-QKD based on the protocol proposed by Curty et al. which removes from the original TF-QKD scheme the need for post-selection on the matching of a global phase, and can deliver nearly an order of magnitude higher secret key rate. Furthermore, we overcome the major difficulty in the practical implementation of TF-QKD, namely, the need to stabilize the phase of the quantum state over kilometers of fiber. A Sagnac loop structure is utilized to ensure excellent phase stability between the different parties. Using decoy states, we demonstrate secret-key generation rates that beat the PLOB bound when the channel loss is above 40 dB.

Tables

Table 1. Table 1: List of intensity sets for the four different values of the overall system loss 38.0 dB, 46.7 dB, 49.4 dB and 55.1 dB. | α | 2 superscript 𝛼 2 |\alpha|^{2} is the average photon number (per pulse) of the coherent states in the X 𝑋 X basis. μ 𝜇 \mu , ν 𝜈 \nu and ω 𝜔 \omega are the average photon number (per pulse) of the decoy states in the Z 𝑍 Z basis. The uncertainty of each intensity refers to the measurement of its statistical fluctuation. *: 5-km fiber spool F A (F B ) is inserted between Alice (Bob) and Charlie. Note that the fiber spool F A-B is fixed inside the loop for all the four cases to ensure that Alice and Bob are physically separated with kilometers of fiber.

Loss	Fiber Inserted∗	Intensities
		${\left|\alpha \right|}^2$	$\mu$	$\nu$	$\omega$
$38.0$ dB	No	$0.0256\pm 0.0001$	$0.087\pm 0.001$	$0.0088\pm 0.0002$	$\left(1.0\pm 0.2\right)\times {10}^{-4}$
$46.7$ dB	No	$0.02495\pm 0.00005$	$0.0978\pm 0.0008$	$0.0099\pm 0.0001$	$\left(7.5\pm 0.2\right)\times {10}^{-5}$
$49.4$ dB	Yes	$0.0183\pm 0.0001$	$0.02005\pm 0.00002$	$0.00828\pm 0.00007$	$\left(9.2\pm 1.0\right)\times {10}^{-6}$
$55.1$ dB	No	$0.0175\pm 0.0002$	$0.0382\pm 0.0004$	$0.00790\pm 0.00007$	$\left(6.5\pm 1.0\right)\times {10}^{-5}$

Table 2. Table 2: List of experimental results for the four different values of the overall system loss considered, 38.0 dB, 46.7 dB, 49.4 dB and 55.1 dB. QBER is the experimental quantum bit error rate observed at detectors D0 and D1 when Alice and Bob choose the X 𝑋 X basis. The experimental secret key rate includes three cases, i.e. the case where intensity fluctuations are disregarded and the worst and best case scenarios where intensity fluctuations are taken into account. These three cases are indicated with the notation R mean , R min and R max respectively. Also, for comparison purposes, this table includes the PLOB bound  plob corresponding to each system loss. *: 5-km fiber spool F A (F B ) is inserted between Alice (Bob) and Charlie. Note that the fiber spool F A-B is fixed inside the loop for all the four cases to ensure that Alice and Bob are physically separated with kilometers of fiber.

Loss	Fiber Inserted∗	QBER		Experimental Secret Key Rates			PLOB Bound
		$D_0$	$D_1$	$R_{mean}$	$R_{min}$	$R_{max}$
$38.0$ dB	No	$0.0032$	$0.0036$	$2.6484\times {10}^{-4}$	$1.9917\times {10}^{-4}$	$3.4765\times {10}^{-4}$	$2.2867\times {10}^{-4}$
$46.7$ dB	No	$0.0058$	$0.0032$	$7.8389\times {10}^{-5}$	$6.9058\times {10}^{-5}$	$8.8458\times {10}^{-5}$	$3.0845\times {10}^{-5}$
$49.4$ dB	Yes	$0.0059$	$0.0056$	$3.6306\times {10}^{-5}$	$2.4061\times {10}^{-5}$	$5.4130\times {10}^{-5}$	$1.6564\times {10}^{-5}$
$55.1$ dB	No	$0.0116$	$0.0108$	$1.7542\times {10}^{-5}$	$1.0516\times {10}^{-5}$	$2.5652\times {10}^{-5}$	$4.4584\times {10}^{-6}$

