Hertz-rate metropolitan quantum teleportation

Quantum teleportation can transfer an unknown quantum state between distant quantum nodes, which holds great promise in enabling large-scale quantum networks. To advance the full potential of quantum teleportation, quantum states must be faithfully transferred at a high rate over long distance. Despite recent impressive advances, a high-rate quantum teleportation system across metropolitan fiber networks is extremely desired. Here, we demonstrate a quantum teleportation system which transfers quantum states carried by independent photons at a rate of 7.1 ± 0.4 Hz over 64-km-long fiber channel. An average single-photon fidelity of ≥90.6 ± 2.6% is achieved, which exceeds the maximum fidelity of 2/3 in classical regime. Our result marks an important milestone towards quantum networks and opens the door to exploring quantum entanglement based informatic applications for the future quantum internet.


Introduction
Quantum teleportation [1] enables the 'disembodied' transfer of an unknown quantum state to a remote location by using quantum entanglement resource with the help of quantum measurement and classical communication.It lies at the heart of the realization of quantum information technologies such as quantum network [2][3][4] and distributed quantum computation [5].Since its initial proposal by Bennett et al. in 1993 [1], quantum teleportation has been demonstrated in various platforms, including atomic ensembles [6], single atoms [7], trapped ions [8,9], solid-state quantum systems [10], nuclear magnetic resonance [11] and quantum optics .Teleportation systems based on quantum optics offer a promising avenue towards quantum networks, which can be realized in continuous-variable (CV) and discrete-variable (DV) systems, respectively.For instance, the transfer and retrieval for both coherent states [28][29][30][31] and nonclassical states [32] have been experimentally realized with optical modes in CV systems, providing a method to realize deterministic quantum teleportation.However the distance of CV system is limited to around ten kilometers [30,31], due to the possible increased fragility with respect to the losses of quantum channels [34].For global-scale quantum networks [2,3], the distribution range of quantum states needs to be greatly extended to thousands of kilometers using quantum teleportation in DV systems.Till now, this has been realized with multiple degrees of freedom over several meters to more than one thousand kilometers, from the table-top experiments [12][13][14][15][16][17] to real-world demonstrations [18][19][20][21][22][23][24][25][26].Especially, by using a low-Earth orbit Micius satellite [35], quantum teleportation over 1200 km has been achieved [25,26].Despite impressive results, a high-rate quantum teleportation system has yet to be demonstrated, which is desired for advancing the development of quantum networks.
Here we report an experimental realization of a Hertz-rate quantum teleportation system through fiber over a metropolitan range.Our demonstration relies on a high-performance time-bin entangled quantum light source with a single piece of fiber-pigtailed periodically poled lithium niobate (PPLN) waveguide.The quantum states to be teleported are carried by a weak coherent single-photon source with decoy states.The indistinguishability of photons after prior quantum states distribution through fiber channels is ensured with a fully running feedback system.As an important feature of our demonstration, photonic time-bin qubits are teleported at a rate of 7.1 ± 0.4 Hz over a 64-km-long fiber channel.An average single-photon fidelity of 90.6 ± 2.6% is achieved with the decoy state method.
Our implementation establishes an important milestone towards quantum internet.

