Teleportation Systems Towards a Quantum Internet

Raju Valivarthi,1, 2 Samantha Davis,1, 2 Cristián Peña,1, 2, 3 Si Xie,1, 2 Nikolai Lauk,1, 2 Lautaro Narváez,1, 2

Jason P. Allmaras,4 Andrew D. Beyer,4 Yewon Gim,2, 5 Meraj Hussein,2 George Iskander,1 Hyunseong Linus

Kim,1, 2 Boris Korzh,4 Andrew Mueller,1 Mandy Rominsky,3 Matthew Shaw,4 Dawn Tang,1, 2 Emma E.

Wollman,4 Christoph Simon,6 Panagiotis Spentzouris,3 Neil Sinclair,1, 2, 7 Daniel Oblak,6 and Maria Spiropulu1, 2

1Division of Physics, Mathematics and Astronomy,California Institute of Technology, Pasadena, CA 91125, USA

2Alliance for Quantum Technologies (AQT), California Institute of Technology, Pasadena, CA 91125, USA3Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

4Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA5AT&T Foundry, Palo Alto, CA 94301, USA

6Institute for Quantum Science and Technology, and Department of Physics & Astronomy,University of Calgary, Calgary, AB T2N 1N4, Canada

7John A. Paulson School of Engineering and Applied Sciences,Harvard University, Cambridge, MA 02138, USA

(Dated: July 29, 2020)

Quantum teleportation is essential for many quantum information technologies including long-distance quantum networks. Using fiber-coupled devices, including state-of-the-art low-noise super-conducting nanowire single photon detectors and off-the-shelf optics, we achieve quantum teleporta-tion of time-bin qubits at the telecommunication wavelength of 1536.5 nm. We measure teleportationfidelities of ≥ 90% that are consistent with an analytical model of our system, which includes realis-tic imperfections. To demonstrate the compatibility of our setup with deployed quantum networks,we teleport qubits over 22 km of single-mode fiber while transmitting qubits over an additional 22km of fiber. Our systems, which are compatible with emerging solid-state quantum devices, providea realistic foundation for a high-fidelity quantum internet with practical devices.

I. INTRODUCTION

Quantum teleportation [1], one of the most captivat-ing predictions of quantum theory, has been widely in-vestigated since its seminal demonstrations over 20 yearsago [2–4]. This is due to its connections to fundamentalphysics [5–14], and its central role in the realization ofquantum information technology such as quantum com-puters and networks [15–19]. The goal of a quantumnetwork is to distribute qubits between different loca-tions, a key task for quantum cryptography, distributedquantum computing and sensing. A quantum networkis expected to form part of a future quantum internet[20–22]: a globally distributed set of quantum proces-sors, sensors, or users there-of that are mutually con-nected over a network capable of allocating quantum re-sources (e.g. qubits and entangled states) between loca-tions. Many architectures for quantum networks requirequantum teleportation, such as star-type networks thatdistribute entanglement from a central location or quan-tum repeaters that overcome the rate-loss trade-off ofdirect transmission of qubits [19, 23–26].

Quantum teleportation of a qubit can be achieved byperforming a Bell-state measurement (BSM) between thequbit and another that forms one member of an entan-gled Bell state [1, 18, 27]. The quality of the teleporta-tion is often characterized by the fidelity F = 〈ψ| ρ |ψ〉of the teleported state ρ with respect to the state |ψ〉accomplished by ideal generation and teleportation [15].This metric is becoming increasingly important as quan-tum networks move beyond specific applications, such as

quantum key distribution, and towards the quantum in-ternet.

Qubits encoded by the time-of-arrival of individualphotons, i.e. time-bin qubits [28], are useful for net-works due to their simplicity of generation, interfacingwith quantum devices, as well as independence of dy-namic transformations of real-world fibers. Individualtelecom-band photons (around 1.5 µm wavelength) areideal carriers of qubits in networks due to their abilityto rapidly travel over long distances in deployed opticalfibers [17, 29–31] or atmospheric channels [32], amongother properties. Moreover, the improvement and grow-ing availability of sources and detectors of individualtelecom-band photons has accelerated progress towardsworkable quantum networks and associated technologies,such as quantum memories [33], transducers [34, 35], orquantum non-destructive measurement devices [36].

Teleportation of telecom-band photonic time-binqubits has been performed inside and outside the labora-tory with impressive results [29–31, 37–42]. Despite this,there has been little work to increase F beyond ∼ 90% forthese qubits, in particular using practical devices that al-low straightforward replication and deployment of quan-tum networks (e.g. using fiber-coupled and commerciallyavailable devices). Moreover, it is desirable to developteleportation systems that are forward-compatible withemerging quantum devices for the quantum internet.

In the context of Caltech’s multi-disciplinary multi-institutional collaborative public-private research pro-gram on Intelligent Quantum Networks and Technologies(IN-Q-NET) founded with AT&T as well as Fermi Na-

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tional Accelerator Laboratory and Jet Propulsion Labo-ratory in 2017, we designed, built, commissioned and de-ployed two quantum teleportation systems: one at Fer-milab, the Fermilab Quantum Network (FQNET), andone at Caltech’s Lauritsen Laboratory for High EnergyPhysics, the Caltech Quantum Network (CQNET). TheCQNET system serves as an R&D, prototyping, andcommissioning system, while FQNET serves as an ex-pandable system, for scaling up to long distances and isused in multiple projects funded currently by DOE’s Of-fice of High Energy Physics (HEP) and Advanced Scien-tific Research Computing (ASCR). Material and deviceslevel R&D in both systems is facilitated and funded bythe Office of Basic Energy Sciences (BES). Both systemsare accessible to quantum researchers for R&D purposesas well as testing and integration of various novel de-vices, such as for example on-chip integrated nanopho-tonic devices and quantum memories, needed to up-grade such systems towards a realistic quantum inter-net.Importantly both systems are also used for improve-ments of the entanglement quality and distribution withemphasis on implementation of protocols with complexentangled states towards advanced and complex quan-tum communications channels. These will assist in stud-ies of systems that implement new teleportation proto-cols whose gravitational duals correspond to wormholes[43], error correlation properties of wormhole teleporta-tion, on-chip codes as well as possible implementation ofprotocols on quantum optics communication platforms.Hence the systems serve both fundamental quantum in-formation science as well as quantum technologies.