Equations

$${\rho }_A=\frac{1}{2\pi }{\int }_0^{2\pi }d{\phi }_A{|{\beta }_A{e}^{i{\phi }_A}⟩}_A⟨{\beta }_A{e}^{i{\phi }_A}|$$

Full text

Keywords: quantum cryptography, quantum key distribution, twin-field quantum key distribution, secret key rate, PLOB bound, phase encoding

Introduction - Quantum key distribution (QKD) R1 ; R2 ; R3 ; R4 ; Ekert1991 makes it possible to distribute secret keys to remote users with information-theoretic security, which means that its security is independent of an attacker’s computational power R5 ; R6 ; R7 ; R8 ; R9 ; R10 ; R11 ; NJP11 ; Mayer_proof ; secure . Experimentally, QKD has been performed over 421 km of fiber R13 , as well as over 1000 km of free space through satellite to ground links R14 ; R15 . Towards the construction of a global quantum internet, performing long distance QKD is a crucial step R16 ; R17 ; R18 ; R19 . However, there is a fundamental limit on the point-to-point secret key rate versus channel transmittance plob ; R21 ; R22 that can be achieved by two remote parties using QKD without an intermediate node. This limit, also called the PLOB bound plob , states that the secret key rate scales basically linearly with the channel transmittance.

To overcome the PLOB bound, besides using quantum repeaters R23 ; R24 ; R25 ; R26 , it has been proposed to employ measurement-device-independent (MDI) QKD R29 in combination with quantum memories PRA89 ; NJP16 or in combination with quantum non-demolition measurements R27 located at the untrusted intermediate node that is used in MDI-QKD. While promising, all these approaches are however far away from our current experimental capabilities. Remarkably, more recently it has been theoretically proven R39 ; R40 ; R41 ; R42 ; tf-qkd_marcos ; R44 ; finite_decoy_tfqkd that variations of the twin-field (TF) QKD protocol proposed by Lucamarini et al. tf-qkd_original can beat the PLOB bound with the help of just one untrusted intermediate node (Charlie) performing a simple interferometric measurement. This shows that intercity QKD could be feasible with today’s technology without the need for quantum memories or quantum non-demolition measurements. In TF-QKD, two users (Alice and Bob) send two optical fields to produce a single-photon interference on a beam splitter located at Charlie. A successful result corresponds to Charlie observing a single-photon detection event, which measures the relative phase between the two optical beams. The fact that TF-QKD uses singles (i.e. single-photon detection events) results in a secret key rate that scales as the square-root of the channel transmittance instead of linearly. This is because only one photon (either from Alice or from Bob) has to arrive at Charlie. Importantly, since TF-QKD has a similar structure as MDI-QKD, in the sense that it uses an untrusted node to interfere Alice and Bob’s signals, all the advantages of MDI-QKD R29 ; PRA89 ; NJP16 ; R27 ; R30 ; R31 ; R32 ; R33 ; R34 , such as its immunity to any possible attack against the measurement unit and its readiness for star networks, are retained by TF-QKD. In this regard, TF-QKD can be considered as a MDI-QKD scheme based on singles, rather than on coincidences.

With the foundations of TF-QKD firmly established, it is now very important to demonstrate the viability of TF-QKD experimentally. One main drawback of the original TF-QKD protocol for practical implementations is that it requires long-distance subwavelength path-length phase stability, which is a new requirement in QKD and is much more demanding to achieve than two-photon interference as needed in standard MDI-QKD. Another drawback is that the original protocol requires to perform a post-selection step based on the matching of a global phase between Alice and Bob, which results in a reduction of the secret key rate by about an order of magnitude. To overcome these limitations, various variants of the original TF-QKD protocol have been very recently proposed and investigated R39 ; R40 ; R41 ; R42 ; tf-qkd_marcos ; R44 ; finite_decoy_tfqkd . For example, ref. R39 analyzes the security of a modified TF-QKD scheme by exploiting a quantum coin idea R6 ; R10 . Ref. R40 , on the other hand, introduces a phase-matching QKD which releases the requirement of active global phase randomization, but still needs a phase post-selection step. In Refs. tf-qkd_marcos ; R44 ; finite_decoy_tfqkd the need for such post-selection step is removed. In summary, the security of some variants of TF-QKD have now been firmly established and their key rates beat the PLOB bound. So, it is now important to implement experimentally a TF-QKD protocol to demonstrate their practicality.