Results
Experimental setup.Figure 1(a) shows an aerial photography of the campus of University of Electronic Science and Technology of China (UESTC) indicating the distances between the locations Alice, Bob and Charlie.Figure 1(b) shows the scheme of our teleportation system, and Fig. 2 depicts its experimental setup.To be compatible with the structure of quantum networks [36], the quantum state to be teleported should be carried by an independent single-photon source, which is different with the Rome scheme [13].In our demonstration, the quantum bit (qubit) sender, Alice, located at a switching room of the backbone network of the campus, prepares a weak coherent single-photon source, which is used to encode photonic time-bin qubit, i.e., a single-photon wavepacket in a coherent The synchronization of the teleportation system is made through the CCs (see Methods).
All fiber spools used in our system are non-zero dispersion-shifted single-mode optical fiber (G.655,Yangtze Optical Fibre and Cable).
Prior entanglement distribution.The property of prior entanglement distribution is measured before performing the BSM of quantum teleportation.In the experiment, we With our fully running feedback system, we measure the Hong-Ou-Mandel (HOM) interference curve [37] with the time-bin qubits from Alice and Bob, respectively.The result given in Fig. 3 With the measured visibilities, the fidelity for the equatorial states can be calculated as It is noted that the classical fidelity bound of 2/3 is only applied when Alice's initial states carried with genuine single photons, rather than weak coherent states prepared with attenuated laser pulses.Here we utilize the decoy state method (DSM) [40][41][42] to estimate the performance of our system given that genuine single photons are used [24].In the experiment, we prepare quantum states |e , |l , |+ , and |+i with varying the mean photon number per qubit at Alice among three values ( µ s A = 0.088, µ d A = 0.029 and µ v A = 0, where µ s A , µ d A and µ v A are the mean photon numbers of the signal, decoy and vacuum state, respectively) and perform quantum teleportation, with details shown in Tables S. III and IV of Supplementary Materials.Based on these results, we calculate the lower bounds of F 1 e/l and F 1 +/+i as shown in Fig. 5, with F 1 Avg 90.6 ± 2.6%, which significantly violates the classical bound of 2/3 by more than 9 standard deviations, clearly demonstrating the capability of our system for high-fidelity teleportation.We present an analytical model of our teleportation system [24], and observe a good quantitative agreement between theory and experiment (see Notes S4 and S5 in Supplementary Materials).Finally, we conclude the key metrics of our teleportation system in Table I, where the state-of-the-art teleportation systems in DV with photonic qubits sent by an independent source are summarized as a comparison.
Note that, in Table 1, the state-transfer distance corresponds to the total length of quantum channel between Alice and Bob, while the teleportation distance is defined as the bee-line spatial separation between the location of the BSM station and the signal photon at the time of the BSM projection [24].

