Teleportation Systems Toward a Quantum Internet

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020317 (2020)

Raju Valivarthi

1,2

Samantha I. Davis,

1,2

Cristián Peña,

1,2,3

Si Xie

1,2

Nikolai Lauk,

1,2

Lautaro Narváez

1,2

Jason P. Allmaras

Andrew D. Beyer,

Yewon Gim,

2,5

Meraj Hussein,

George Iskander

Hyunseong Linus Kim

1,2

Boris Korzh

Andrew Mueller,

Mandy Rominsky,

Matthew Shaw,

Dawn Tang

1,2

Emma E. Wollman,

Christoph Simon,

Panagiotis Spentzouris,

Daniel Oblak,

Neil Sinclair,

1,2,7

and Maria Spiropulu

1,2,

Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena,

California 91125, USA

Alliance for Quantum Technologies (AQT), California Institute of Technology, Pasadena, California 91125, USA

Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA

AT&T Foundry, Palo Alto, California 94301, USA

Institute for Quantum Science and Technology, and Department of Physics and Astronomy, University of Calgary,

Calgary, Alberta T2N 1N4, Canada

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge,

Massachusetts 02138, USA

(Received 28 July 2020; accepted 16 October 2020; published 4 December 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 superconducting

nanowire single-photon detectors and off-the-shelf optics, we achieve conditional quantum teleportation

of time-bin qubits at the telecommunication wavelength of 1536.5 nm. We measure teleportation fidelities

≥

90% that are consistent with an analytical model of our system, which includes realistic imperfec-

tions. 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 22 km of fiber. Our sys-

tems, which are compatible with emerging solid-state quantum devices, provide a realistic foundation for

a high-fidelity quantum Internet with practical devices.

DOI:

10.1103/PRXQuantum.1.020317

I. INTRODUCTION

Quantum teleportation [

], one of the most captivating

predictions of quantum theory, has been widely investi-

gated since its seminal demonstrations over 20 years ago

[

–

]. This is due to its connections to fundamental physics

[

–

] and its central role in the realization of quantum

information technology such as quantum computers and

networks [

–

]. The goal of a quantum network is to

distribute qubits between different locations, a key task for

quantum cryptography, distributed quantum computing,

and sensing. A quantum network is expected to form part

of a future quantum Internet [

–

]: a globally distributed

smaria@caltech.edu

Published by the American Physical Society under the terms of

the

Creative Commons Attribution 4.0 International

license. Fur-

ther distribution of this work must maintain attribution to the

author(s) and the published article’s title, journal citation, and

DOI.

set of quantum processors, sensors, or users thereof that

are mutually connected over a network capable of allocat-

ing quantum resources (e.g., qubits and entangled states)

between locations. Many architectures for quantum net-

works require quantum teleportation, such as star-type net-

works that distribute entanglement from a central location

or quantum repeaters that overcome the rate-loss trade-off

of direct transmission of qubits [

–

Quantum teleportation of a qubit can be achieved by

performing a Bell-state measurement (BSM) between the

qubit and another that forms one member of an entan-

gled Bell state [

]. The quality of the teleportation

is often characterized by the fidelity

of the

teleported state

with respect to the state

accom-

plished by ideal generation and teleportation [

]. This

metric is becoming increasingly important as quantum net-

works move beyond specific applications, such as quantum

key distribution, and toward the quantum Internet.

Polarization qubits have been preferred for demonstra-

tions of quantum teleportation over free-space channels

2691-3399/20/1(2)/020317(16)

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Published by the American Physical Society

R A J U VA L I VA RT H I

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[

–

], including the recent ground-to-satellite quantum

teleportation [

], because of their ease of preparation and

measurement, as well as the lack of polarization rotation

in free space. Qubits encoded by the time of arrival of

individual photons, i.e., time-bin qubits [

], are useful

for fiber networks due to their simplicity of generation,

interfacing with quantum devices, as well as the inde-

pendence of dynamic polarization transformations of real-

world fibers. Individual telecom-band photons (around

1.5

m wavelength) are ideal carriers of qubits in networks

due to their ability to rapidly travel over long distances in

deployed optical fibers [

–

] or atmospheric chan-

nels [

], among other properties. Moreover, the improve-

ment and growing availability of sources and detec-

tors of individual telecom-band photons has accelerated

progress toward workable quantum networks and associ-

ated technologies, such as quantum memories [

], trans-

ducers [

], or quantum nondestructive measurement

devices [

Teleportation of telecom-band photonic time-bin qubits

has been performed inside and outside the laboratory,

with impressive results [

–

]. Despite this, there

has been little work to increase

beyond approximately

90% for these qubits, in particular using practical devices

that allow straightforward replication and deployment of

quantum networks (e.g., using fiber-coupled and com-

mercially available devices). Moreover, it is desirable to

develop teleportation systems that are forward compat-

ible with emerging quantum devices for the quantum

Internet.