Here we perform quantum teleportation of time-binqubits at a wavelength of 1536.5 nm with an averageF ≥ 90%. This is accomplished using a compact setupof fiber-coupled devices, including low-dark-count sin-gle photon detectors and off-the-shelf optics, allowingstraight-forward reproduction for multi-node networks.To illustrate network compatibility, teleportation is per-formed with up to 44 km of single-mode fiber betweenthe qubit generation and the measurement of the tele-ported qubit, and is facilitated using semi-autonomouscontrol, monitoring, and synchronization systems, withresults collected using scalable acquisition hardware. Oursystems, which operates at a clock rate of 90 MHz, canbe run remotely for several days without interruption andyield teleportation rates of a few Hz using the full lengthof fiber. Our qubits are also compatible with erbium-doped crystals, e.g. Er:Y2SiO5, that are used to developquantum network devices like memories and transduc-ers [44–46]. Finally, we develop an analytical model ofour system, which includes experimental imperfections,predicting that the fidelity can be improved further to-wards unity by well-understood methods (such as im-provement in photon indistinguishability). Our demon-strations provide a step towards a workable quantum net-work with practical and replicable nodes, such as theambitious U.S. Department of Energy quantum researchnetwork envisioned to link the U.S. National Laborato-

ries.In the following we describe the components of our sys-

tems as well as characterization measurements that sup-port our teleportation results, including the fidelity of ourentangled Bell state and Hong-Ou-Mandel (HOM) inter-ference [47] that underpins the success of the BSM. Wethen present our teleportation results using both quan-tum state tomography (QST) [48] and projection mea-surements based on a decoy state method [49], followedby a discussion of our model. We conclude by consid-ering improvements towards near-unit fidelity and GHzlevel teleportation rates.

II. SETUP

Our fiber-based experimental system is summarizedin the diagram of Fig. 1. It allow us to demonstratea quantum teleportation protocol in which a photonicqubit (provided by Alice) is interfered with one memberof an entangled photon-pair (from Bob) and projected(by Charlie) onto a Bell-state whereby the state of Al-ice’s qubit can be transferred to the remaining memberof Bob’s entangled photon pair. Up to 22 (11) km ofsingle mode fiber is introduced between Alice and Char-lie (Bob and Charlie), as well as up to another 11 kmat Bob, depending on the experiment (see Sec. III). Allqubits are generated at the clock rate, with all of theirmeasurements collected using a data acquisition (DAQ)system. Each of the Alice, Bob, Charlie subsystems arefurther detailed in the following subsections, with theDAQ subsystem described in Appendix A 1.

A. Alice: single-qubit generation

To generate the time-bin qubit that Alice will teleportto Bob, light from a fiber-coupled 1536.5 nm continuouswave (CW) laser is input into a lithium niobate intensitymodulator (IM). We drive the IM with one pulse, or twopulses separated by 2 ns. Each pulse is of ∼65 ps fullwidth at half maximum (FWHM) duration. The pulsesare produced by an arbitrary waveform generator (AWG)and amplified by a 27 dB-gain high-bandwidth amplifierto generate optical pulses that have an extinction ratioof up to 22 dB. We note that this method of creatingtime-bin qubits offers us flexibility not only in terms ofchoosing a suitable time-bin separation, but also for syn-chronizing qubits originating from different nodes in anetwork. A 90/10 polarization-maintaining fiber beamsplitter combined with a power monitor (PWM) is usedto apply feedback to the DC-bias port of the IM so asto maintain a constant 22 dB extinction ratio [50]. Inorder to successfully execute the quantum teleportationprotocol, photons from Alice and Bob must be indistin-guishable in all degrees of freedom (see Sec. III B). Hence,the optical pulses at the output of the IM are band-passfiltered using a 2 GHz-bandwidth (FWHM) fiber Bragg

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FIG. 1. Schematic diagram of the quantum teleportation system consisting of Alice, Bob, Charlie, and the data acquisition(DAQ) subsystems. See the main text for descriptions of each subsystem. One cryostat is used to house all SNSPDs, it isdrawn as two for ease of explanation. Detection signals generated by each of the SNSPDs are labelled 1-4 and collected atthe TDC, with 3 and 4 being time-multiplexed. All individual components are labeled in the legend, with single-mode opticalfibers (electronic cables) in grey (green), and with uni- and bi-chromatic (i.e. unfiltered) optical pulses indicated.

grating (FBG) centered at 1536.5 nm to match the spec-trum of the photons from the entangled pair-source (de-scribed in Sec. II B). Furthermore, the polarization ofAlice’s photons is determined by a manual polarizationcontroller (POC) in conjunction with a polarizing beamsplitter (PBS) at Charlie. Finally, the optical pulses fromAlice are attenuated to the single photon level by a vari-able optical attenuator (VOA), to approximate photonic

time-bin qubits of the form |A〉 = γ |e〉A +√

1− γ2 |l〉A,where the late state |l〉A arrives 2 ns after the early state|e〉A, γ is real and set to be either 1, 0, or 1/

√2 to

generate |e〉A, |l〉A, or |+〉A = (|e〉A + |l〉A)/√

2, respec-tively, depending on the experiment. The complex rel-ative phase is absorbed into the definition of |l〉A. Theduration of each time bin is 800 ps.