Discussion
Metrics for a quantum network are of course the rate, fidelity and distance of quantum teleportation.Although our work has moved one important step closer to high-speed quantum teleportation over a metropolitan area, further increases of teleportation rate in our system could be reached by increasing the repetition rate of system, the efficiencies of SNSPDs and BSM, and using multiple spectral channel [43].Further insights into photonic quantum information encoding, the use of multiple degrees of freedom [44] or multiple qubits [45] will also certainly increase the information capacity of quantum teleportation system based on hyperentanglement Bell-state analysis [46,47].The deviations of the fidelity from unity in our system are mostly due to multiphoton events of quantum light sources and the remaining distinguishability of the two photons undergoing the BSM.We may replace the SNSPDs with photon-number resolving SNSPDs [48,49] to allow post-selection of multiphoton events.
Alternatively, another promising solution to multiphoton events from Alice is applying sin-gle quantum emitters that can generate individual photons deterministically [50].Further, the indistinguishability between the photons from Alice and Bob could be improved by using narrower FBGs (see Supplementary Materials Note S2).To extend the teleportation distance, the combination of low-Earth-orbit satellite links [25,26] and quantum repeater architecture [51,52] may provide a prospective avenue for the long distances beyond 5000 km or so [53].It is also noted that the signal photons in our system, centered at 1531.87 nm, both in terms of wavelength and spectral width, are compatible with quantum memory in erbium-doped materials [54][55][56].This, in conjunction with entanglement swapping, constitutes an elementary link of a quantum network, which has been realized recently between two solid states quantum memories [57][58][59].
In conclusion, we have demonstrated a quantum teleportation system over metropolitan area, where a 7.1 ± 0.4 Hz teleportation rate is achieved with up to 64 km state-transfer distance.An average fidelity of 86.4 ± 4.5% is measured using QST.Using the DSM, we obtain an average single-photon fidelity of 90.6 ± 2.6%.Our results are further supported by an analytical model which is consistent with measurements of the quantum teleportation system.Finally, our work establishes the possibility of the high-speed quantum information transmission, which serves as a blueprint for the construction of metropolitan quantum network and eventually towards the global quantum internet.
Methods PPLN module design.The entangled photon pairs are generated using cascaded nonlinear processes of second harmonic generation (SHG) and spontaneous parameter down conversion (SPDC) in a periodically poled lithium niobate (PPLN) waveguide module [43,60,61].By fiber-integrating the PPLN waveguide with noise-rejecting filters [61], the spontaneous Raman scattering noise photons generated in the module are greatly reduced, and entangled photon pairs with a high rate under the same coincidence-to-accidental ratio (CAR) are obtained.More details about parameters of our PPLN module are listed in Table S.I of Supplementary Materials.
Synchronization.The master clock of the teleportation system is generated from an AWG at Bob and converted into optical pulses by using distributed feedback (DFB) lasers.The optical pulses are sent through the CCs from Bob to Charlie, and from Charlie to Alice.
Charlie and Alice receive the optical pulses and convert them into electrical signals using photon detectors (PDs), the outputs of which are used for synchronization at both stations.
As shown in Fig. 2, Charlie is connected to Alice and Bob through dark fibers of campus backbone networks.Among them, the fiber that carries photonic qubits is referred to as the quantum channel (QC) and the fiber that transmits optical pulses is referred to as the CC.
In addition, both unbalanced interferometers at Alice and Bob are calibrated and stabilized by single photon interference.This method permits us to carry out the preparation and measurement of time-bin qubits with stable phases.
Stabilization to ensure the indistinguishability of photon.A successful Bell-state measurement (BSM) relies on the indistinguishability of the two photons, which are generated by independent sources and have been distributed through a 22-km-long fiber channel for each.To do that, the spatial indistinguishability is ensured by using single-mode optical fibers.The spectral indistinguishability of photons from Alice and Bob is ensured by spectral filtering with separate 10-GHz-wide temperature-stabilized FBGs.However, in a real-world quantum teleportation system, the length and birefringence of the optical fiber are influenced by external environments, such as the strain and temperature fluctuations, which make photons distinguishable in degrees of temporal mode and polarization mode.
To overcome these challenges, we develop the following experimental techniques to stabilize the QCs (see Supplementary Materials     5: Individual and average fidelities of four teleported states with ideal state, obtained with quantum state tomography (QST) method and the decoy state method (DSM).Red bars are fidelities measured using QST.Blue bars are fidelities obtained with DSM.Both fidelities from the two methods exceed the classical limit of 2/3, i.e., the dashed gray line.For the QST and DSM we set µ SPDC = 0.042.Error bars are calculated using Monte Carlo simulation, assuming Poissonian detection statistics (see Supplementary Materials Note S5 and Tables S. III and IV for more calculation and statistics details).Assuming the input fields â1 and â2 are ideal thermal and coherent field, respectively, g 1 (0) = 2 and g 2 (0) = 1 are obtained.We can derive the visibility of HOM interference from Eq. ( 3): Supposing the mean photon number for the two input fields are identical with n1 = n2 , a theoretical upper bound of HOM interference visibility V theory = 40% is obtained.The HOM visibility measured in our experiment is V exp = 35.3± 1.0% with identical mean photon number from Alice and Bob (see Fig. with the calculation results.For all the calculations and measurements, the mean photon pair number µ SPDC is 0.042.