In this paper we perform a proof-of-principle experimental implementation of a TF-QKD protocol introduced by Curty et al. tf-qkd_marcos . This protocol does not need a post-selection step based on the matching of a global phase, and can provide a secret key rate which is about an order of magnitude higher than previous proposals qcrypt2018 . The key idea is to use coherent states for key generation and photon number states as the complementary basis to prove security NJP11 . The latter type of states can be simulated by means of phase-randomized coherent states in combination with the decoy-state method R35 ; R36 ; R37 ; R38 . In our experiment, to stabilize the phase of the quantum states over kilometers of fiber, an auto-compensating set-up, which provides excellent phase stability, is built up. We remark that an auto-compensating set-up, also known as “plug-and-play” system, is widely used in QKD plug-play ; NJP4 and is the workhorse of a widely deployed commercial QKD system manufactured by the company ID Quantique. Security proofs for such “plug-and-play” QKD systems have been developed in PRA77 ; NJP12 . More concretely, we use a Sagnac loop where an optical pulse (generated by a single laser in our experiment) travels through either a clockwise path or an anti-clockwise path and then the two paths interfere with each other. In one path, the pulse is modulated in phase by Alice whereas in the other path, the pulse is modulated in phase by Bob. To implement the decoy-state method, intensity modulators are employed to modulate the pulses leaving Bob and Alice’s stations. Our experimental results confirm an achievable secret key rate well above the PLOB bound, and constitute a crucial step towards demonstrating the practical experimental feasibility of TF-QKD.

Protocol and Experiment - The TF-QKD protocol introduced in tf-qkd_marcos is composed of the following five steps.

Step 1: Each of Alice and Bob prepares a weak coherent state. Alice (Bob) chooses the $X$ basis with probability $P_{X}$ and the $Z$ basis with probability $P_{Z}=1-P_{X}$. If the $X$ basis is chosen, Alice (Bob) randomly prepares a coherent state $\left|\alpha\right\rangle_{A}$ ($\left|\alpha\right\rangle_{B}$) for the bit value $b_{A}=0$ ($b_{B}=0$) or $\left|-\alpha\right\rangle_{A}$ ($\left|-\alpha\right\rangle_{B}$) for the bit value $b_{A}=1$ ($b_{B}=1$). If the $Z$ basis is chosen, Alice (Bob) prepares a phase-randomized coherent state

[TABLE]

($\rho_{B}$ has the same expression as (1) with all subscripts changed to $B$). The value of the intensity $|\beta_{A}|^{2}$ ($|\beta_{B}|^{2}$) is chosen at random from a set $S=\left\{\mu,\nu,\omega\right\}$ containing say three possible intensities.

Step 2: Alice and Bob send their states to the middle node Charlie through optical channels, each of them with transmittance $\sqrt{\eta}$.

Step 3: Charlie interferes the incoming states with a 50:50 beam splitter followed by two single-photon detectors, $D_{0}$ and $D_{1}$. He records the result, i.e. which detector clicks at each expected arrival time slot.

Step 4: Once the quantum communication phase of the protocol has finished, Charlie announces all the results obtained, and Alice and Bob declare the bases used.

Step 5: Based on the information announced, Alice and Bob estimate the bit and phase error rate and distill a secret key from those instances where they used the $X$ basis and Charlie declared one detection click. More precisely, whenever Charlie reports one click event in say $D_{0}$ ($D_{1}$) and both Alice and Bob choose the $X$ basis, $b_{A}$ and $b_{B}$ ($b_{B}\oplus 1$) are regarded as their raw keys.