B. Bob: entangled qubit generation andteleported-qubit measurement

Similar to Alice, one (two) optical pulse(s) with aFWHM of ∼ 65 ps is (and separated by 2 ns are) cre-ated using a 1536.5 nm CW laser in conjunction with alithium niobate IM driven by an AWG, while the 90/10beam splitter and PWM are used to maintain an extinc-tion ratio of at least 20 dB. An Erbium-Doped FiberAmplifier (EDFA) is used after the IM to boost the pulsepower and thus maintain a high output rate of photonpairs.

The output of the EDFA is sent to a Type-0 period-ically poled lithium niobate (PPLN) waveguide for sec-ond harmonic generation (SHG), upconverting the pulsesto 768.25 nm. The residual light at 1536.5 nm is re-moved by a 768 nm band-pass filter with an extinctionratio ≥ 80 dB. These pulses undergo spontaneous para-

4

metric down-conversion (SPDC) using a Type-II PPLNwaveguide coupled to a polarization-maintaining fiber(PMF), approximately producing either a photon pair|pair〉B = |ee〉B , or the time-bin entangled state |φ+〉B =(|ee〉B + |ll〉B)/

√2, if one or two pulses, respectively, are

used to drive the IM.

The ordering of the states refers to so-called signal andidler modes of the pair of which the former has parallel,and the latter orthogonal, polarization with respect tothe axis of the PMF. As before, the relative phase isabsorbed into the definition of |ll〉B . Each photon is sep-arated into different fibers using a PBS and spectrallyfiltered with FBGs akin to that at Alice. Note the band-width of the FBG is chosen as a trade-off between spec-tral purity and generation rate of Bob’s photons [51].

The photon in the idler mode is sent to Charlie forteleportation or HOM measurements (see Sec. III B), orto the MZI (see below) for characterizations of the en-tangled state (see Sec. III A), with its polarization de-termined using a POC.The photon in the signal modeis sent to a Mach Zehnder interferometer (MZI) by wayof a POC (and an additional 11 km of single-mode fiberfor some measurements), and is detected by supercon-ducting nanowire single photon detectors (SNSPDs) [52]after high-pass filtering (HPF) to reject any remaining768.25 nm light. The MZI and detectors are used forprojection measurements of the teleported state, charac-terization of the time-bin entangled state, or measuringHOM interference at Charlie. The time-of-arrival of thephotons is recorded by the DAQ subsystem using a time-to-digital converter (TDC) referenced to the clock signalfrom the AWG.

All SNSPDs are installed in a compact sorption fridgecryostat [53], which operates at a temperature of 0.8 Kfor typically 24 h before a required 2 h downtime. OurSNSPDs are developed at the Jet Propulsion Laboratoryand have detection efficiencies between 76 and 85%, withlow dark count rates of 2-3 Hz. The FWHM temporalresolution of all detectors is between 60 and 90 ps whiletheir recovery time is ∼50 ns. A detailed descriptionof the SNSPDs and associated setup is provided in Ap-pendix A 2.

The MZI has a path length difference of 2 ns and isused to perform projection measurements of |e〉B , |l〉B ,and (|e〉B + eiϕ |l〉B)/

√2, by detecting photons at three

distinct arrival times in one of the outputs, and varyingthe relative phase ϕ [28]. Detection at the other out-put yields the same measurements except with a relativephase of ϕ + π. Using a custom temperature-feedbacksystem, we slowly vary ϕ for up to 15 hour time intervalsto collect all measurements, which is within the cryostathold time. Further details of the MZI setup is describedin Appendix A 3.

C. Charlie: Bell-state measurement

Charlie consists of a 50/50 polarization-maintainingfiber beam splitter (BS), with relevant photons from theAlice and Bob subsystems directed to each of its inputsvia a PBSs and optical fiber. The photons are detectedat each output with an SNSPD after HPFs, with theirarrival times recorded using the DAQ as was done atBob. Teleportation is facilitated by measurement of the|Ψ−〉AB = (|el〉AB − |le〉AB)/

√2 Bell state, which cor-

responds to the detection of a photon in |e〉 at one de-tector followed by the detection of a photon in |l〉 at theother detector after Alice and Bob’s (indistinguishable)qubits arrive at the BS [54]. Projection on the |Ψ−〉ABstate corresponds to teleportation of |A〉 up to a knownlocal unitary transformation, i.e. our system produces−iσy |A〉, with σy being the Pauli y-matrix.

III. EXPERIMENTAL RESULTS

Prior to performing quantum teleportation, we mea-sure some key parameters of our system that underpinthe teleportation fidelity. Specifically, we determine thefidelity of the entangled state produced by Bob by mea-suring the entanglement visibility Vent [55], and also de-termine to what extent Alice and Bob’s photons are in-distinguishable at Charlie’s BS using the HOM effect [47].