In the context of the California Institute of Technology

(Caltech) multidisciplinary multi-institutional collabora-

tive public-private research program on Intelligent Quan-

tum Networks and Technologies (IN-Q-NET) founded in

2017 with AT&T as well as the Fermi National Accelera-

tor Laboratory and the Jet Propulsion Laboratory, we have

designed, built, commissioned, and deployed two quantum

teleportation systems: one at Fermilab, the Fermilab Quan-

tum Network (FQNET), and one at Caltech’s Lauritsen

Laboratory for High Energy Physics, the Caltech Quan-

tum Network (CQNET). The CQNET system serves as a

research and development (R&D), prototyping, and com-

missioning system, while FQNET serves as an expandable

system, for scaling up to long distances, and is used in

multiple projects funded currently by the United States

(U.S.) Department of Energy’s Office of High Energy

Physics (HEP) and Advanced Scientific Research Comput-

ing (ASCR). The material- and devices-level R&D of both

systems is facilitated and funded by the Office of Basic

Energy Sciences (BES). Both systems are accessible to

quantum researchers for R&D purposes as well as testing

and integration of various novel devices, such as, for exam-

ple, on-chip integrated nanophotonic devices and quantum

memories, needed to upgrade such systems toward a real-

istic quantum Internet. Importantly, both systems are also

used for improvements of the entanglement quality and

distribution, with emphasis on implementation of proto-

cols with complex entangled states toward advanced and

complex quantum communications channels. These will

assist in studies of systems that implement new teleporta-

tion protocols the gravitational duals of which correspond

to wormholes [

], error-correlation properties of worm-

hole teleportation, and on-chip codes as well as possible

implementation of protocols on quantum optics commu-

nication platforms. Hence the systems serve both funda-

mental quantum information science as well as quantum

technologies.

Here, we perform quantum teleportation of time-bin

qubits, conditioned on a successful BSM, at a wavelength

of 1536.5 nm with an average

≥

90%. This is accom-

plished using a compact setup of fiber-coupled devices,

including low-dark-count single-photon detectors and off-

the-shelf optics, allowing straightforward reproduction for

multinode networks. To illustrate network compatibility,

teleportation is performed with up to 44 km of single-

mode fiber between the qubit generation and the mea-

surement of the teleported qubit, and is facilitated using

semiautonomous control, monitoring, and synchronization

systems, with results collected using scalable acquisition

hardware. Our system, which operates at a clock rate of 90

MHz, can be run remotely for several days without inter-

ruption and can yield teleportation rates of a few Hertz

using the full length of fiber. Our qubits are also compatible

with erbium-doped crystals, e.g., Er : Y

SiO

, which are

used to develop quantum network devices such as mem-

ories and transducers [

–

]. The 1536.5-nm operating

wavelength is within the low-loss (

-band) telecommuni-

cation window for long-haul communication and where a

variety of off-the-shelf equipment is available. Finally, we

develop an analytical model of our system, which includes

experimental imperfections, predicting that the fidelity

can be improved further toward unity by well-understood

methods (such as improvement in photon indistinguisha-

bility). Our demonstrations provide a step toward a work-

able quantum network with practical and replicable nodes,

such as the ambitious U.S. Department of Energy quan-

tum research network envisioned to link the U.S. National

Laboratories.

In the following, we describe the components of

our systems as well as characterization measurements

that support our teleportation results, including the

fidelity of our entangled Bell state and the Hong-Ou-

Mandel (HOM) interference [

] that underpins the suc-

cess of the BSM. We then present our teleportation

results using both quantum state tomography (QST)

[

] and projection measurements based on a decoy-

state method (DSM) [

], followed by a discussion of

our model. We conclude by considering improvements

toward near-unit fidelity and gigahertz-level teleportation

rates.

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II. SETUP

Our fiber-based experimental system is summarized in

Fig.

. It allows us to demonstrate a quantum teleportation

protocol in which a photonic qubit (provided by Alice) is

interfered with one member of an entangled photon pair

(from Bob) and projected (by Charlie) onto a Bell state,

whereby the state of Alice’s qubit can be transferred to

the remaining member of Bob’s entangled photon pair. Up

to 22 (11) km of single-mode fiber is introduced between

Alice and Charlie (Bob and Charlie), as well as up to

another 11 km at Bob, depending on the experiment (see

Sec.

III

). All qubits are generated at the clock rate, with

all of their measurements collected using a data acquisi-

tion (DAQ) system. Each of the Alice, Bob, and Charlie

subsystems are further detailed in the following

subsections, with the DAQ subsystem described in

Appendix

A. Alice: single-qubit generation

To generate the time-bin qubit that Alice will teleport

to Bob, light from a fiber-coupled 1536.5-nm continuous-

wave (cw) laser is input into a lithium niobate intensity

modulator (IM). We drive the IM with one pulse, or two

pulses separated by 2 ns. Each pulse is of approximately

65 ps full width at half maximum (FWHM) duration. The

pulses are produced by an arbitrary-waveform generator

(AWG) and amplified by a 27-dB-gain high-bandwidth

amplifier to generate optical pulses that have an extinc-

tion ratio of up to 22 dB. We note that this method of

creating time-bin qubits offers us flexibility not only in

Charlie

DAQ

Alice

CIR

VOA

POC

AMP

Bob

PWM

90|10

TDC

Data Analysis

and Storage

Clock

FBG

FIS

LAS

FIS

CIR

SPDC

SHG

MZI

EDFA

PWM

AMP

Cryostat

PBS

50|50

BPF

FBG

1536 nm

signal

idler

POC

FIS

AWG

Clock

90|10

PWM = Powermeter

PBS = Polarizing

Beam Splitter

Controller

POC = Polarization

MZI = Mach-Zehnder

Interferometer

LAS = Laser

IM = Intensity Modulator

HPF = High-Pass Filter

EDFA = Erbium-Doped

Fiber Amplifier

FIS = Fiber Spool

SPDC = Spontaneous

Parametric Down-Conversion

SNSPD = Superconducting

Nanowire Single-Photon Detector

VOA = Variable Optical

Attenuator

SHG = Second-Harmonic

Generation

TDC = Time-To-Digital

Converter

FBG

1536 nm

1510 nm

768 nm

1536 nm

AMP = Amplifier

AWG = Arbitrary-

Waveform Generator

BS = Beam Splitter

CIR = Circulator

FBG = Fiber Bragg Grating

BPF = Band-Pass Filter

Bandwidth: 20 nm

Cryostat

50|50

PBS

HPF

1510 nm

HPF

1510 nm

FIG. 1.

