Infrastructure for the Quantum Internet

Seth Lloyd,∗

Jeffrey H. Shapiro,†

and Franco N. C. WongMassachusetts Institute of Technology

Research Laboratory of Electronics77 Massachusetts Avenue

Cambridge, MA 02139

{slloyd@, jhs@, franco@ncw2}.mit.edu

Prem Kumar, Selim M. Shahriar,and Horace P. YuenNorthwestern University

Department of Electrical andComputer Engineering

Center for Photonic Communicationand Computing

2145 North Sheridan RoadEvanston, IL 60208

{kumarp, shahriar, yuen}@ece.northwestern.edu

ABSTRACTA team of researchers from the Massachusetts Institute ofTechnology (MIT) and Northwestern University (NU) is de-veloping a system for long-distance, high-fidelity qubit tele-portation. Such a system will be required if future quantumcomputers are to be linked together into a quantum Inter-net. This paper presents recent progress that the MIT/NUteam has made, beginning with a review of the teleportationarchitecture and its loss-limited performance analysis.

Categories and Subject DescriptorsB.4.3 [Input/Output and Data Communications]: In-terconnection (Subsystems); C.2.1 [Computer Commu-nication Networks]: Network Architecture and Design;C.2.5 [Computer Communication Networks]: Localand Wide-Area Networks

KeywordsQuantum communication, teleportation, qubits, entangle-ment, quantum memory

1. INTRODUCTIONThe physical world is, at bottom, quantum mechanical. Al-though not directly visible in ordinary life at the human sizescale, quantum mechanics has already had a huge effect onour technological society. The quantum states of semicon-ductor band structure have—with lots of work to developunderstanding and the technology—given us integrated cir-cuits and semiconductor lasers. These in turn have fueled

∗Also Department of Mechanical Engineering.†Also Department of Electrical Engineering and ComputerScience

the computation and communication revolution that havecreated the Information Age we live in today. But anotherrevolution may be in the offing, one that could bring us intothe Quantum Information Age.

The digital abstraction, on which modern computation andcommunication rests, admits to only two possible states: aclassical on-off system must be in either state 0 or state1, representing a single bit of information. Quantum me-chanics is quite different. A two-level quantum system—thereader unfamiliar with basic quantum mechanics should con-sult the Appendix for a quick review of all that is neededto understand this paper—can be characterized by two or-thogonal basis states (vectors in a Hilbert space) |0〉 and |1〉.The system itself, however, may be in an arbitrary super-position of these two states, α|0〉 + β|1〉, where α and β arecomplex numbers that satisfy the normalization condition|α|2 + |β|2 = 1. Such arbitrary superpositions represent asingle quantum bit of information, i.e., a qubit.

At the micro scale, qubit superposition is innocuous. Forexample, suppose that the quantum system in question isthe polarization of a single photon. Then |0〉 and |1〉 mightbe horizontal and vertical polarization states, respectively.Their equal superpositions, (|0〉 ± |1〉)/

√2, then represent

±45◦ linear-polarization states. At the macro scale, on theother hand, things are not so palatable. Take the Schodinger’scat paradox, in which |0〉 is a live cat and |1〉 is a dead cat.What possible reality can we ascribe to (|0〉 ± |1〉)/

√2, viz.,

to equal superpositions of life and death?

As if the quantum weirdness of one qubit were not enough,more strange behavior emerges when we move on to two (ormore) qubits. The classical behavior of two correlated butrandom bits is easy to understand, and can be translatedinto a corresponding qubit situation. But two qubits may beentangled, a thoroughly non-classical behavior in which theyexist in a superposition of two distinct two-qubit states. En-tanglement leads to non-local correlations—spooky action ata distance seemingly in violation of causality—that led Ein-stein, Podolsky, and Rosen to suggest quantum mechanicsmight not be complete [1]. Yet, in every experiment in which

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 20049

superposition and entanglement have been put to the test,quantum mechanics always proves out, which brings us, atlast, to quantum computation and quantum communication.

Superposition and entanglement lead to algorithms—for quan-tum computation—whose performance outstrips the bestthat can be possibly done with a classical computationalalgorithm. This advantageous behavior derives from quan-tum parallelism, whereby superposition allows many com-putation paths to be explored simultaneously. For example,Grover’s algorithm [2], for querying an unsorted database

of size N on a quantum computer, affords a√N speedup

when compared to its best classical computation competi-tor. Even more impressive is Shor’s factoring algorithm [3],which provides an exponential speedup in comparison withthe best known classical approach to factoring. Moreover,just as the Internet opened the door to a vast array of net-worked applications of classical computing, so too might anetwork of quantum computers lead to a host of new quan-tum applications, see e.g., the recent suggestion [4] that thePublic Goods Game has a quantum network solution whichdoes not require a trusted third party.