A. Entanglement visibility

The state |pair〉B (and hence the entangled state|φ+〉B) described in Sec. II B is idealized. In real-ity, the state produced by Bob is better approximatedby a two-mode squeezed vacuum state |TMSV〉B =√

1− p∑∞n=0

√pn |nn〉B after the FBG filter and neglect-

ing loss [56]. Here, n is the number of photons per tem-poral mode (or qubit), p is the emission probability ofa single pair per mode (or qubit), with state orderingreferring to signal and idler modes. However, |TMSV〉Bapproximates a photon pair for p

5

The results shown in Fig. 2 are fit proportional to1+Vent sin (ωT + Φ), where Vent = (Rx−Rn)/(Rx+Rn),with Rx(n) denoting the maximum (minimum) rate ofcoincidence events [55], ω and Φ are unconstrained con-stants, and T is the temperature of the MZI, findingVent = 96.4± 0.3%.

The deviation from unit visibility is mainly due to non-zero multi photon emissions [57], which is supported byan analytical model that includes experimental imperfec-tions [58]. Nonetheless, this visibility is far beyond the1/3 required for non-separability of a Werner state [59]

and the locality bound of 1/√

2 [55, 60]. Furthermore, itpredicts a fidelity Fent = (3Vent+1)/4 = 97.3± .2% withrespect to |φ+〉 [59], and hence is sufficient for quantumteleportation.

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FIG. 2. Entanglement visibility. The temperature of the in-terferometer is varied to reveal the expected sinusoidal vari-ations in the rate of coincidence events. A fit reveals theentanglement visibility Vent = 96.4± 0.3%, see main text fordetails. Uncertainties here and in all measurements are cal-culated assuming Poisson statistics.

B. HOM interference visibility

The BSM relies on quantum interference of photonsfrom Alice and Bob. This is ensured by the BS at Charlie,precise control of the arrival time of photons with IMs,identical FBG filters, and POCs (with PBSs) to providethe required indistinguishabiliy. The degree of interfer-ence is quantified by way of the HOM interference visibil-ity VHOM = (Rd−Ri)/Rd, with Rd(i) denoting the rate ofcoincident detections of photons after the BS when thephotons are rendered as distinguishable (indistinguish-able) as possible [47]. Completely indistinguishable sin-gle photons from Alice and Bob may yield VHOM = 1.However in our system, Alice’s qubit is approximated

from a coherent state |α〉A = e−|α|2/2

∑∞n=0

αn√n!|n〉A

with α

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FIG. 3. Hong-Ou-Mandel (HOM) interference. A relative dif-ference in arrival time is introduced between photons from Al-ice and Bob at Charlie’s BS. HOM interference produces a re-duction of the three-fold coincidence detection rate of photonsas measured with SNSPDs after Charlie’s BS and at Bob. Afit reveals a) VHOM = 70.9±1.9% and b) VHOM = 63.4±5.9%when lengths of fiber are added, see main text for details.

BSM (see Sec. II C). Since measurement of |+〉 in oursetup by symmetry is equivalent to any state of theform (|e〉 + eiϕ |l〉)/

√2 (and in particular the remaining

three basis states (|e〉 − |l〉)/√

2 and (|e〉 ± i |l〉)/√

2), wemay determine the average teleportation fidelity Favg =(Fe + Fl + 4F+)/6 of any time-bin qubit.

First, we prepare |e〉A and |l〉A with µA = 3.53×10−2,with Bob’s idler bypassing the MZI to be detected bya single SNSPD. We measure Fe = 95 ± 1% and Fl =96 ± 1%, conditioned on a successful measurement of|Ψ−〉AB at Charlie, with fidelity limited by multipho-ton events in Alice and Bob’s qubits and dark countsof the SNSPDs [58]. We then repeat the measurementwith µA = 9.5× 10−3 after inserting the aforementioned44 km length of fiber as before to emulate Alice, Charlieand parts of Bob being separated by long distances. Thisgives Fe = 98 ± 1% and Fl = 98 ± 2%, with no reduc-

tion from the additional fiber loss owing to our low noiseSNSPDs.

Next, we prepare |+〉A with µA = 9.38 × 10−3, in-sert the MZI and, conditioned on the BSM, we measureF+ = (1 + V+)/2 = 84.9 ± 0.5% by varying ϕ. Here,V+ = 69.7 ± 0.9% is the average visibility obtained byfits to the resultant interference measured at each out-put of the MZI, as shown in Fig. 4a. The reduction infidelity from unity is due to multiphoton events and dis-tinguishability, consistent with that inferred from HOMinterference, as supported by further measurements andanalytical modelling in Sec. IV.

The measurement is repeated with the additional longfiber, giving V+ = 58.6±5.7% and F+ = 79.3±2.9% withresults and corresponding fit shown in Fig. 4b. The re-duced fidelity is likely due to aforementioned polarizationvariations over the long fibers, consistent with the reduc-tion in HOM interference visibility, and exacerbated hereowing to the less than ideal visibility of the MZI over longmeasurement times (see Sec. A 3).

The results yield Favg = 89 ± 1% (86 ± 3%) without(with) the additional fiber, which is significantly abovethe classical bound of 2/3, implying strong evidence ofquantum teleportation [62], and limited from unity bymultiphotons events, distinguishability, and polarizationvariations, as mentioned [58].