A 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 is drawn as two for

ease of explanation. The detection signals generated by each of the SNSPDs are labelled 1–4 and collected at the TDC, with 3 and

4 being time multiplexed. All individual components are labeled in the legend, with single-mode optical fibers (electronic cables) in

gray (green), and with uni- and bichromatic (i.e., unfiltered) optical pulses indicated.

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terms of choosing a suitable time-bin separation but also

for synchronizing qubits originating from different nodes

in a network. A 90:10 polarization-maintaining fiber beam

splitter (BS) combined with a power monitor (PWM) is

used to apply feedback to the dc-bias port of the IM so as

to maintain a constant 22 dB extinction ratio [

]. In order

to successfully execute the quantum teleportation protocol,

photons from Alice and Bob must be indistinguishable in

all degrees of freedom (see Sec.

III B

). Hence, the opti-

cal pulses at the output of the IM are band-pass filtered

using a 2-GHz-bandwidth (FWHM) fiber Bragg grating

(FBG) centered at 1536.5 nm to match the spectrum of the

photons from the entangled pair source (described in Sec.

II B

). Furthermore, the polarization of Alice’s photons is

determined by a manual polarization controller (POC) in

conjunction with a polarizing beam splitter (PBS) at Char-

lie. Finally, the optical pulses from Alice are attenuated

to the single-photon level by a variable optical attenua-

tor (VOA), to approximate photonic time-bin qubits of the

form

√

−

, where the late state

arrives 2 ns after the early state

and

is real and

set to be either 1, 0, or 1

√

2 to generate

,or

(

√

2, respectively, depending on the

experiment. The complex relative phase is absorbed into

the definition of

. The duration of each time bin is

800 ps.

B. Bob: entangled qubit generation and

teleported-qubit measurement

Similar to Alice, one (two) optical pulse(s) with a

FWHM of approximately 65 ps is (and separated by 2 ns

are) created using a 1536.5-nm cw laser in conjunction

with a lithium niobate IM driven by an AWG, while the

90:10 BS and PWM are used to maintain an extinction

ratio of at least 20 dB. An erbium-doped fiber amplifier

(EDFA) is used after the IM to boost the pulse power and

thus maintain a high output rate of photon pairs.

The output of the EDFA is sent to a type-0 periodi-

cally poled lithium niobate (PPLN) waveguide for second-

harmonic generation (SHG), up-converting the pulses to

768.25 nm. The residual light at 1536.5 nm is removed

by a 768-nm band-pass filter with an extinction ratio

≥

dB. These pulses undergo spontaneous parametric down-

conversion (SPDC) using a type-II PPLN waveguide cou-

pled to a polarization-maintaining fiber (PMF), approxi-

mately producing either a photon pair

pair

the time-bin entangled state

(

√

if one or two pulses, respectively, are used to drive the IM.

The ordering of the states refers to so-called signal and

idler modes of the pair of which the former has parallel,

and the latter orthogonal, polarization with respect to the

axis of the PMF. As before, the relative phase is absorbed

into the definition of

. Each photon is separated into

different fibers using a PBS and spectrally filtered with

FBGs akin to that at Alice. Note that the bandwidth of the

FBG is chosen as a trade-off between spectral purity and

the generation rate of Bob’s photons [

The photon in the idler mode is sent to Charlie for

teleportation or HOM measurements (see Sec.

III B

)or

to a Mach-Zehnder interferometer (MZI) for characteri-

zations of the entangled state (see Sec.

III A

), with its

polarization determined using a POC. The photon in the

signal mode is sent to a MZI by way of a POC (and

an additional 11 km of single-mode fiber for some mea-

surements) and is detected by superconducting nanowire

single-photon detectors (SNSPDs) [

] after high-pass fil-

tering (HPF) to reject any remaining 768.25-nm light. The

MZI and detectors are used for projection measurements of

the teleported state, characterization of the time-bin entan-

gled state, or measuring HOM interference at Charlie. The

time of arrival of the photons is recorded by the DAQ sub-

system using a time-to-digital converter (TDC) referenced

to the clock signal from the AWG.

All SNSPDs are installed in a compact-sorption fridge

cryostat [

], which operates at a temperature of 0.8 K

for typically 24 h before a required 2-h downtime. Our

SNSPDs are developed at the Jet Propulsion Laboratory

and have detection efficiencies between 76 and 85%, with

low dark-count rates of 2–3 Hz. The FWHM temporal res-

olution of all detectors is between 60 and 90 ps, while their

recovery time is approximately 50 ns. A detailed descrip-

tion of the SNSPDs and the associated setup is provided in

Appendix

The MZI has a path-length difference of 2 ns and is

used to perform projection measurements of

,and

(

√

2, by detecting photons at three distinct

arrival times in one of the outputs and varying the rela-

tive phase

[

]. Detection at the other output yields the

same measurements except with a relative phase of

Using a custom temperature-feedback system, we slowly

vary

for up to 15-h time intervals to collect all mea-

surements, which is within the cryostat hold time. Further

details of the MZI setup are described in Appendix

C. Charlie: Bell-state measurement

Charlie consists of a 50:50 polarization-maintaining

fiber BS, with relevant photons from the Alice and Bob

subsystems directed to each of its inputs via a PBSs

and optical fiber. The photons are detected at each out-

put with an SNSPD after HPFs, with their arrival times

recorded using the DAQ, as was done at Bob. Teleportation

is facilitated by measurement of the

−

(

−

√

2 Bell state, which corresponds to the detection

of a photon in

at one detector followed by the detection

of a photon in

at the other detector after Alice and Bob’s

(indistinguishable) qubits arrive at the BS [

]. Projection

on the

−

state corresponds to teleportation of

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to a known local unitary transformation, i.e., our system

produces

−

, with

being the Pauli

matrix.