Note S3: Tomography of teleported state
We use quantum state tomography (QST) to reconstruct the density matrix of the quantum state after teleportation, and then calculate the quantum teleportation fidelity.Based on the scheme in Ref. [38], the density matrix of a single time-bin qubit can be represented by Stokes parameters: The projection measurement counts are N e , N l , N + , N − , N +i and N −i , respectively.
Using these counts, we obtain the Stokes parameters as follows: Substituting Eq. ( 7) into Eq.( 5), the density matrix ρ after teleportation can be obtained.The teleportation fidelity is calculated with the expected state |ψ by: Note S4: Analytical model of teleportation system We apply the analytical model in Ref. [24] to figure out the main parameters in our experiment, thus improving the performance of our teleportation system.In this method, the sum of the fidelity and the error rate is equal to 1. Hence, the fidelity of the teleported state can be predicted with the probability of three-fold coincidence counts for successful teleportation (P S ) and that for failure teleportation (P F ) Notice that in the following experiments, all the measured fidelities are calculated by using Eq. ( 9), with P S obtained by the maximum three-fold coincidence counts and P F obtained by the minimum three-fold coincidence counts.
Since the average photon number per qubit in our system is much less than 1, we ignore the contribution of higher-order terms to the predicted results.Here, we only consider the following cases: n A 2, n i 2, n A + n i 2 and n s 2 ( n A and n i denote the number of photons arriving at the BS from Alice and Bob.n s is the signal photon number generated by entangled photon pairs).The three-fold coincidence counts probability per qubit P (n A , n i , n s ) for different cases can be expressed by: where µ SPDC represents the average entangled photon pair number per qubit, µ A is the average photon number per qubit at Alice, η A is the transmission probability of quantum channel from Alice to Charlie (QC A→C ), η i represent the transmission probability of quantum channel from Bob to Charlie (QC B→C ), η s is the transmission probability of signal photons (stored in a fiber spool), ξ BSM is the detection efficiency of SNSPDs used for BSM, ξ s is the detection efficiency of SNSPDs for signal photons.All of the experimental parameters in the teleportation system are listed in Table S. II.
The teleportation fidelity of input states on the equator of the Bloch sphere is given by: ζ [P (1, 1, 1) + P (1, 1, 2)] 2 [P (1, 1, 1) + P (1, 1, 2) + P (0, 2, 2) + P (2, 0, 1)] , where ζ represents the degree of indistinguishability between photons from Alice and Bob (see Note S2).The teleportation rate can be expressed as: where R rep represents the repetition rate of teleportation system.From Eqs. ( 10) and ( 11 The teleportation rate decays exponentially as the state-transfer distance increases with the length of fiber spool at Bob.The decoy state method (DSM) is originally put forward to defend against photon number splitting attack in quantum key distribution by preparing sender's source with multiple intensity levels [40][41][42].We apply this method and follow Ref. [24] to extract the fidelity and rate of single photon in our teleportation system.In the experiment, the teleported states at Alice are prepared by attenuated laser pulses with different average photon numbers (denoted as signal state µ s A , decoy state µ d A , and vacuum state µ v A ).The error rate E (1) for the single photon component of the weak coherent single-photon source is upper bounded by [42] E (µ d A ) and E (0) are the error rates when Alice's teleported state is encoded in the decoy state and vacuum state, respectively.Q (µ d A ) is the corresponding gain, i.e., the probability for three-fold coincidence counts when a weak coherent state with mean photon number µ d A is prepared at Alice.Y (0) is the yield for a vacuum state, and Y (1) L is the lower bound of yield for the genuine single photon state.From Eq. ( 13), we can estimate the upper bound of the error rate of the teleported state prepared using genuine single photons.The parameters except Y (1) L in Eq. ( 13) can be measured experimentally.Assuming a weak coherent state is created at Alice with a mean photon number of µ, the corresponding gain can be expressed as: superposition of two time bins.The time-bin qubit is obtained by passing the single-photon wavepacket through an unbalanced Mach-Zehnder interferometer (UMZI) with path-length difference ∆τ .The time-bin qubit can be written as |ψ A = α |e + βe iφ |l , where |e represents the early time bin (i.e., a photon having passed through the short arm of the interferometer); |l is the late time bin (i.e., a photon having passed through the long arm); φ is a relative phase between |e and |l , and α 2 + β 2 = 1.Alice sends the created quantum states carried by single-photon wavepackets to Charlie, located at a laboratory at a flight distance of 400 m away, through a quantum channel (QC) of 22 km, i.e., QC A→C , including 2 km deployed fiber in field and 20 km fiber spool.The quantum information receiver, Bob, located at another laboratory, 210 m from Charlie, shares with Charlie a pair of time-bin entangled photonic qubits in the state of |Φ + = 2 −1/2 (|ee + |ll ), with one at 1549.16 nm (idler) and the other at 1531.87 nm (signal).The idler photons are distributed through another 22 km QC to Charlie, i.e., QC B→C , including 2 km deployed fiber in field and 20 km fiber spool.Charlie performs the joint Bell-state measurement (BSM) between the qubits sent by Alice and Bob, using a 50:50 fiber beam splitter (BS).We select only projections onto the singlet state of |ψ − = 2 −1/2 (|el − |le ), which can be realized by the detection of one photon in each output port of BS with a time difference of 625 ps.When a |ψ − has been successfully detected, the BSM result is sent to Bob over a classical channel (CC) by means of an optical pulse.In this case, the signal photons at Bob (stored in a 20-km-long fiber spool) are projected onto the state of |ψ B = σ y |ψ A , with σ y being a Pauli matrix.
distribute the idler photons through QC B→C to Charlie while the signal photons are held by a 20 km spool of fiber at Bob.The distributed time-bin entanglement property is characterized with the Franson interferometer, with details shown in Supplementary Materials Note S1.The visibilities of two-photon interference fringes are 94.3 ± 0.1% and 93.5 ± 0.1%, respectively, as shown in Fig.3(a).The error bars of visibilities are calculated by Monte Carlo simulation assuming Poissonian detection statistics.This result indicates that the quantum entanglement property still maintains after being distributed over 42 km fiber channels.Furthermore, it also allows us to ensure that parameters of two UMZIs can be remotely set as the same in our setup, which is a crucial requirement for the quantum teleportation processes.Indistinguishability of photons atCharlie.Alice's and Bob's photons need to be indistinguishable at Charlie for a successful BSM, which is difficult in long distance quantum teleportation.The spatial and spectral indistinguishabilities are ensured by using singlemode fibers and identical fiber Bragg grating (FBG) filters for both photons.The pathlength difference and polarization of the photons are stabilized with an active and automatic feedback system (see Methods).The experimental results of indistinguishabilities at Charlie are shown in Figs.3(b) and (c).