To glean more information about our teleportation sys-tem beyond the fidelity, we reconstruct the density matri-ces of the teleported states using a maximum-likelihoodQST [48] described in Appendix C. The results of theQST with and without the additional fiber lengths aresummarized in Figs. 8 and 9, respectively. As can beseen, the diagonal elements for |+〉 are very close to theexpected value indicating the preservation of probabili-ties for the basis states of |e〉 and |l〉 after teleportation,while the deviation of the off-diagonal elements indicatethe deterioration of coherence between the basis states.The decoherence is attributed to multiphoton emissionsfrom our entangled pair source and distinguishability,consistent with the aforementioned teleportation fideli-ties of |+〉A, and further discussed in Sec. IV. Finally,we do also extract the teleportation fidelity from thesedensity matrices, finding the results shown in Fig. 5,and Favg = 89 ± 1% (88 ± 3%) without (with) the fiberspools, which are consistent with previous measurementsgiven the similar µA used for QST.

We point out that the 2/3 classical bound may only beapplied if Alice prepares her qubits using genuine singlephotons, i.e. |n = 1〉, rather than using |α

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FIG. 4. Quantum teleportation of |+〉. Teleportation is per-formed b) with and a) without an additional 44 km of single-mode fiber inserted into the system. The temperature of theinteferometer is varied to yield a sinusoidal variation of thethree-fold coincidence rate at each output of the MZI (blueand red points). A fit of the visibilities (see Sec. III A) mea-sured at each output (V+,1, V+,2) of the MZI gives an averagevisibility V+ = (V+,1 +V+,2)/2 of a) 69.7±0.91% without theadditional fiber and b) 58.6± 5.7% with the additional fiber.

state method [49] and follow the approach of Refs.[29, 64]. Decoy states, which are traditionally usedin quantum key distribution to defend against photon-number splitting attacks, are qubits encoded into co-herent states |α〉 with varying mean photon numberµA = |α|2. Measuring fidelities of the teleported qubitsfor different µA, the decoy-state method allows us to cal-culate a lower bound F dA on the teleportation fidelity ifAlice had encoded her qubits using |n = 1〉.

We prepare decoy states |e〉A, |l〉A, and |+〉A with vary-ing µA, as listed in Table I, and perform quantum telepor-tation both with and without the added fiber, with tele-portation fidelities shown in Table I. From these resultswe calculate F dA as shown in Fig. 5, with F

davg ≥ 93± 4%

(F davg ≥ 89± 2%) without (with) the added fiber, whichsignificantly violate the classical bound and the boundof 5/6 given by an optimal symmetric universal cloner[65, 66], clearly demonstrating the capability of our sys-

|e |l | + Average0.0

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Single-photon fidelity from DSMFidelity from QST

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b) Single-photon fidelity from DSMFidelity from QST

FIG. 5. Quantum teleportation fidelities for |e〉A, |l〉A, and|+〉A, including the average fidelity. The dashed line rep-resents the classical bound. Fidelities using quantum statetomography (QST) are shown using blue bars while the min-imum fidelities for qubits prepared using |n = 1〉, F de , F dl ,and F d+, including the associated average fidelity F

davg, respec-

tively, using a decoy state method (DSM) is shown in grey.Panels a) and b) depicts the results without and with addi-tional fiber, respectively. Uncertainties are calculated usingMonte-Carlo simulations with Poissonian statistics.

tem for high-fidelity teleportation. As depicted in Fig.5 these fidelities nearly match the results we obtainedwithout decoy states within statistical uncertainty. Thisis due to the suitable µA, as well as low µB and SNSPDdark counts in our previous measurements [58].

IV. ANALYTICAL MODEL AND SIMULATION

As our measurements have suggested, multi-photoncomponents in, and distinguishability between, Alice andBob’s qubits reduce the values of key metrics includ-ing HOM interference visibility and, consequently, quan-tum teleportation fidelity. To capture these effects inour model, we employ a Gaussian-state characteristic-function method developed in Ref. [58], which was en-

8

qubit without long fiber with long fiberµA (×10−3) F dA (%) µA (×10−3) F dA (%)

|e〉A 3.53 95.2 ± 1 26.6 95.7 ± 1.51.24 86.7 ± 2 9.01 98.4 ± 1.1

0 52.8 ± 3.4 - -|l〉A 3.53 95.9 ± 1 32.9 98.6 ± 0.7

1.24 90.5 ± 2 9.49 98.4 ± 1.60 52.8 ± 3.4 - -

|+〉A 9.38 84.7 ± 1.1 29.7 73.6 ± 3.02.01 83.2 ± 3.6 10.6 82.21 ± 3.9

0 52.8 ± 3.4 - -

TABLE I. Teleportation fidelities with (right column) andwithout (center column) the 44 km-length of fiber for Alice’squbit states prepared with varying µA. Mean photon numbersand fidelities for vacuum states with fiber are assumed to bezero and 50%, respectively.

abled by work in Ref. [67]. This approach is well-suitedto analyze our system because the quantum states, oper-ations, and imperfections (including losses, dark counts,etc.) of the experiment can be fully described usingGaussian operators, see e.g. Ref. [68]. We now brieflyoutline the model of Ref. [58], and employ it to estimatethe amount of indistinguishability ζ between Alice andBob’s qubits in our measurements of HOM interferenceand quantum teleportation.

Distinguishability in any degree-of-freedom may bemodelled by introducing a virtual beam splitter of trans-mittance ζ into the paths of Alice and Bob’s relevant pho-tons. As shown in Fig. 6, indistinguishable componentsof incoming photon modes are directed towards Charlie’sBS where they interfere, whereas distinguishable compo-nents are mixed with vacuum at the BS and do not con-tribute to interference. Here ζ = 1 (ζ = 0) correspondsto the case when both incoming photons are perfectlyindistinguishable (distinguishable). Now we may calcu-late the probability of a three-fold coincidence detection

event P3f between D1, D2 (Charlies’ detectors), and D3

FIG. 6. Schematic depiction of distingushability between Al-ice and Bob’s photons at Charlie’s BS. Distinguishability ismodeled by means of a virtual beam splitter with a transmit-tance ζ. Indistinguishable photons contribute to interferenceat the Charlie’s BS while distinguishable photons are mixedwith vacuum, leading to a reduction of HOM visibility andteleportation fidelity. See main text for further details.