III. EXPERIMENTAL RESULTS

Prior to performing quantum teleportation, we measure

some key parameters of our system that underpin the tele-

portation fidelity. Specifically, we determine the fidelity

of the entangled state produced by Bob by measuring the

entanglement visibility

ent

[

] and also determine to

what extent Alice and Bob’s photons are indistinguishable

at Charlie’s BS using the HOM effect [

A. Entanglement visibility

The state

pair

(and hence the entangled state

) described in Sec.

II B

is idealized. In real-

ity, the state produced by Bob is better approximated

by a two-mode squeezed vacuum state

TMSV

√

−

∑

∞

√

after the FBG filter and neglect-

ing loss [

]. Here,

is the number of photons per temporal

mode (or qubit) and

is the emission probability of a

single pair per mode (or qubit), with the state ordering

referring to the signal and idler modes. However,

TMSV

approximates a photon pair for

1, with a

≈

mean number of pairs per mode (or qubit), conditioned on

measurement of a pair such that the

0 term is elimi-

nated. As a compromise between the pair-creation rate

∝

and the quality of entanglement, here and henceforth we

set the mean photon number of our pair source to be

(

8.0

0.4

)

−

per time bin, which is feasible because

of the exceptionally low dark counts of our SNSPDs. The

measurement of

is outlined in Appendix

We generate

and measure

ent

by directing the

idler photon to the second input port of the MZI, slightly

modifying the setup of Fig.

. The idler photon is delayed

compared to the signal, allowing unambiguous measure-

ment of each qubit. We vary

and project each qubit of

the entangled state onto phase-varied superpositions of

and

by accumulating coincidence events of photons at

both the outputs of the interferometer [

The results shown in Fig.

are fitted proportional to

ent

sin

(ω

)

, where

ent

(

−

)/(

)

with

(

)

denoting the maximum (minimum) rate of coin-

cidence events [

and

are unconstrained constants,

and

is the temperature of the MZI, finding

ent

96.4

0.3%.

The deviation from unit visibility is mainly due to

nonzero multiphoton emissions [

], which is supported

by an analytical model that includes experimental imper-

fections [

]. Nonetheless, this visibility is far beyond the

3 required for nonseparability of a Werner state [

]

and the locality bound of 1

√

]. Furthermore, it

predicts a fidelity

ent

(

ent

97.3

.2% with

respect to

[

] and hence is sufficient for quantum

teleportation.

Coincidences per 30 min

CQNET/FQNET 2020

FIG. 2. Entanglement visibility. The temperature of the inter-

ferometer is varied to reveal the expected sinusoidal variations

in the rate of coincidence events. A fit reveals the entangle-

ment visibility

ent

96.4

0.3% (see the main text for details).

The uncertainties here and in all measurements are calculated

assuming Poisson statistics.

B. HOM interference visibility

The BSM relies on quantum interference of photons

from Alice and Bob. This is ensured by the BS at Char-

lie, precise control of the arrival time of photons with IMs,

identical FBG filters, and POCs (with PBSs) to provide the

required indistinguishability. The degree of interference

is quantified by way of the HOM interference visibility

HOM

(

−

, with

(

)

denoting the rate of coin-

cident detections of photons after the BS when the photons

are rendered as distinguishable (indistinguishable) as pos-

sible [

]. Completely indistinguishable single photons

from Alice and Bob may yield

HOM

1. However, in

our system, Alice’s qubit is approximated from a coher-

ent state

−|

∑

∞

(α

√

)

with

akin to how Bob’s pair is approximated from

TMSV

(see Sec.

III A

), with

being Alice’s mean photon

number per mode (or qubit) [

]. Therefore, the contri-

bution of undesired photons from Alice and Bob lowers

the maximum achievable

HOM

below unity, with a fur-

ther reduction if the interfering photons are not completely

indistinguishable. The dependence of

HOM

with varied

and

, including effects of losses or distinguishable pho-

tons in our system, is analytically modeled in Ref. [

]and

briefly discussed in Sec.

We measure

HOM

by slightly modifying the setup of

Fig.

. We prepare

with

2.6

−

and

Bob as

pair

and direct Alice’s photon and Bob’s idler

to Charlie, with Bob’s signal bypassing the MZI to be

directly measured by an SNSPD. Alice’s IM is used to

introduce distinguishability by way of a relative difference

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in arrival time

of Alice and Bob’s photons at Char-

lie’s BS. Using Charlie’s SNSPDs and the third detector

at Bob, a threefold coincidence detection rate is measured

for varying

, with results shown in Fig.

3(a)

. Since the

temporal profiles of our photons are approximately Gaus-

sian, we fit our results to

−

HOM

exp

(

−

)

where

is the maximum coincidence rate when the pho-

tons are completely distinguishable and

300 ps is the

temporal duration of the optical pulses [

], find-

ing

HOM

70.9

1.9%. The maximum

HOM

for this

experiment is 83.5% if the photons are completely indis-

tinguishable [

], with the difference ascribed to slight

distinguishability between our photons as supported by the

further measurements and analytical modeling in Sec.

Improvements to our system to remove this distinguisha-

bility are discussed in Sec.

To test our system for quantum teleportation over long

distances, we introduce the aforementioned 22-, 11-, and

11-km lengths of single-mode fiber between Alice and

Charlie, Bob and Charlie, and in the path of Bob’s sig-

nal photon, respectively, repeat our measurement of

HOM

and fit the results as before [see Fig.

3(b)

]. We find

HOM

63.4

5.9%, which is consistent with the maximum

HOM

that we expect when including the impact of the additional

5.92-(2.56-)dB loss between Charlie and Alice (Bob) as

well as the effect of photon distinguishability (analyzed

in Sec.