Fig. 1 shows the schematic diagram of our experimental set-up. It is a two-way QKD system consisting of a Sagnac interferometer. It is similar to the “plug-and-play” QKD system employed in plug-play ; NJP4 . The Sagnac arrangement is chosen to overcome the main challenges in implementing TF-QKD, namely, to share a phase reference between Alice and Bob and to achieve single-photon interference at Charlie, which requires phase stability. The common-path nature of the Sagnac loop automatically compensates for phase fluctuations of the two fields from Alice and Bob. In this set-up, the laser source is in Charlie’s hands and is shared by Alice and Bob. This ensures that the three parties have the same phase reference. Charlie uses his intensity modulator (IMC) and his optical attenuator (AttC) to create weak coherent pulses from a cw DFB laser (PRO 800, wavelength 1552.6 nm). The IMC has an extinction ratio of $>$30 dB. The pulses have a FWHM width of 900 ps and a repetition rate of 10 MHz. They go through an optical circulator and then enter the Sagnac loop through a 50:50 fiber-based beam splitter. Clockwise and counter-clockwise traveling pulses each go through an optical variable attenuator (and a 5-km fiber spool FB or FA respectively in one case) before arriving at Alice’s or Bob’s station. The clockwise (counter-clockwise) pulses go through Bob’s (Alice’s) station without being modulated. Therefore, no information is directly communicated between Alice and Bob. The clockwise (counter-clockwise) pulses then undergo a 7-km fiber spool FA-B and reach Alice’s (Bob’s) station. This is to ensure that Alice and Bob are physically separated with kilometers of fiber. Inside the station of Alice (Bob), there is a phase modulator PMA (PMB) and an intensity modulator IMA (IMB). If Alice (Bob) chooses the $X$ basis, she (he) uses her (his) PMA (PMB) to randomly add a [math] or $\pi$ phase to the pulse. If Alice (Bob) chooses the $Z$ basis, then she (he) uses her (his) PMA (PMB) to add a random phase between [math] and $2\pi$ to the pulse. The intensity modulator is applied to set the average number of photons per pulse to be, either $|\alpha|^{2}$ for the signal states in the $X$ basis, or one of the intensities in the set $S=\left\{\mu,\nu,\omega\right\}$ for the decoy states in the $Z$ basis. After the phase and intensity modulations, Alice (Bob) sends the pulses through a variable optical attenuator and, in some cases, through a 5-km spool of actual fiber FA (FB), before reaching the beam splitter of Charlie. The loss between Alice (Bob) and Charlie is adjusted to simulate the loss due to the communication channel. To demonstrate the practicality of the scheme, we add the 5-km fiber spool FA (FB) between Alice (Bob) and Charlie, in addition to the attenuator. The pulses coming from Alice and Bob interfere at Charlie’s beam splitter. One output of this beam splitter is directed to a single-photon detector (SPD) $D_{0}$ via the circulator, while the other output is followed directly by another SPD, $D_{1}$. The SPDs are commercial free-run avalanche photodiodes (ID220) with an efficiency of 11.7% and a dark count rate of 750 Hz. The SPDs have a time jitter on the order of 200 ps, matching the optical pulse width. Charlie records each click event (within a 900 ps window where the detection is expected), and publicly announces the result. Afterward, Alice and Bob declare their bases choices and use the instances where they both selected the $X$ basis and Charlie announced a single-click event to distill a secure secret key.

Whether in Alice’s station or in Bob’s station, both clockwise and counter-clockwise traveling pulses pass through the IM and PM. It is crucial to ensure that Alice (Bob) only modulates the clockwise (counter-clockwise) traveling pulse. This is achieved by using appropriate fiber lengths, between Alice (Bob) and Charlie and between Alice and Bob, so that the two counter propagating pulses never overlap with each other at any modulator inside the Sagnac loop. Note that this is not a practical limitation, since in practice Alice and Bob can measure the fiber lengths in the link and add or remove small lengths of fiber within their own set-up. All modulators used in our set-up are driven and synchronized by a high-speed arbitrary waveform generator (AWG, Keysight M8195A). The delay times of the driving signal of Alice’s (Bob’s) IMA (IMB) and PMA (PMB), relative to the driving signal of Charlie’s IMC, are well adjusted to ensure that Alice and Bob modulate the intended pulses. As in any practical system, there are unintended reflections and backscattering from the channel, causing unintended “clicks” in the detectors. Fortunately, these unintended signals do not arrive at the detectors at the same time as the signals from Alice and Bob. With precise synchronization, we can eliminate the unintended clicks by choosing the appropriate time windows for detection in Charlie’s station. To further reduce the errors in the detection results, the fiber lengths within Charlie’s station are adjusted to guarantee that the unintended signals due to reflections / backscattering do not overlap with the real signals at the detectors, i.e., do not fall inside the detection window.