(detects Bob’s signal photon) for a given qubit state ρABfrom Alice and Bob:

P3f = Tr{ρAB(I− (|0〉 〈0|)⊗3

â1,â2,â3)

⊗ (I− (|0〉 〈0|)⊗3

b̂1,b̂2,b̂3)⊗ (I− (|0〉 〈0|)ĉ)}, (1)

where the â and b̂ operators refer to modes, which origi-nate from Alice and Bob’s virtual beam splitters and aredirected to D1 and D2, respectively, and ĉ correspondsto Bob’s idler mode, which is directed to D3, see Fig. 6.This allows the derivation of an expression for the HOMinterference visibility

VHOM (ζ) = [P3f (0)− P3f (ζ)]/P3f (0), (2)

consistent with that introduced in Sec. III B. SinceAlice and Bob ideally produce ρAB = (|α〉 〈α|) ⊗(|TMSV〉 〈TMSV|), and recognizing that all operators inP3f are Gaussian, we analytically derive

P3f (ζ) = 1− 2exp(−µA/2[1+(1−ζ

2)ηiµB/2]1+ηiµB/2

)

1 + ηiµB/2− 1

1 + ηsµB+

exp(−µA)1 + ηiµB

− exp(−µA)1 + (1− ηs)ηiµB + ηsµB

+ 2exp(−µA/2[1+(1−ζ

2)(1−ηs)ηiµB/2+ηsµB ]1+(1−ηs)ηiµB/2+ηsµB )

1 + (1− ηs)ηiµB/2 + ηsµB, (3)

for varied ζ, where ηi and ηs are the transmission ef-ficiencies of the signal and idler photons, including de-tector efficiencies. We similarly calculate the impact of

distinguishability on the teleportation fidelity of |+〉:

F (ζ) = P3f (ζ, ϕmax)/[P3f (ζ, ϕmax) + P3f (ζ, ϕmin)],(4)

where ϕmax (ϕmin) is the phase of the MZI added intothe path of the signal photon, corresponding to maximum

9

(minimum) three-fold detection rates.To compare the model to our measurements, we use the

experimental mean photon numbers for the photon-pairsource ηi = 1.2 × 10−2 and ηs = 4.5 × 10−3 as deter-mined by the method described in Appendix B. We thenmeasure the teleportation fidelity of |+〉 and HOM inter-ference visibility (keeping the MZI in the system to en-sure ηs remains unchanged) for different values µA. Theresults are plotted in Fig. 7. The data is then fitted tothe expressions VHOM (ζ) and F (ζ) derived in our modeland graphed in Fig. 7. The fitted curves are in very goodagreement with our experimental values and consistentlyyield a value of ζ = 90% for both measurements types.This implies that we have only a small amount of resid-ual distinguishability between Alice and Bob’s photons.Potential effects leading to this distinguishability are dis-cussed in Sec. V.

Overall, our analytic model is consistent with our ex-perimental data [58] in the regime of µA 100 GHz) and those generated at Al-ice by the IM (15 GHz), leading to nonidentical filteringby the FBG. This can be improved by narrower FBGsor by using a more broadband pump at Alice (e.g. us-ing a mode locked laser or a higher bandwidth IM, e.g >50 GHz, which is commercially available). Alternatively,pure photon pairs may be generated by engineered phasematching, see e.g. Ref. [71]. Distinguishability owing tononlinear modulation during the SHG process could alsoplay a role [72]. The origin of distinguishability in oursystem, whether due to imperfect filtering or other deviceimperfections (e.g. PBS or BS) will be studied in futurework. Coupling loss can be minimized to less than afew dB overall by improved fiber-to-chip coupling, lower-loss components of the FBGs (e.g. the required isolator),spliced fiber connections, and reduced losses within ourMZI. Note that our current coupling efficiency is equiva-lent to ∼50 km of single mode fiber, suggesting that oursystem is well-suited for quantum networks provided loss

10

is reduced.While the fidelities we demonstrate are sufficient for

several applications, the current ∼Hz teleportation rateswith the 44 km length of fiber are still low. Higher repe-tition rates (e.g. using high-bandwidth modulators withwide-band wavelength division multiplexed filters andlow-jitter SNSPDs [73]), improvements to coupling anddetector efficiencies, enhanced BSM efficiency with fast-recovery SNSPDs [74], or multiplexing in frequency [64]will all yield substantial increases in teleportation rate.Note that increased repetition rates permits a reductionin time bin separation which will allow constructing theMZI on chip, providing exceptional phase stability andhence, achievable fidelity. Importantly, the aforemen-tioned increases in repetition rate and efficiency are af-forded by improvements in SNSPD technology that arecurrently being pursued with our JPL, NIST and otheracademic partners.

Upcoming system-level improvements we plan to in-vestigate and implement include further automation bythe implementation of free-running temporal and polar-ization feedback schemes to render the photons indistin-guishable at the BSM [29, 30]. Furthermore, several elec-trical components can be miniaturized, scaled, and mademore cost effective (e.g. field-programmable gate arrayscan replace the AWG).We note that our setup prototypewill be easily extended to independent lasers at differentlocations, also with appropriate feedback mechanisms forspectral overlap [75, 76]. These planned improvementsare compatible with the data acquisition and control sys-tems that were built for the systems and experiments atFQNET and CQNET presented in this work.