). This suggests that, importantly, the additional

fiber does not introduce any further distinguishability (that

we cannot account for), thereby supporting our system’s

use in quantum networking. Overall, the presence of clear

HOM interference suggests that our system (with or with-

out the additional fiber) introduces relatively little in the

way of imperfections that can negatively impact the BSM

and hence the fidelity of quantum teleportation.

C. Quantum teleportation

We now perform quantum teleportation of the time-bin

qubit basis states

and

, so as to measure the

teleportation fidelities,

,and

, respectively, of the

teleported states with respect to their ideal counterparts,

up to the local unitary introduced by the BSM (see Sec.

II C

). Since measurement of

in our setup by symme-

try is equivalent to any state of the form

(

√

(and, in particular, the remaining three basis states

(

−

√

2and

(

√

2), we may determine the aver-

age teleportation fidelity

avg

(

6ofany

time-bin qubit.

First, we prepare

and

with

3.53

−

with Bob’s idler bypassing the MZI to be detected by

a single SNSPD. We measure

1% and

1%, conditioned on a successful measurement of

−

at Charlie, with the fidelity limited by multipho-

ton events in Alice and Bob’s qubits and dark counts of

the SNSPDs [

]. We then repeat the measurement with

Threefold coincidences per 10 min

CQNET/FQNET 2020

(a)

(b)

FIG. 3. Hong-Ou-Mandel (HOM) interference. A relative dif-

ference in arrival time is introduced between photons from Alice

and Bob at Charlie’s BS. HOM interference produces a reduc-

tion of the threefold coincidence detection rate of photons as

measured with SNSPDs after Charlie’s BS and at Bob. A fit

reveals (a)

HOM

70.9

1.9% and (b)

HOM

63.4

5.9%

when lengths of fiber are added (see the main text for details).

9.5

−

after inserting the aforementioned 44-km

length of fiber as before to emulate Alice, Charlie, and

parts of Bob being separated by long distances. This gives

1% and

2%, with no reduction from

the additional fiber loss owing to our low-noise SNSPDs.

Next, we prepare

with

9.38

−

, insert

the MZI and, conditioned on the BSM, we measure

(

84.9

0.5% by varying

. Here,

69.7

0.9% is the average visibility obtained by fits to the

resultant interference measured at each output of the MZI,

as shown in Fig.

4(a)

. The reduction in fidelity from unity

is due to multiphoton events and distinguishability, con-

sistent with that inferred from HOM interference, as sup-

ported by further measurements and analytical modeling in

Sec.

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Threefold coincidences per 12 min

Threefold coincidences per 40 min

CQNET/FQNET 2020

(a)

(b)

FIG. 4.

The quantum teleportation of

. Teleportation is

performed (b) with and (a) without an additional 44 km of

single-mode fiber inserted into the system. The temperature of

the inteferometer is varied to yield a sinusoidal variation of the

threefold coincidence rate at each output of the MZI (blue and

red points). A fit of the visibilities (see Sec.

III A

) measured

at each output (

)

of the MZI gives an average vis-

ibility

(

2 of (a) 69.7

0.91% without the

additional fiber and (b) 58.6

5.7% with the additional fiber.

The measurement is repeated with the additional long

fiber, giving

58.6

5.7% and

79.3

2.9%,

with the results and the corresponding fit shown in

Fig.

4(b)

. The reduced fidelity is likely due to aforemen-

tioned polarization variations over the long fibers, consis-

tent with the reduction in HOM interference visibility, and

is exacerbated here owing to the less than ideal visibility of

the MZI over long measurement times (see Appendix

The results yield

avg

(

)

without

(with) the additional fiber, which is significantly above the

classical bound of 2

3, implying strong evidence of quan-

tum teleportation [

] and limited from unity by multipho-

ton events, distinguishability, and polarization variations,

as mentioned [

To glean more information about our teleportation sys-

tem beyond the fidelity, we reconstruct the density matrices

of the teleported states using a maximum-likelihood QST

[

] described in Appendix

. The results of the QST with

and without the additional fiber lengths are summarized

in Figs.

and

, respectively. As can be seen, the diago-

nal elements for

are very close to the expected value,

indicating the preservation of probabilities for the basis

states of

and

after teleportation, while the devia-

tion of the off-diagonal elements indicates the deterioration

of coherence between the basis states. The decoherence

is attributed to multiphoton emissions from our entan-

gled pair source and distinguishability, consistent with the

aforementioned teleportation fidelities of

, and is fur-

ther discussed in Sec.

. Finally, we do also extract the

teleportation fidelity from these density matrices, finding

the results shown in Fig.

and

avg

(

)

CQNET/FQNET 2020

(a)

(b)

FIG. 5. Quantum teleportation fidelities for

,and

, including the average fidelity. The dashed line represents

the classical bound. Fidelities using QST are shown using blue

bars, while the minimum fidelities for qubits prepared using

,and

, including the associated average fidelity

avg

, respectively, using a DSM are shown in gray. (a),(b)

The results without and with additional fiber, respectively. The

uncertainties are calculated using Monte Carlo simulations with

Poissonian statistics.

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without (with) the fiber spools, which are consistent with

previous measurements given the similar

used for QST.

The use of weak coherent states instead of ideal single

photons reduces the teleportation fidelity because of mul-

tiphoton events. This effect is further discussed in Sec.

We also point out that the 2

3 classical bound may only

be applied if Alice prepares her qubits using genuine sin-

gle photons, i.e.,

, rather than using

as we

do in this work [

]. As a way to account for the photon

statistics of Alice’s qubits, we turn to an analysis using

decoy states.