Building a large-scale quantum computer is, at present, anextraordinary and as yet unmet challenge. However, evenwere many such computers available, linking them togetheras a quantum network cannot be done with standard fiber-optic Internet infrastructure. The problem is the inscrutable,fragile nature of the qubit. An unknown qubit α|0〉 + β|1〉cannot be measured perfectly, i.e., there is no measurementthat can divine α and β for an arbitrary qubit. An unknownqubit cannot be cloned [5], i.e., we cannot make multiplecopies of an arbitrary α|0〉 + β|1〉 from a single exemplar.Qubits are also physically fragile, i.e., the state of a two-level quantum system is degraded by interaction with its sur-rounding environment, such as when our qubit-carrying pho-ton is absorbed or scattered. Collectively, these difficultiesplace a seemingly impossible hurdle in the path of quantumcomputer networks. Sharing a quantum algorithm across anetwork requires that qubits associated with intermediatecalculations be shared across that network. This is so be-cause measuring these intermediate-stage qubits will destroythe superpositions and entanglement that give quantum al-gorithms their speedups. Furthermore, fragility precludesreliable direct long-distance transmission of intermediate-stage qubits. Nevertheless, there is a way out of this bind.If the remote quantum computers can entangle a pair ofqubits—one at each site—then with local operations andclassical communication they can transfer an arbitrary qubitfrom one to the other by some more quantum weirdness:qubit teleportation [6].

In qubit teleportation the sender (call her Alice) makes aspecial joint measurement on the qubit to be teleported,α|0〉 + β|1〉, together with her qubit from the entangledpair she has shared with the receiver (call him Bob). Al-ice’s measurement yields two bits of classical information,namely a message that is equally likely to be any elementof {00, 01, 10, 11}, but no information that she can employto deduce α|0〉 + β|1〉. Indeed, after the measurement, thetwo qubits in Alice’s possession will be in a state that istotally independent of the values of α and β. But, entan-glement is a very special property. The classical informa-

tion obtained by Alice from her joint measurement is exactly

what is needed by Bob to transform his qubit from the orig-inal entangled pair into the state α|0〉 + β|1〉. Amazingly,the four transformations that Bob must employ—dependingon which classical message {00, 01, 10, 11} he receives fromAlice—are completely independent of α and β. Thus, Bobtoo has no information about the qubit that has been tele-ported.

In summary, qubit teleportation does not constitute a per-fect measurement of an unknown qubit, as neither Alice norBob have learned anything about the unknown qubit. Nordoes teleportation violate the no-cloning theorem, becauseBob cannot create his replica until after Alice’s measure-ment has destroyed the original. Moreover, qubit fragilitycan be tolerated, because environmental effects only limitthe speed with which Alice and Bob can build up a reser-voir of entangled qubits for use in teleportation, and preciousintermediate-stage computation qubits α|0〉 + β|1〉 are onlymanipulated within the confines of the quantum communi-cation element inside Alice’s quantum computer. Finally,qubit teleportation does not contradict causality, becauseclassical, light-speed limited, communication from Alice toBob is intrinsic to the protocol.

In what follows we will describe the approach to long-distancequbit teleportation being pursued by a team of researchersfrom the Massachusetts Institute of Technology (MIT) andNorthwestern University (NU), and review the recent progressthat this team has made. Successful completion of thisprogram would provide the basic infrastructure on whicha Quantum Internet might be built.

2. TELEPORTATION ARCHITECTUREAs a prelude to our discussion of the MIT/NU architecture,let us quantify the Bennett et al. protocol for qubit telepor-tation [6], whose qualitative structure was spelled out in theIntroduction. In advance of undertaking the teleportationoperation, Alice and Bob arrange to share a singlet state oftwo qubits,

|ψ−〉AB = (|01〉AB − |10〉AB)/√

2, (1)

where

|01〉AB ≡ |0〉A ⊗ |1〉B , (2)

etc. (We shall see, below, how this sharing can be accom-plished with an entangled photon source and trapped-atomquantum memories.) Charlie then supplies Alice with anunknown qubit

|Ψ〉C ≡ α|0〉C + β|1〉C , (3)

asking her to teleport that state to Bob. Alice proceeds asfollows. First, she measures the Bell-state observable,

B ≡4X

n=0

n|Bn〉ACAC〈Bn|, (4)

on the tensor-product Hilbert space HA ⊗HC , where

|B0〉AC ≡ |10〉AC − |01〉AC√2

, |B1〉AC ≡ |10〉AC + |01〉AC√2

,

|B2〉AC ≡ |11〉AC − |00〉AC√2

, |B3〉AC ≡ |11〉AC + |00〉AC√2

,

(5)