Overall, our high-fidelity teleportation systems achiev-ing state-of-the-art teleporation fidelities of time-binqubits serve as a blueprint for the construction of quan-tum network test-beds and eventually global quantumnetworks towards the quantum internet. In this work,we present a complete analytical model of the telepora-tion system that includes imperfections, and compare itwith our measurements. Our implementation, using ap-proaches from High Energy Physics experimental systemsand real-world quantum networking, features near fully-automated data acquisition, monitoring, and real-timedata analysis. In this regard our Fermilab and CaltechQuantum Networks serve as R& D laboratories and pro-totypes towards real-world quantum networks. The highfidelities achieved in our experiments using practical andreplicable devices are essential when expanding a quan-tum network to many nodes, and enable the realizationof more advanced protocols, e.g. [18, 77, 78].

ACKNOWLEDGEMENTS

R.V., N.L., L.N., C.P., N.S., M.S. and S.X. acknowl-edge partial and S.D. full support from the Alliancefor Quantum Technologies (AQT) Intelligent QuantumNetworks and Technologies (IN-Q-NET) research pro-

gram. R.V., N.L., L.N., C.P., N.S., M.S. S.X. andA.M. acknowledge partial support from the U.S. De-partment of Energy, Office of Science, High EnergyPhysics, QuantISED program grant, under award num-ber de-sc0019219. A.M. is supported in part by theJPL President and Directors Research and Develop-ment Fund (PDRDF). C.P. further acknowledges par-tial support from the Fermilab’s Lederman Fellowshipand LDRD. D.O. and N.S. acknowledge partial sup-port from the Natural Sciences and Research Council ofCanada (NSERC). D.O. further acknowledges the Cana-dian Foundation for Innovation, Alberta Innovates, andAlberta Economic Development, Trade and TourismsMajor Innovation Fund. J.A. acknowledges support bya NASA Space Technology Research Fellowship. Partof the research was carried out at the Jet PropulsionLaboratory, California Institute of Technology, under acontract with the National Aeronautics and Space Ad-ministration (80NM0018D0004). We thank Jason Trevor(Caltech Lauritsen Laboratory for High Energy Physics),Nigel Lockyer and Joseph Lykken (Fermilab), VikasAnant (PhotonSpot), Aaron Miller (Quantum Opus), In-der Monga and his ESNET group at LBNL, the groups ofWolfgang Tittel and Christoph Simon at the Universityof Calgary, the groups of Nick Hutzler, Oskar Painter,Andrei Faraon, Manuel Enders and Alireza Marandiat Caltech, Marko Loncar’s group at Harvard, Ar-tur Apresyan and the HL-LHC USCMS-MTD Fermilabgroup; Marco Colangelo (MIT); Tian Zhong (Chicago);AT&T’s Soren Telfer, Rishi Pravahan, Igal Elbaz, AndreFeutch and John Donovan. We acknowledge the enthusi-astic support of the Kavli Foundation on funding QIS&Tworkshops and events and the Brinson Foundation sup-port especially for students working at FQNET andCQNET. M.S. is especially grateful to Norm Augustine(Lockheed Martin), Carl Williams (NIST) and Joe Broz(SRI, QED-C); Hartmut Neven (Google Venice); AmirYacoby and Misha Lukin (Harvard); Ned Allen (Lock-heed Martin); Larry James and Ed Chow (JPL); theQCCFP wormhole teleportation team especially DanielJafferis (Harvard) and Alex Zlokapa (Caltech), Mark Ka-sevich (Stanford), Ronald Walsworth (Maryland), JunYeh and Sae Woo Nam (NIST); Irfan Siddiqi (Berkeley);Prem Kumar (Northwestern), Saikat Guha (Arizona),Paul Kwiat (UIUC), Mark Saffman (Wisconcin), JelenaVuckovic (Stanford) Jack Hidary (X), and the quantumnetworking teams at ORNL, ANL, and BNL, for produc-tive discussions and interactions on quantum networksand communications.

Appendix A: Detailed description of experimentalcomponents

1. Control systems and data acquisition

Our system is built with a vision towards future repli-cability, with particular emphasis on systems integra-

http://arxiv.org/abs/de-sc/0019219

11

tion. Each of the Alice, Bob and Charlie subsystemsis equipped with monitoring and active feedback stabi-lization systems (e.g. for IM extinction ratio), or has ca-pability for remote control of critical network parameters(e.g. varying the qubit generation time). Each subsystemhas a central classical processing unit with the followingfunctions: oversight of automated functions and work-flows within the subsystem, data acquisition and man-agement, and handling of input and output synchroniza-tion streams. As the quantum information is encodedin the time domain the correct operation of the classicalprocessing unit depends critically on the recorded time-of-arrival of the photons at the SNSPDs. Thus signifi-cant effort was dedicated to build a robust DAQ subsys-tem capable of recording and processing large volumes oftime-tagged signals from the SNSPDs and recorded byour TDCs at a high rate. The DAQ is designed to en-able both real-time data analysis for prompt data qualitymonitoring as well as post-processing data analysis thatallows to achieve the best understanding of the data.