1. Teleportation fidelity using decoy states

To determine the minimum teleportation fidelity of

qubits prepared using single photons, we use a DSM [

]

and follow the approach of Refs. [

]. Decoy states,

which are traditionally used in quantum key distribution to

defend against photon-number splitting attacks, are qubits

encoded into coherent states

with varying mean photon

number

. By measuring fidelities of the tele-

ported qubits for different

, the DSM allows us to

calculate a lower bound

on the teleportation fidelity if

Alice had encoded her qubits using

We prepare decoy states

,and

with varying

, as listed in Table

, and perform quantum teleportation

both with and without the added fiber, with teleporta-

tion fidelities shown in Table

. From these results, we

calculate

as shown in Fig.

, with

avg

≥

(

avg

≥

2%) without (with) the added fiber, which

significantly violates the classical bound and the bound

of 5

6 given by an optimal symmetric universal cloner

[

], clearly demonstrating the capability of our system

for high-fidelity teleportation. As depicted in Fig.

, these

fidelities nearly match the results we obtain without decoy

TABLE I.

The teleportation fidelities with (right column) and

without (center column) the 44-km length of fiber for Alice’s

qubit states prepared with varying

. The mean photon numbers

and fidelities for vacuum states with fiber are assumed to be zero

and 50%, respectively.

Qubit

Without long fiber

With long fiber

(

−

)

(%)

(

−

)

(%)

3.53

95.2

26.6

95.7

1.5

1.24

86.7

9.01

98.4

1.1

52.8

3.4

...

3.53

95.9

32.9

98.6

0.7

1.24

90.5

9.49

98.4

1.6

52.8

3.4

...

9.38

84.7

1.1

29.7

73.6

3.0

2.01

83.2

3.6

10.6

82.21

3.9

52.8

3.4

...

states within statistical uncertainty. This is due to the suit-

able

,aswellaslow

and SNSPD dark counts in our

previous measurements [

IV. ANALYTICAL MODEL AND SIMULATION

As our measurements have suggested, multiphoton

components in, and distinguishability between, Alice and

Bob’s qubits reduce the values of key metrics includ-

ing HOM interference visibility and, consequently, quan-

tum teleportation fidelity. To capture these effects in our

model, we employ a Gaussian-state characteristic-function

method developed in Ref. [

], which is enabled by work

in Ref. [

]. This approach is well suited to analyze

our system because the quantum states, operations, and

imperfections (including losses, dark counts, etc.) of the

experiment can be fully described using Gaussian oper-

ators (see, e.g., Ref. [

]). We now briefly outline the

model of Ref. [

] and employ it to estimate the amount

of indistinguishability

between Alice and Bob’s qubits

in our measurements of HOM interference and quantum

teleportation.

The distinguishability in any degree of freedom may

be modeled by introducing a virtual BS of transmittance

into the paths of Alice and Bob’s relevant photons.

As shown in Fig.

, the indistinguishable components of

incoming photon modes are directed toward Charlie’s BS,

where they interfere, whereas distinguishable components

are mixed with vacuum at the BS and do not contribute to

interference. Here,

0) corresponds to the case

when both incoming photons are perfectly indistinguish-

able (distinguishable). Now we may calculate the probabil-

ity of a threefold coincidence detection event

between

(Charlies’ detectors) and

(detects Bob’s signal

photon) for a given qubit state

from Alice and Bob:

{

(

I

−

(

)

⊗

)

⊗

(

I

−

(

)

⊗

)

⊗

(

I

−

(

)

}

,(1)

where the

and

operators refer to modes, which origi-

nate from Alice and Bob’s virtual BSs and are directed to

and

, respectively, and

corresponds to Bob’s idler

mode, which is directed to

(see Fig.

). This allows

the derivation of an expression for the HOM interference

visibility

HOM

(ζ )

[

(

)

−

(ζ )

]

(

)

,(2)

consistent with that introduced in Sec.

III B

. Since Alice

and Bob ideally produce

(

)

⊗

(

TMSV

)

and recognizing that all operators in

are

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FIG. 6.

A schematic depiction of the distingushability between

Alice and Bob’s photons at Charlie’s BS. The distinguishabil-

ity is modeled by means of a virtual BS with a transmittance

Indistinguishable photons contribute to interference at the Char-

lie’s BS while distinguishable photons are mixed with vacuum,

leading to a reduction of HOM visibility and teleportation fidelity

(see the main text for further details).

Gaussian, we analytically derive

(ζ )

−

exp

(

−

(μ

)

(

−

)η

)

−

exp

(

−

)

−

exp

(

−

)

(

−

)η

exp

(

−

(μ

)

(

−

)(

−

)η

]

(

−

)η

)

(

−

)η

(3)

for varied

, where

and

are the transmission effi-

ciencies of the signal and idler photons, including the

detector efficiencies. We similarly calculate the impact of

distinguishability on the teleportation fidelity of

(ζ )

(ζ

max

[

(ζ

max

)

(ζ

min

)

], (4)

where

max

(

min

) is the phase of the MZI added into

the path of the signal photon, corresponding to maximum

(minimum) threefold detection rates.

To compare the model to our measurements, we use the

experimental mean photon numbers for the photon-pair

source

1.2

−

and

4.5

−

, as deter-

mined by the method described in Appendix

. We then

measure the teleportation fidelity of

and HOM inter-

ference visibility (keeping the MZI in the system to ensure

that

remains unchanged) for different values

.The

results are plotted in Fig.

. The data are then fitted to the

expressions

HOM

(ζ )

and

(ζ )

derived in our model and

graphed in Fig.

. The maximum teleportation fidelity and

CQNET/FQNET 2020

(a)

(b)

FIG. 7. The evaluation of the photon indistinguishability using

an analytical model. (a) The quantum teleportation fidelity of

. (b) The HOM interference visibility, each with varied mean

photon number

of Alice’s qubits. Fits of analytical models to

the data reveal

90% indistinguishability between Alice and

Bob’s photons at Charlie’s BS. Bob produces

photon pairs

on average, while

and

are the probabilities for an individual

idler (signal) photon to arrive at Charlie’s BS and be detected at

Bob’s detector, respectively.