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200410

are the Bell states. Next, she sends Bob the result of hermeasurement—one of the integers {0, 1, 2, 3}—as two bits ofclassical information. This message contains all the informa-tion that Bob needs to complete the teleportation process.To see that this is true, we use a little algebra to show thatthe Alice, Bob, Charlie state prior to Alice’s Bell-observablemeasurement is

|ψ〉ABC = |ψ−〉AB ⊗ |Ψ〉C

=α|0〉B + β|1〉B

2⊗ |B0〉AC

− α|0〉B − β|1〉B2

⊗ |B1〉AC

− α|1〉B + β|0〉B2

⊗ |B2〉AC

+α|1〉B − β|0〉B

2⊗ |B3〉AC . (6)

So, if Alice’s Bell-observable outcome is 0, then Bob’s stateis

|Ψ〉B = α|0〉B + β|1〉B , (7)

and nothing needs to be done to complete the teleportation.If her Bell-observable outcome is 1, then Bob completes theteleportation process with a phase-flip operation, i.e., chang-ing the sign of the |1〉B component of his qubit relative toits |0〉B component. If Alice’s outcome is 2, then Bob mustperform the bit-flip operation, swapping the |0〉B and |1〉Bcomponents of his qubit. Finally, if Alice’s outcome is 3,Bob must use both the phase-flip and bit-flip operations.Bob need not know anything about α and β to completethe teleportation operation once he is in receipt of Alice’smessage. Likewise, the Alice-Charlie state after she makesthe Bell-observable measurement is one of the {|Bn〉}, allof which possibilities have pre-measurement probabilities of1/4, so that Alice gleans no information about α and β fromher observation.

2.1 Architectural ElementsAn initial demonstration of the Bennett et al. qubit telepor-tation protocol was reported by Bouwmeester et al. [7],[8],but this work was a table-top experiment in which only oneof the Bell states was measured. As a result, only whenphotodetection clicks heralded the occurrence of that Bellstate—a probability 1/4 event—could the teleportation becompleted. Furthermore, this experiment did not includeany quantum memory element, so that the teleported statecould not be saved for later use. Indeed its presence wasdemonstrated by an appropriate, but annihilative, measure-ment procedure.

The Quantum Internet must connect distant quantum com-puters via teleportation with complete Bell-observable mea-surements, so as not to sacrifice precious intermediate-stagequbits. Also, it needs quantum memory, for storing a reser-voir of entangled qubits and to serve as input and outputregisters for the teleportation itself. The MIT/NU architec-ture addresses all of these requirements. Nonlinear opticsare used to produce a stream of polarization-entangled pho-ton pairs, i.e., signal and idler photon states of the singletform

|ψ−〉SI ≡ (|HV 〉SI − |V H〉SI)/√

2. (8)

The signal and idler photons are routed down L-km-longlengths of standard telecommunication optical fiber—thesignal photons proceeding down one fiber and the idler pho-tons down the other—to a pair of quantum memories com-prised of 87Rb atoms that are trapped inside high-Q opticalcavities [9]. One of these memories belongs to Alice and theother to Bob; the Bell-observable measurements are made inAlice’s memory, and the teleportation-completing transfor-mations are performed in Bob’s memory. Without delvingtoo much into details, see [10]–[13] for more information, theoverall operation can be described in terms of architecturalelements shown in Figs. 1–3.

Figure 1(a) shows a simplified energy diagram for the atomiclevels in 87Rb that are used in storing the signal and idlerqubits from |ψ−〉SI in Alice’s memory and Bob’s memory, re-spectively. A photon of arbitrary polarization—expressed asa mixture of left-circular and right-circular components (σ−

and σ+, respectively) can be absorbed by a rubidium atomthat is in the ground state A, transferring its coherence—the α and β values that together characterize its polarizationstate—to the energy-degenerate excited levels B. By coher-ently pumping the B-to-D transition with an appropriatelaser field, this quantum coherence is transferred to long-lived D levels for storage and processing. Whether or not aphoton has been captured in this manner can be verified—ina non-destructive manner—by subsequently pumping the A-to-C cycling transition with another laser field. If the atomhas been transferred to the D states, then no fluorescencewill be seen on this cycling transition. Thus, Alice and Bobrun a time-slotted memory loading protocol in which theyrepeatedly try to absorb photons, starting over at the endof each trial in which one or both of them see cycling transi-tion fluorescence [12]. By employing an appropriate latticeof such trapped atoms, Alice and Bob can sequentially ac-cumulate a reservoir of shared entanglement for teleportingquantum states between their respective quantum comput-ers.