A number of fiber-based polarization controllers are installed inside the Sagnac loop (see Fig. 1) to ensure that the pulses are aligned in polarization after they travel through the entire loop and interfere at Charlie’s beam splitter. All the fiber spools are stored in sealed boxes, but no active polarization stabilization is applied. The sealed boxes mimic the environment of buried fiber cables. The voltage of the driving signal of Alice’s (Bob’s) PMA (PMB) is tuned to Vπ (4.5V) to maximize the interference visibility. Since the two pulses coming from Alice and Bob travel exactly the same path, they undergo the same phase drift and their relative phase is therefore stable without active phase stabilization. Due to the combination of the aforementioned measures, even after traveling kilometers of fiber, the interference visibility of our system is kept well above 99% for an extensive duration, as shown in Fig. 2 (a). When the fiber spool FA (FB) is taken out of the loop and Alice (Bob) and Charlie is connected only through an attenuator (blue squares in Fig. 2 (a)), the system is more stable and the average interference visibility is about 99.8%. When the 5-km fiber spool FA (FB) is added between Alice (Bob) and Charlie (red circles in Fig. 2 (a)), the interference visibility is slightly lower compared to the cases without the fibers spools FA and FB. We attribute this degradation of the visibility to polarization fluctuations, the depolarization effect, as well as low-levels of Rayleigh backscattering in long fiber spools. Nonetheless, the interference visibility in this latter case is still stable for at least 40 minutes and the average value is about 99.7%. In order to keep the high visibility and improve the performance of the system, we stop the experiment every 40 minutes for polarization realignment. When a random phase is required (for the decoy state signal), the voltage of the driving signal is randomly chosen from [math] to Vπ. In this scenario, the photons should be detected by the detectors $D_{0}$ and $D_{1}$ with equal probability. Fig. 2 (b) shows the ratio of the total number of photon counts at $D_{0}$ to the total number of photon counts at $D_{1}$ when phase randomization is applied continuously at both Alice’s and Bob’s stations for 40 minutes. Note that the total number of photon counts at detector $D_{0}$ is calibrated to compensate for the circulator loss. As illustrated in Fig. 2 (b), both cases (with or without the 5-km fiber spools FA and FB) maintain a stable ratio close to 1, which indicates that phase randomization can be effectively implemented.

Results and Discussion - We implement the experiment for four different values of the overall system loss between Alice and Bob, 38.0 dB, 46.7 dB, 55.1 dB and 49.4 dB, respectively. Optical attenuators are applied to simulate the channel loss for the first three system losses. For the 49.4 dB loss, 5-km fiber spools are inserted between Alice (as well as Bob) and Charlie in addition to the attenuator. As discussed above, the detector efficiency is about 11.7%, which is equivalent to a 9.3 dB loss that we include in the overall system loss. For different values of the system loss, we choose different intensity sets $\left\{|\alpha|^{2},\mu,\nu,\omega\right\}$. The selection of the signal intensity $|\alpha|^{2}$ is done by optimizing a priori the secret key rate formula for a channel model that approximately simulates the expected behavior of an experimental realization, based on the devices’ parameters. In the asymptotic regime, given that the weakest decoy intensity $\omega$ is sufficiently small, the selection of the other two decoy intensities $\mu$ and $\nu$ (within certain limits) is not so crucial and its effect on the resulting secret key rate turns out to be small. So, we take the values of $\mu$ and $\nu$ shown in Table 1 as an example and for experimental convenience.