FIG. 1 :FIG. 2 :
FIG.1: Three-node quantum teleportation system.(a) Aerial view of the teleportation system.Alice 'A' is located at network's switching room, Bob 'B' and Charlie 'C' are located at two separated laboratories.All fibers connecting the three nodes belong to the UESTC backbone network.During the experiment, only the signals created by Alice, Bob and Charlie are transferred through these 'dark' fibers.(b) Scheme of the teleportation system.Alice prepares the initial state |ψ A with a weak coherent single-photon source and sends it to Charlie through a quantum channel (QC A→C ).An entanglement source at Bob generates a pair of entangled photons in the state |Φ + and then sends the idler photon to Charlie via another quantum channel (QC B→C ).The signal photon is stored in a fiber spool.Charlie implements a joint Bell-state measurement (BSM) between the qubit sent by Alice and Bob, projecting them onto one of the four Bell states |ψ − .Then the BSM result is sent to Bob via a classical channel (CC), who performs a unitary (U) transformation on the signal photon to recover the initial state (see Methods).

FIG. 3 :FIG. 4 :
FIG.3: Experimental results of prior entanglement distribution, indistinguishability of photons at Charlie, and teleportation of equatorial states.(a) Two-photon Franson interference fringes of time-bin entanglement source after distribution.Blue and red circles show the coincidence counts for the phase of UMZI1 on idler path set at 0 and π/2, respectively.The visibilities of the fitting curves are 94.3 ± 0.1% and 93.5 ± 0.1%, with the uncertainties calculated using the Monte Carlo method.(b) Automatic timing control on QC A→C and QC B→C , respectively.Red (blue) circles represent the drifts of Alice's (Bob's) photons arrival times with respect to the system clock.Green circles represent the coincidence counts of HOM interference per 100 seconds with active feedback.(c) Automatic polarization feedback on QC A→C .Red (Blue) lines correspond to relative fluctuations of QC A→C with feedback off (on).(d) Normalized HOM interference curve after fiber transmission of photons at Charlie.The visibility of the HOM curve is 35.3 ± 1.0%, corresponding to a single-photon indistinguishability of 88.8 ± 2.4% at Charlie, with details shown in Supplementary Materials Note S2.(e) Teleportation of equatorial states.The red and blue circles represent the three-fold coincidence counts from the two outputs of UMZI2 at Bob.The visibilities of the fitting curves are 61.4 ± 4.0% and 60.0 ± 3.9%, respectively, indicating the coherence property of Alice's state is successfully teleported to the signal photons.All error bars are calculated by Monte Carlo simulation assuming Poissonian detection statistics.
photon number from Alice and Bob (see Fig.3(d)  in the main text), which indicates that the single-photon indistinguishability of single-photon wavepacket at Charlie is V exp /V theory = 88.8 ± 2.4%, i.e., a residual distinguishability of 11.2±2.4%.The measured second order auto-correlation value of the idler photons (1549.16nm) after 10 GHz spectral filtering is 1.88 ± 0.04.Thus we argue that the reduction in the indistinguishability from the ideal value of 1 could be attributed to the imperfection of spectral shape of the filter, which can be further improved.By using Eq.(4), we calculate the predicted HOM visibility as a function of the mean photon number of teleported qubits µ A , shown by the blue line in Fig.S. 5(a).The red circles in Fig.S. 5(a) shows the experimental HOM visibility versus µ A , which agrees well