Figure 1(b) sketches the structure of the entangled photonsource in the MIT/NU architecture. It consists of two op-tical parametric amplifiers (OPAs), viz., resonant optical

cavities each containing a second-order (χ(2)) nonlinear crys-tal in which photon pairs are produced whenever a photonfrom a strong pump laser of frequency ωP fissions into asignal photon at frequency ωS and an idler photon at fre-quency ωI . Energy conservation, at the photon level, re-quires that ωS + ωI = ωP . Momentum conservation, atthe photon level, requires that the wave vectors associated

with the pump, signal, and idler obey ~kS + ~kI = ~kP . Weassume type-II phase matching, in Fig. 1(b), which forcesthe signal and idler photons to be orthogonally polarized,as indicated by the bullets and arrows. With proper choiceof nonlinear material, each OPA can be made to operateat frequency degeneracy, i.e., the center frequencies of thesignal and idler will both be ωP /2. Making ωP /2 a cav-ity resonance for both the signal and the idler polarizationsthen dramatically increases the resulting signal-idler pho-ton flux within the narrow (∼15MHz) bandwidth of the87Rb atomic line. Finally, by combining the outputs fromtwo anti-phased, coherently-pumped OPAs on a polarizingbeam splitter (PBS), we obtain the stream of polarization-entangled (singlet-state) photon pairs that are needed.

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200411

C

B

A

D

OPA 1

OP

A 2

S1S1

S2

S2I1

I1

I 2

I 2

PBS

σ– σ+

(a)

(b)

Figure 1: Essential components of the MIT/NUquantum communication architecture: (a) simpli-fied energy-level diagram for the trapped rubidiumatom; (b) source of polarization-entangled photonpairs.

The 87Rb-atom quantum memory has its A-to-B absorp-tion line at 795 nm, but low-loss fiber propagation occursin the 1.5-µm-wavelength window. Furthermore, standardtelecommunication fiber does not preserve the polarizationstate of the light propagating through it. These obstaclesto long-distance distribution of polarization-entangled pho-tons to the Rb-atom memories are accounted for, within theMIT/NU architecture, by quantum-state frequency conver-sion [14] and time-division-multiplexed (TDM) polarizationrestoration (cf. [15]), as shown in Figs. 2 and 3. In particu-

χχ(2)

795 nm OUTPUT1570 nm PUMP

1608 nm INPUT

Figure 2: Schematic diagram of quantum-state fre-quency conversion.

lar, by applying a strong pump beam at 1570 nm to anothersecond-order nonlinear crystal—chosen to satisfy the appro-priate phase-matching condition—we can convert a qubitphoton received at 1608 nm (in the low-loss fiber transmis-sion window) to a qubit photon at the 795 nm wavelength ofthe 87Rb quantum memory. For polarization restoration wepostpone the PBS combining, shown in Fig. 1(b), until af-ter fiber propagation. This is accomplished by transmitting

time slices from the signal beams from our two OPAs downone fiber in the same linear polarization but in nonoverlap-ping time slots, accompanied by a strong out-of-band laserpulse. By tracking and restoring the linear polarization ofthe strong pulse, we can restore the linear polarization of thesignal-beam time slices at the far end of the fiber. After thislinear-polarization restoration, we then reassemble a time-epoch of the full vector signal beam by delaying the first timeslot and combining it on a polarizing beam splitter with thesecond time slot after the latter has had its linear polar-ization rotated by 90◦. A similar procedure is performedto reassemble idler time-slices after they have propagateddown the other fiber.

S2 S1 λP

λP

λP

WDM

MUX

WDM

DEMUX

S2 S2

S1

S1S2

HWP

2 x 2

switch

S2 S2S1 S1

S2 S2

S1

HWP2 x 2

switch

pol. & timing

measurements

pol.controller

Figure 3: Schematic diagram of time-division-multiplexed polarization restoration for the signalbeam. HWP = half-wave plate, WDM MUX =wavelength division multiplexer, WDM DEMUX =wavelength division demultiplexer.

Once Alice and Bob have entangled the atoms within theirrespective memories by absorbing an entangled pair of pho-tons, the rest of the qubit teleportation protocol is accom-plished as follows. Charlie’s qubit message is stored in an-other 87Rb atom, which is trapped in the same optical cavityas Alice’s memory atom. By a coherence transfer procedure[16], Charlie inserts his qubit into Alice’s memory atom ina manner that permits the Bell-observable measurement tobe accomplished by determining in which of four possiblestates—not shown in Fig. 1(a)—that memory atom now re-sides. Alice sends the result of her Bell-observable measure-ment to Bob, who completes the teleportation process bystandard atomic level manipulations that realize the phase-flip and bit-flip qubit operations. For details, see [11].