HOM visibility are approximately 0.85 and approximately

0.65, respectively, and are maximized around

0.001.

The teleportation fidelity and HOM visibility decrease to

0.5 and 0.0, respectively, when

0 corresponding to a

vacuum input from Alice. The decrease at large values of

is predominantly due to the multiphoton events from

Alice.

The fitted curves are in very good agreement with

our experimental values and consistently yield a value of

90% for both measurement types. This implies that

we have only a small amount of residual distinguisha-

bility between Alice and Bob’s photons. The potential

effects leading to this distinguishability are discussed in

Sec.

Overall, our analytical model is consistent with our

experimental data [

] and our experimental setup allows

high-fidelity teleportation of arbitrary time-bin qubits in

the regime of

1, which is the parameter space most

often used in quantum networking protocols (e.g., key

distribution). Our model thus offers a practical way to

determine any underlying distinguishability in a deployed

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network where a full characterization of the properties of

Alice and Bob’s photons may not be possible.

V. DISCUSSION AND OUTLOOK

We demonstrate quantum teleportation systems for pho-

tonic time-bin qubits at a quantum channel- and device-

compatible wavelength of 1536.5 nm using a fiber-based

setup comprising state-of-the-art SNSPDs and off-the-shelf

components. We measure an average fidelity above 90%

using QST and a decoy-state analysis with up to 44 km

of single-mode fiber in our setup. Our results are fur-

ther supported by an analytical model in conjunction with

measurements of the entanglement and HOM interference

visibilities.

The decoy-state analysis indicates that the maximum

teleportation fidelity is currently restricted by that of

the teleported qubits prepared in superposition states, for

which a 10% distinguishability between the qubits under-

going BSM and the contribution of multiple photon pairs

play the largest role. Our model predicts that the average

fidelity will increase to 95% with completely indistin-

guishable photons, while fidelities close to unity can be

achieved with a lowered mean number of photon pairs.

Alternatively, we may replace our SNSPDs with efficient

photon-number-resolving (PNR) SNSPDs [

] to allow

postselection of multiphoton events at the MZI or BSM

[

]. Both approaches must be accompanied by increased

coupling efficiencies of the pair source beyond the current

approximately 1%, either to maintain realistic teleportation

rates (above the system noise and current rate of phase

drift of the MZI) or to derive any advantage from PNR

capability.

As suggested by the width of our HOM interference

fringe—which predicts an average photon bandwidth of

0.44

/σ

∼

1.5 GHz (see Sec.

III B

), i.e., less than the 2 GHz

bandwidth of our FBGs—the indistinguishability in our

system could be limited by the large difference in the band-

width between the photons originating from the SPDC

(

100 GHz) and those generated at Alice by the IM

(15 GHz), leading to nonidentical filtering by the FBG.

This can be improved by narrower FBGs or by using a

more broadband pump at Alice [e.g., using a mode-locked

laser or a higher-bandwidth IM (say,

50 GHz), which is

commercially available]. Alternatively, pure photon pairs

may be generated by engineered phase matching (see, e.g.,

Ref. [

]). Distinguishability owing to nonlinear modula-

tion during the SHG process could also play a role [

The origin of distinguishability in our system, whether due

to imperfect filtering or other device imperfections (e.g.,

PBS or BS) will be studied in future work. The coupling

loss can be minimized to less than a few decibels overall by

improved fiber-to-chip coupling, lower-loss components

of the FBGs (e.g., the required isolator), spliced fiber con-

nections, and reduced losses within our MZI. Note that our

current coupling efficiency is equivalent to approximately

50 km of single-mode fiber, suggesting that our system is

well suited for quantum networks provided that the loss is

reduced.

While the fidelities that we demonstrate are sufficient

for several applications, the current teleportation rates

with the 44 km length of fiber, which are approxi-

mately in the Hertz range, are still low. Higher repetition

rates (e.g., through the use of high-bandwidth modula-

tors with wide-band wavelength-division-multiplexed fil-

ters and low-jitter SNSPDs [

]), improvements to cou-

pling and detector efficiencies, enhanced BSM efficiency

with fast-recovery SNSPDs [

], or multiplexing in fre-

quency [

] will all yield substantial increases in the

teleportation rate. Note that increased repetition rates per-

mit a reduction in the time-bin separation, which will allow

us to construct the MZI on chip, providing exceptional

phase stability and, hence, achievable fidelity. Impor-

tantly, the aforementioned increases in the repetition rate

and efficiency are afforded by improvements in SNSPD

technology that are currently being pursued with our Jet

Propulsion Laboratory (JPL), NIST, and other academic

partners.

Upcoming system-level improvements that we plan to

investigate and implement include further automation by

the implementation of free-running temporal and polariza-

tion feedback schemes to render the photons indistinguish-

able at the BSM [

]. Furthermore, several electrical

components can be miniaturized, scaled, and made more

cost effective (e.g., field-programmable gate arrays can

replace the AWG). We note that our setup prototype will be

easily extended to independent lasers at different locations,

also with appropriate feedback mechanisms for spectral

overlap [

]. These planned improvements are com-

patible with the data acquisition and control systems that

were built for the systems and experiments at FQNET and

CQNET presented in this work.

Overall, our high-fidelity teleportation systems—

achieving state-of-the-art teleportation fidelities of time-

bin qubits—serve as a blueprint for the construction of

quantum network test beds and eventually global quan-

tum networks toward the quantum Internet. In this work,

we present a complete analytical model of the teleporation

system that includes imperfections and compare it with

our measurements. Our implementation, using approaches

from HEP experimental systems and real-world quantum

networking, features near fully automated data acquisition,

monitoring, and real-time data analysis. In this regard, our

Fermilab and Caltech Quantum Networks serve as R&D

laboratories and prototypes toward real-world quantum

networks. The high fidelities achieved in our experiments

using practical and replicable devices are essential when

expanding a quantum network to many nodes and enable

the realization of more advanced protocols (see, e.g., Refs.