The DAQ system is built on top of the standaloneLinux library of our commercial TDC. It records timetags whenever a signal is detected in any channel in co-incidence with the reference 90 MHz clock. Time tagsare streamed to a PC where they are processed in real-time and stored to disk for future analysis. A graphicaluser interface has been developed, capable of real-timevisualization and monitoring of photons detected whileexecuting teleportation. It also allows for easy control ofthe time-intervals used for each channel and to configurerelevant coincidences between different photon detectionevents across all TDC channels. We expect our DAQ sub-system to serve as the foundation for future real-worldtime-bin quantum networking experiments (see Sec. V).

2. Superconducting nanowire single photondetectors

We employ amorphous tungsten silicide SNSPDs man-ufactured in the JPL Microdevices Laboratory for allmeasurements at the single photon level (see Sec. II B)[52]. The entire detection system is customized for opti-mum autonomous operation in a quantum network. TheSNSPDs are operated at 0.8 K in a closed-cycle sorptionfridge [53]. The detectors have nanowire widths between140 to 160 nm and are biased at a current current of 8to 9 µA. The full-width at half maximum (FWHM) tim-ing jitter (i.e. temporal resolution) for all detectors isbetween 60 and 90 ps (measured using a Becker & HicklSPC-150NXX time-tagging module). The system detec-tion efficiencies (as measured from the fiber bulkhead ofthe cryostat) are between 76 and 85 %. The SNSPDs fea-ture low dark count rates between 2 and 3 Hz, achieved byshort-pass filtering of background black-body radiationthrough coiling of optical fiber to a 3 cm diameter withinthe 40 K cryogenic environment, and an additional band-pass filter coating deposited on the detector fiber pigtails

(by Andover Corporation). Biasing of the SNSPDs is fa-cilitated by cryogenic bias-Ts with inductive shunts toprevent latching, thus enabling uninterrupted operation.The detection signals are amplified using Mini-CircuitsZX60-P103LN+ and ZFL-1000LN+ amplifiers at roomtemperature, achieving a total noise figure of 0.61 dBand gain of 39 dB at 1 GHz, which enables the low sys-tem jitter. Note that FWHM jitter as low as 45 ps isachievable with the system, by biasing the detectors atapproximately 10 µA, at the cost of an elevated DCR onthe order of 30 cps. Using commercially available com-ponents, the system is readily scalable to as many as 64channels per cryostat, ideal for star-type quantum net-works, with uninterrupted 24/7 operation. The bulkiestcomponent of the current system is an external heliumcompressor, however, compact rack-mountable versionsare readily available [53].

3. Interferometer and phase stabilization

We use a commercial Kylia 04906-MINT MZI, whichis constructed of free-space devices (e.g mirrors, beamspliters) with small form-factor that fits into a hand-heldbox. Light is coupled into and out of the MZI usingpolarization maintaining fiber with loss of ∼2.5 dB. Theinterferometer features an average visibility of 98.5% thatwas determined by directing |+〉 with µA = 0.07 into oneof the input ports, measuring the fringe visibility on eachof the outputs using an SNSPD. The relative phase ϕ iscontrolled by a voltage-driven heater that introduces asmall change in refractive index in one arm of the MZI.However, this built-in heater did not permit phase stabil-ity sufficient to measure high-fidelity teleportation, withthe relative phase following the slowly-varying ambienttemperature of the room. To mitigate this instability,we built another casing, thermally isolating the MZI en-closure from the laboratory environment and controlledthe temperature via a closed-loop feedback control sys-tem based on a commercial thermoelectric cooler and aLTC1923 PID-controller. The temperature feedback isprovided by a 10 kΩ NTC thermistor while the set-pointis applied with a programmable power supply. This con-trol system permits us to measure visbilities by slowlyvarying ϕ over up to 15 hour timescales. We remarkthat no additional methods of phase control were usedbeyond that of temperature.

Appendix B: Estimation of mean number of photonpairs and transmission efficiencies of signal and idler

photons

Using a method described in Ref. [55], we measurethe mean number of photon pairs produced by Bob µBas a function of laser excitation power before the PPLNwaveguide used for SHG. To this end, we modify thesetup of Fig. 1 and direct each of Bob’s signal and idler

12

photons to a SNSPD. We then measure detection eventswhile varying the amplification of our EDFA by way ofan applied current. We extract events when photon pairswhich originated from the same clock cycle are measuredin coincidence, and when one photon originating from acycle is measured in coincidence with a photons origi-nated from a preceding or following clock cycle, in otherwords we measure the so-called coincidence and acciden-tal rates. The ratio of accidentals to coincidences ap-proximates µB

13

0

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al P

art

CQNET/FQNET Prelim. 2020Teleportation of |e

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inar

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CQNET/FQNET Prelim. 2020Teleportation of |l

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FIG. 8. Elements of the density matrices of teleported |e〉,|l〉, and |+〉 states with the additional 44 km of fiber in thesystem.The black points are generated by our teleportationsystem and the blue bars with red dashed lines are the valuesassuming ideal teleportation.

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CQNET/FQNET Prelim. 2020Teleportation of | +

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Teleportation Systems Towards a Quantum InternetAbstractI IntroductionII SetupA Alice: single-qubit generationB Bob: entangled qubit generation and teleported-qubit measurementC Charlie: Bell-state measurement

III Experimental ResultsA Entanglement visibilityB HOM interference visibilityC Quantum teleportation1 Teleportation fidelity using decoy states

IV Analytical model and simulationV Discussion and Outlook AcknowledgementsA Detailed description of experimental components1 Control systems and data acquisition2 Superconducting nanowire single photon detectors3 Interferometer and phase stabilization

B Estimation of mean number of photon pairs and transmission efficiencies of signal and idler photonsC Quantum State tomography References