2.2 Basic Performance AnalysisTwo key figures of merit can be used to characterize the per-formance of the MIT/NU qubit teleportation architecture:throughput and fidelity. The former is the number of singlet-states per second that can be accumulated by sequential op-eration of the quantum-memory loading protocol applied toa lattice of trapped atoms. The latter is the accuracy withwhich an unknown qubit is replicated by the teleportationprocess. Figure 4 comprises a throughput/fidelity assess-ment under rather idealized conditions [12], in which the po-larization restoration, Alice’s Bell-observable measurement,

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200412

and Bob’s phase-flip and bit-flip operations are flawless. In-cluded in the calculations underlying Fig. 4 is the prop-agation loss in the fibers, as well as fixed losses associ-ated with incomplete quantum-state frequency conversion,etc., in each source-to-memory path and the fact that OPAsources can produce multiple photon pairs within a giventrial of the memory loading protocol. Fidelity is severelydegraded when Alice’s memory and Bob’s memory absorbphotons from different—hence not entangled—photon pairs.The probability of such multiple-pair events is reduced byreducing the pumping level on the OPAs, but this comes atthe cost of reducing the throughput, as the rate of single-pair events is also decreased. Figures 4(a) and (b) plot the

10

100

1000

10000

0 20 40 60 80 100

Thro

ughput (s

ingle

ts/s

ec)

End-to-end path length, 2L (km)

0.75

0.8

0.85

0.9

0.95

1

0 20 40 60 80 100

Tele

port

ation fid

elit

y

End-to-end path length, 2L (km)

(a)

(b)

Figure 4: Throughput (singlets/sec) (a) and tele-portation fidelity (b) versus end-to-end path lengthfor the MIT/NU architecture. Assumptions and pa-rameter values for these plots are given in the text.

throughput and fidelity, respectively, versus end-to-end pathlength 2L. These curves assume that: the loading proto-col is run at 500 kHz; the OPAs are pumped at 1% of os-cillation threshold; the OPA cavity linewidth is twice thememory cavity linewidth; the fiber loss is 0.2 dB/km; andthere is 5 dB of fixed loss in each source-to-memory path.We see that in this ideal, loss-limited performance regimethe MIT/NU architecture can sustain throughputs in excessof 100 pairs/sec out to 50 km end-to-end path lengths withteleportation fidelities above 97%.

2.3 Sensitivity AnalysisAlthough the idealized throughput/fidelity behavior of theMIT/NU architecture is quite promising, it is important toassess how sensitive this performance is to imperfections inthe system. Toward that end, we have developed error mod-els [13] that account for a variety of possible degradationmechanisms. Some of the results that we have obtained

from these models are shown in Figs. 5 and 6. In particular,Fig. 5 shows that modest errors in polarization restorationdo not appreciably degrade the throughput or the fidelityof the MIT/NU architecture. Here cos(θS) and cos(θI) aretransmission factors projecting the imperfectly polarizationrestored signal and idler photons onto the reference polariza-tion in Fig. 3. Interestingly, polarization-restoration errorshave virtually no effect on fidelity, although they must bekept well below 1 rad to minimize throughput loss.

Figure 6(a) addresses the impact of improperly phased pumpbeams—∆ψ = 0 is the desired anti-phased situation thatproduces singlet states—in the dual-OPA source shown inFig. 1(b). At ∆ψ = π the pumps are in phase, thus produc-ing a stream of triplet states,

|ψ+〉SI ≡ (|HV 〉SI + |V H〉SI)/√

2, (9)

from the dual-OPA. Alice and Bob, however, are perform-ing teleportation under the assumption that they are shar-ing a singlet state; the phase flip that changes the singlet|ψ−〉SI into |ψ+〉SI is therefore not accounted for and theresult is a disastrous loss of fidelity. However, keeping thepump-phase error well below 1 rad—something that is easilyaccomplished experimentally—will ensure that near-peak fi-delity is maintained. Figure 6(b) shows the fidelity loss thatarises from there being Gaussian-distributed pump gain fluc-tuations in the dual-OPA source. These fluctuations aretaken to have variance σ2

G about a mean gain G = 0.1, cor-responding to operation at 1% of oscillation threshold. Wesee that realistic 1% pump-power fluctuations, correspond-ing to σ2

G = 10−4, lead to minimal performance loss.

3. PROGRESS ON ENTANGLEMENTGENERATION AND TRANSMISSION

3.1 χ(2) SourcesThe dual-OPA entanglement source of Fig. 1(b) can be eas-

ily implemented using a single χ(2) nonlinear optical crystalwith bidirectional pumping. The dual-pump single-crystalconfiguration ensures that the two counter-propagating lightbeams generated from the two coherently driven OPAs aretruly identical, thus allowing the outputs to be combined in-terferometrically at the polarizing beam splitter in Fig. 1(b).We have recently demonstrated this dual-pump system usinga periodically-poled potassium titanyl phosphate (PPKTP)crystal for generating polarization-entangled photons witha flux of over 10 000 detected pairs/s/mW of pump powerwith a quantum-interference visibility of over 90% [17]. ThePPKTP source has signal and idler outputs at ∼795 nm forloading local Rb-based quantum memories and a spectralbandwidth of 500 GHz.