where σ i
is the Pauli matrix and S i represents the Stokes parameter.Stokes parameters can be calculated by projecting the states to the basis of |e e|, |l l|, |+ +| , |− −|, | + i +i| and | − i −i| , where |+ , |− , | + i and | − i are represented by a linear combination of |e and |l : ), we predict the fidelity of |+ state as a function of µ A -the blue curve in Fig.S. 5(b).An increase of the fidelity with µ A is observed.The fidelity reaches a maximum value when the probabilities of receiving one photon from Alice and Bob are equal at Charlie[67].With further increasing of µ A , the multiphoton events decrease the fidelity.The teleportation rate predicted by our model as a function of state-transfer distance is shown in Fig.S. 5(c).
photon pairs.To verify the prediction of our model, we carry out experiments under different situations.Red circles in Fig.S. 5(b) show the measured fidelities of |+ state with different µ A .The fidelities of |+ state and teleportation rates with different state-transfer distances of 44, 64 and 84 km are illustrated by red circles in Figs.S. 5(c) and (d), respectively.The measured fidelities of |+ states with different µ SPDC are shown by the red circles in Fig.S. 5(e).From the above results, we observe an excellent quantitative agreement between experiments and model.

Fig.S. 2 :Fig.S. 3 :Fig.S. 4 :
Fig.S. 2: Properties of entangled photon pairs.(a) and (b) Idler and signal photon counts versus pump power.The black circles represent measured idler and signal counts with different pump powers.The green line is the quadratic polynomial fitting curve of the measured counts.The quadratic and liner parts are shown as the red and blue lines, respectively.(c) and (d) Coincidence counts and accidental coincidence counts versus pump power.The red circles are measured coincidence counts within a coincidence window of 200 ps for 30 seconds under different pump power levels.The blue circles are measured accidental coincidence counts within a coincidence window of 200 ps for 30 seconds under different pump power levels.
The fidelity F e/l can be calculated byF e/l = R c / (R c + R w ), where R c and R w represent the probability of detecting the correct and wrong state in the pole basis, respectively.The measured fidelity for the |e input state is 92.2±1.0%and for the |l input state 92.4±1.1%.Assuming that the performance of equatorial states is the same, i.e., F + = F − = F +i = F −i = F equator , we apply F avg = (4F equator + F e + F l ) /6 to avg = (2 (F + + F +i ) + F e + F l ) /6 is 86.4 ± 4.5%, showing the quantum nature of the disembodied state transfer from Alice to Bob.
to-digital converter (TDC, ID900, ID Quantique).Charlie's setup.The photons from Alice and Bob are projected onto the |ψ − Bell state using a 50:50 BS and two SNSPDs with 60% detection efficiency.To ensure the indistinguishability of the two photons distributed through a 22-km-long fiber channel for each, we actively stabilize the arrival times and polarization with an active and automatic feedback system on both QC A→C and QC B→C channels.

TABLE I :
Comparison between the state-of-the-art results and our work.

TABLE S .
I: Parameters of PPLN module.

TABLE S .
II: Experimental parameters in our system.

TABLE S .
III: Gains[Hz] for different input states and mean photon number.

TABLE S .
IV: Fidelities for different input states and mean photon number.

TABLE S .
V: Gains [Hz] of equatorial states for different µ SPDC .

TABLE S .
VI: Fidelities of equatorial states for different µ SPDC .