[

]).

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ACKNOWLEDGMENTS

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 AQT IN-

Q-NET research program. R.V., N.L., L.N., C.P., N.S.,

M.S. S.X., and A.M. acknowledge partial support from

the U.S. Department of Energy, Office of Science, High

Energy Physics, QuantISED program grant, under Award

No. DE-SC0019219. A.M. is supported in part by the

JPL President and Director’s Research and Development

Fund (PDRDF). C.P. further acknowledges partial support

from Fermilab’s Lederman Fellowship and Lab Directed

R&D (LDRD). D.O. and N.S. acknowledge partial sup-

port from the Natural Sciences and Research Council of

Canada (NSERC). D.O. further acknowledges the Cana-

dian Foundation for Innovation, Alberta Innovates, and

Alberta Economic Development, Trade and Tourism’s

Major Innovation Fund. J.A. acknowledges support by a

NASA Space Technology Research Fellowship. Part of

the research was carried out at the Jet Propulsion Labo-

ratory, California Institute of Technology, under a contract

with the National Aeronautics and Space Administration

(80NM0018D0004). We thank Jason Trevor (Caltech Lau-

ritsen Laboratory for High Energy Physics), Nigel Lockyer

and Joseph Lykken (Fermilab), Vikas Anant (PhotonSpot),

Aaron Miller (Quantum Opus), Inder Monga and his

ESnet group at Lawrence Berkeley National Laboratory

(LBNL), the groups of Wolfgang Tittel and Christoph

Simon at the University of Calgary, the groups of Nick

Hutzler, Oskar Painter, Andrei Faraon, Manuel Enders,

and Alireza Marandi at Caltech, Marko Loncar’s group at

Harvard, Artur Apresyan and the HL-LHC USCMS-MTD

Fermilab group; Marco Colangelo (MIT); Tian Zhong

(Chicago); and AT&T’s Soren Telfer, Rishi Pravahan, Igal

Elbaz, Andre Feutch, and John Donovan. We acknowl-

edge the enthusiastic support of the Kavli Foundation

on funding quantum information science and technology

(QIS&T) workshops and events and the Brinson Founda-

tion support, especially for students working at FQNET

and CQNET. M.S. is especially grateful to Norm Augus-

tine (Lockheed Martin), Carl Williams (NIST), and Joe

Broz (SRI, QED-C); Hartmut Neven (Google Venice);

Amir Yacoby and Misha Lukin (Harvard); Ned Allen

(Lockheed Martin); Larry James and Ed Chow (JPL);

the Quantum Communication Channels for Fundamental

Physics (QCCFP) wormhole-teleportation team, especially

Daniel Jafferis (Harvard) and Alex Zlokapa (Caltech),

Mark Kasevich (Stanford), Ronald Walsworth (Maryland),

Jun Yeh and Sae Woo Nam (NIST); Irfan Siddiqi (Berke-

ley); Prem Kumar (Northwestern), Saikat Guha (Arizona),

Paul Kwiat (UIUC), Mark Saffman (Wisconsin), Jelena

Vuckovic (Stanford) Jack Hidary (former googleX), and

the quantum networking teams at Oak Ridge National Lab-

oratory (ORNL), Argonne National Laboratory (ANL),

and Brookhaven National Laboratory (BNL), for produc-

tive discussions and interactions on quantum networks and

communications.

APPENDIX A: DETAILED DESCRIPTION OF

EXPERIMENTAL COMPONENTS

1. Control systems and data acquisition

Our system is built with a vision toward future repli-

cability, with a particular emphasis on systems integra-

tion. Each of the Alice, Bob, and Charlie subsystems is

equipped with monitoring and active-feedback stabiliza-

tion systems (e.g., for the IM extinction ratio) or has

a capability for remote control of the critical network

parameters (e.g., varying the qubit generation time). Each

subsystem has a central classical processing unit with the

following functions: oversight of automated functions and

workflows within the subsystem, data acquisition and man-

agement, and handling of input and output synchronization

streams. As the quantum information is encoded in the

time domain, the correct operation of the classical pro-

cessing unit depends critically on the recorded time of

arrival of the photons at the SNSPDs. Thus a significant

effort is dedicated to building a robust DAQ subsystem

capable of recording and processing the large volumes

of time-tagged signals generated by the SNSPDs and

recorded by our TDCs at a high rate. The DAQ is designed

to enable both real-time data analysis for prompt data-

quality monitoring as well as postprocessing data analysis

that allows us to achieve the best understanding of the

data.

The DAQ system is built on top of the stand-alone

Linux library of our commercial TDC. It records time tags

whenever a signal is detected in any channel in coinci-

dence with the reference 90-MHz clock. The time tags

are streamed to a personal computer, where they are pro-

cessed in real-time and stored to disk for future analysis. A

graphical user interface is developed, capable of real-time

visualization and monitoring of photons detected while

executing teleportation. It also allows for easy control of

the time intervals used for each channel and configuration

of the relevant coincidences between the different pho-

ton detection events across all TDC channels. We expect

our DAQ subsystem to serve as the foundation for future

real-world time-bin quantum networking experiments (see

Sec.

2. Superconducting nanowire single-photon detectors

We employ amorphous tungsten silicide SNSPDs man-

ufactured in the JPL Microdevices Laboratory for all mea-

surements at the single-photon level (see Sec.

II B

)[

The entire detection system is customized for optimum

autonomous operation in a quantum network. The SNSPDs

are operated at 0.8 K in a closed-cycle sorption fridge [

The detectors have nanowire widths between 140 and 160

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