The PPKTP source, however, is not suitable for our long-distance teleportation architecture because fiber-optic trans-mission at 795 nm would be lossy (more than 5 dB/km com-pared with 0.2 dB/km at 1550 nm) and the source is toobroadband for an atom-based quantum memory whose band-width is tens of MHz. To minimize the transmission lossin optical fibers and to increase the spectral brightness ofthe source, we have developed an even more efficient entan-glement source—using a periodically poled lithium niobate(PPLN) crystal—with outputs at 795 nm and 1608 nm [18],shown schematically in Fig. 7. With two vastly different out-

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200413

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Figure 5: Throughput (singlets/sec) (a) and tele-portation fidelity (b) versus polarization-restorationerror angles. These plots assume a 50 km end-to-endpath length; other assumptions and parameter val-ues for these plots are as given for Fig. 4.

put wavelengths, the bidirectionally pumped PPLN source issuitable for loading a local quantum memory at 795 nm anda remote quantum memory accessed via fiber-optic trans-port of the 1608 nm photons and quantum-state frequencyconversion. Fiber-optic delivery offers the additional ad-vantage of confining photons to a well-defined spatial modethat can be matched to what is required for the quantummemories.

The output wavelengths of the PPLN entanglement sourcemay be selected, by choice of the crystal’s grating period,and tuned, by changing its temperature. With high effi-ciency (107 pairs/s) and wavelength tunability, this source[19] is compatible with quantum memories using differentatomic species (each with a different wavelength). The PPLNoutput state can be set to one of the four Bell states bycontrolling the relative phase of the various output pathlengths, accomplished by translating the position of oneof the mirrors in the beam paths. Figure 7(b) shows themaximum and minimum coincidence rates after the lightbeams pass through polarization analyzers at different mir-ror sweep positions (as a function of time). It shows thatit is possible to work at any desired polarization-entangledoutput state, such as the singlet state. Single spatial modefiber delivery of the photons has another intrinsic advan-tage: the amount of light being collected by the ∼10-µm-diameter optical fiber acts as a spectral filter. This occursbecause emission angle is correlated with emission frequency

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Figure 6: Teleportation fidelity versus pump phaseerror (a) and pump-gain variance (b). These plotsassume a 50 km end-to-end path length; other as-sumptions and parameter values for these plots areas given for Fig. 4.

for the PPLN source. We have thus observed a much smaller(∼50GHz) bandwidth for the fiber-coupled PPLN source ata spectral brightness of 320 pairs/s/GHz/mW of pump. Byusing a 100-mW pump and a commercially available first-order grating PPLN crystal (instead of the third-order grat-ing in the present setup) the source can be scaled to yielda polarization-entangled pair generation rate of ∼16 000/sin a 50 MHz bandwidth. Together with quantum-state fre-quency conversion, such a source is suitable for loading re-mote quantum memories.

3.2 χ(3) SourcesDespite the success reported in the previous section, it is stillthe case that efficient coupling into optical fiber of the entan-gled photons produced in bulk nonlinear crystals poses a sig-nificant challenge [21]. Therefore, a source emitting entan-gled photons directly in the fiber is desirable, preferably inthe low-loss transmission band of standard silica fibers near1550 nm wavelength. We have recently developed such asource by exploiting the χ(3) (Kerr) nonlinearity of the fiberitself [22]. When a pump pulse is injected into a fiber at awavelength close to the zero-dispersion point, inelastic four-photon scattering (FPS) is significantly enhanced. In thisprocess, two pump photons at frequency ωp scatter throughthe Kerr nonlinearity of the fiber to create energy-time en-tangled Stokes (signal) and anti-Stokes (idler) photons atfrequencies ωs and ωa, respectively, such that 2ωp = ωs+ωa.

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200414

(a)

PPLN

OPA1outputs

OPA2outputs

Pump532 nm

signal: ~800 nmidler: ~1600 nm

(b)

-0.5

1

2.5

4

0 4 8

Sweep time (s) (piezo scanning)

Coin

ciden

cera

te(k

Hz)

Singlet Singlet

Triplet

Figure 7: (a): Schematic diagram of dual-pumpednondegenerate PPLN downconverter. Signal andidler from each output beam are combined by polar-izing beam splitters after the polarization of one ofthe two beams undergoes 90◦ rotation. (b): Coinci-dence rate versus timed sweep of phase-controllingmirror showing different entanglement states thatcan be achieved. Polarizers at the detectors were setorthogonal and accidental counts were subtracted.