For each value of the system loss and each intensity pair, Alice and Bob each send out $3\times 10^{9}$ coherent pulses. The experimental quantum bit error rates (QBER) observed in both detectors $D_{0}$ and $D_{1}$ when Alice and Bob choose $X$ basis are listed in Table 2. Given the high stability of the system and the high interference visibility, the QBERs observed in the experiment are correspondingly low. Even the maximum QBER observed at the highest system loss is lower than 1.2%. As the overall system loss decreases, the impact of the dark counts of the single-photon detectors diminishes. This is reflected in the lower QBERs obtained for lower system loss. More experimental data, such as the experimentally observed gains, could be found in the supplementary material supplementary . To extract a secure secret key, we use the security analysis and secret key rate formula reported in tf-qkd_marcos (see supplementary material supplementary ). The secret key rates for different system losses, as well as the corresponding PLOB bound, are also listed in Table 2. For each value of the system loss, Table 2 includes three cases, i.e. the case where intensity fluctuations are disregarded and the worst- and best-case scenarios where intensity fluctuations are taken into account. For the worst- and best-case scenarios, we numerically minimize and maximize the secret key rate formula among all possible values for the different intensities (within the reported experimental intervals which include the intensity fluctuations). These three cases are indicated in the table with the notation Rmean, Rmin and Rmax respectively. These results are also illustrated in Fig. 3, which shows the secure key rate (bits per pulse) in logarithmic scale as a function of the overall system loss. The solid red line illustrates the PLOB bound introduced in plob . The solid green line corresponds to a theoretical simulation result realized with the channel model introduced in  tf-qkd_marcos . This channel model, for simplicity, assumes the same experimental parameters (similar to those of the experimental implementation) over all the distances and does not optimize the values of the different decoy intensities which are fixed a priori. The experimental secret key rates are illustrated as black crosses. The vertical line of each cross shows the difference between the worst- and best-case scenarios when intensity fluctuations are taken into account, i.e. the difference between and Rmin and Rmax. The horizontal line of each cross shows the uncertainty of the overall loss. The point where these two lines cross coincides with the secret key rate if intensity fluctuations are disregarded, i.e. with Rmean in Table 2. As depicted in Fig. 3, the experimental secret key rates are reasonably close to the theoretical simulation results, except that the key rate at the system loss 49.4 dB is slightly lower compared with the simulation result. This is because of the 5-km fiber spools FA (FB) that are added in between Alice (Bob) and Charlie in this case, which reduces the performance of the system. Nonetheless, the experimental results, as expected, follow the rate-loss dependence of TF-QKD, scaling with the square-root of the channel transmittance. When the overall system loss between Alice and Bob is around 38 dB, the secret key rate sits around the PLOB bound. However, as the overall system loss increases, the experimental key rate evidently surpasses the PLOB bound even when the minimum key rate in the worst-case scenario is considered. This achievement experimentally proves that TF-QKD can beat the fundamental bounds on the private capacity of point-to-point QKD. Particularly, the observed higher key rate, compared with the PLOB bound, at loss 49.4 dB with real fiber spools (FA and FB) shows the feasibility of the practical application of TF-QKD.

Conclusion - In summary, we have implemented the first twin-field quantum key distribution experiment over 10 km of real optical fibers without active phase stabilization or phase post-selection. The secure key rate of the system scales as the square-root of the overall channel transmittance. In particular, we have observed that the resulting secret key rate at the high loss region clearly beats the PLOB bound even when intensity fluctuations are taken into account. Our work shows the feasibility of overcoming the private capacity of a point-to-point QKD link with current technology. Longer fibers could be added into our system in the future to extend the range of TF-QKD.

Acknowledgments - We thank Olinka Bedroya and Shihan Sajeed for their assistance and enlightening discussions. We also thank financial support from NSERC, CFI, ORF, US Office of Naval Research, MITACS, Royal Bank of Canada and Huawei Technologies Canada, Ltd. M.C. acknowledges support from the Spanish Ministry of Economy and Competitiveness (MINECO), the Fondo Europeo de Desarrollo Regional (FEDER) through grants TEC2014-54898-R and TEC2017-88243-R, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 675662 (project QCALL).