The isotropic nature of the Kerr nonlinearity in fused-silica-glass fiber makes these correlated, scattered photons pre-dominantly co-polarized with the pump. Thus, by coher-ent combination of outputs from two orthogonally-polarizedparametric processes we have produced polarization entan-glement [23]. Indeed, all four Bell states were prepared, andBell’s inequality violations of up to 10 standard deviations ofmeasurement uncertainty were demonstrated [23]. In earlyexperiments with this source, the total number of coinci-dence counts between the Stokes and anti-Stokes photonsexceeded the number of accidental coincidences by only afactor of 2.5 [22]. We have recently shown that spontaneousRaman scattering accompanying FPS causes this problem.By reducing the detuning between the Stokes and pumpphotons, we have been able to show (see Fig. 8) that theaccidental coincidences can reduced to less than 10% of thetrue coincidences at a production rate of about 0.04 photon-pairs/pulse [24]. Hence, this fiber source should yield a two-photon quantum interference visibility in excess of 85% with-

out subtracting the accidental coincidences.

To demonstrate the utility of our fiber-based source of po-larization entangled photons [23] for long-distance entangle-ment distribution, we separated the Stokes and anti-Stokesphotons—which are entangled in polarization but have dif-ferent wavelengths—by use of an optical filter and launched

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gle

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x10

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(a)

Figure 8: Measured coincidence rate versus numberof scattered photons per pump pulse (labelled SingleCounts/Pulse) in the anti-Stokes channel for scat-tered photons co-polarized with the pump. The di-amonds represent the total-coincidence counts pro-duced by a single pump pulse and the triangles rep-resent the coincidence counts produced by two ad-jacent pump pulses. The latter measures the acci-dental coincidences contributing to the total and iswell fitted by the line, which is a plot of the productof the photon counts in the Stokes and anti-Stokeschannels produced by the adjacent, i.e., indepen-dent, pump pulses. The inset shows the number ofscattered photons per pump pulse detected in theanti-Stokes channel as a function of the number ofphotons in the pump pulse (hollow circles). Tak-ing into account the detection efficiency of 6% inthe anti-Stokes channel, at a photon-pair productionrate of 0.04 (0.067) per pulse the ratio between thetotal coincidence rate and the accidental coincidencerate is 13:1 (7.5:1). See [24] for further details.

them into separate 25-km-long spools of commercially avail-able single-mode fibers (Corning SMF-28 and Corning LEAF),as shown schematically in Fig. 9(a). The propagation lossthrough each spool of fiber was measured to be approxi-mately 0.2 dB/km. Fiber polarization controllers were splicedinto the photon propagation path at the end of each spooland test pulses of known polarization states were used toalign the polarization axes (horizontal and vertical) at theinput and output ends of the two fibers. A polarizer wasused at the output end of each fiber to project the polar-ization state of the emerging photon to 45◦ relative to thevertical.

After 25 km of propagation in separate spools of fiber, theemerging Stokes and anti-Stokes photons were detected incoincidence to measure quantum interference as a functionof the relative phase φp between the two pump pulses thatcreated the polarization entanglement. (Appropriate delaysin the photon-counting electronics were introduced to ac-count for the propagation in the two fibers.) The resultsare shown in Fig. 9(b): no interference is observed in thesingle counts, whereas high-visibility (86%) interference isobserved in the coincidence counts. These results clearlyshow that high-fidelity polarization entanglement can sur-

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200415

(a)

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25Km Fiber Spool25Km Fiber Spool

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Figure 9: (a): Schematic of the experimental setupto demonstrate long-distance entanglement distri-bution. FPC, fiber polarization controller; FPBS,fiber-pigtailed polarization beam splitter; APD,avalanche-photodiode based photon-counting detec-tor. (b): Single counts (right ordinate) and co-incidence counts (left ordinate) registered by thedetectors of Stokes and anti-Stokes photons versusthe relative phase between the source’s two pumppulses. The contribution of accidental coincidenceshas been subtracted.

vive 50 km end-to-end fiber propagation [25].

4. QUANTUM-STATE FREQUENCYCONVERSION

The MIT-NU architecture requires that fiber-delivered pho-tons at 1.55–1.6 µm wavelength be upconverted to the 795 nmwavelength of the Rb-atom memory with high efficiency andin a manner that preserves their polarization state. Figure 3shows a time-division-multiplexed scheme for transmittingan arbitrary polarization state down an optical fiber withhigh fidelity. Even though the scheme has yet to be demon-strated for a single-photon input, the required componentsand technologies, such as fast switches and wavelength divi-sion multiplexing and demultiplexing, are readily availablein the telecommunication industry. If the frequency transla-tion step is performed just before the outputs are reassem-bled, the polarizations of the two components (S1 and S2

in Fig. 3) are known and the same, and hence only a singlestage of frequency upconversion is needed.

We have recently demonstrated 90%-efficient single-photonupconversion from 1.55µm to 631 nm by sum-frequency mix-ing [20]; Fig. 10(a) shows the basic scheme. A strong pumplaser at 1064 nm is resonated in a ring cavity to sustain a

high circulating pump power within the cavity, reaching amaximum of 22W. The intracavity pump mixes with aninput photon at 1.55µm in a PPLN crystal to generate aphoton at 631 nm. Our 90% efficiency is limited by inade-quate pump power (∼35 W is required for complete upcon-version), but the results clearly demonstrate the viabilityof our upconversion scheme. Figure 10(b) plots the upcon-version efficiency versus circulating pump power. We haveobserved a significant amount of background photocounts,due to pump-induced fluorescence which is upconverted intothe same spectral and spatial mode as the signal photons.While these background counts do not significantly affectthe teleportation architecture, they can be eliminated byusing a longer-wavelength pump laser.

0 5 10 15 20 250%

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Single-photonupconversionef ficiency

(a)

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631-nm outputPPLN

µ

Figure 10: Frequency upconversion: (a) schematicof sum-frequency mixing upconversion with a ringcavity configuration for the continuous-wave (cw)pump; (b) single-photon upconversion efficiency.

5. TRAPPED ATOM QUANTUM MEMORYAn essential ingredient for the MIT/NU architecture is aquantum memory unit (QMU) embodied by a single 87Rbatom trapped inside a high-finesse optical cavity: it entailsa strong coupling between a single atom and a single photonwith a very small mode volume. Figure 11(a) illustrates thebasic concept behind our single-atom QMU. The atom is tobe caught in a three-dimensional dipole force trap, conven-tionally known as a FORT (far off-resonance trap). Uponcapture in this trap, the atom is to be further cooled downto the ground state of its center-of-mass motion. The cavityis formed by a pair of curved mirrors, each with a reflectivityas high as 99.998%, with a separation of less than 100µm.In Fig. 11(b), we show one of several custom designed opto-mechanical devices we have developed in order to realize thisgeometry. The piezo-electric elements are needed to lock the

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200416

FORT Beam

Rb AtomCavity Field

(a)

FORT beam input port

Piezo

Cavity beam

output port

Cavity mirror

holder

FORT beam input port

Piezo

Cavity beam

output port

Cavity mirror

holder

(b)

Figure 11: (a) Schematic of the trapped-atom QMU;(b) integrated cavity-FORT for realizing the QMU.

cavity to the optical transition of interest. The whole systemhas to be operated under an ultra-high vacuum, in order toensure that the atom does not collide with residual gas andget ejected from the trap.

To isolate and cool a single atom to the point that it canbe trapped in the FORT requires several steps. First, solidrubidium is heated to produce an atomic vapor and thenpassed through a pair of nozzles to produce an atomic beam.The atoms in this beam are slowed to a few m/s using reso-nant radiation pressure from a diode laser, and then cooledand trapped in a magneto-optic trap (MOT). Because theMOT cannot be co-located with the cavity-FORT unit, wetransfer the atoms from one to the other using the followinglaunch-and-catch approach. We apply a pulse of radiationpressure to create an atomic fountain that is arranged tomake the atoms come to rest at the center of the cavity-FORT unit, at which point the FORT is enabled. A colli-sional blockade, or its equivalent, is then used to ensure thatthe FORT contains only one atom. Finally, a Raman cool-ing process is used to cool the atom to its motional groundstate.

We have built two separate vacuum chambers in order torealize the two QMU’s needed for demonstrating the qubitteleportation process. In each chamber, the atoms are cooledand trapped in a MOT in an identical fashion after whichthey undergo launch-and-catch into the cavity-FORT units.The distance between the MOT and the cavity-FORT isgreater in one chamber than it is in the other, so we haveadded a quadrupolar magnetic-wire guide to the longer-distance QMU to ensure a significant atomic flux into itscavity. Figure 12 shows a typical launch-and-catch pro-cess. Figure 12(a) shows a cloud of atoms after they werelaunched into a FORT. Figure 12(b) shows the line density

Figure 12: Launch-and-catch process using a FORTbeam: (a) launched atomic cloud after the FORTis turned on; (b) atomic density integrated horizon-tally.

(a) (b) (c)

Figure 13: Preliminary performance of thequadrupolar wire guide: (a) atomic cloud withoutguide; (b) atomic cloud with guide; (c) differencebetween the clouds in (a) and (b).

of atoms integrated horizontally, with the spike occurringat the FORT location. As time progresses, most of theatoms fall, except for the ones caught in the FORT. Fig-ure 13 shows the preliminary performance of our quadrupo-lar wire guide. Figure 13(a) shows the atom cloud afterlaunch, looking from the side, without the guide turned on.Figure 13(b) shows the same cloud with the guide turnedon. Figure 13(c) is the difference between these two cases,showing a compression of the atomic density towards thecenter. The guiding shown here is not very pronounced,because the guiding wires were far apart at the measure-ment location. We are currently reconfiguring our imagingsystem to see the atoms at a location where the wires arecloser together. The next step is to install the cavity-FORTintegrated structure inside the vacuum chamber. Once thisis achieved, we still face the challenge of ensuring that onlya single atom is trapped in each FORT, and that it can becooled to the motional ground state.

6. CONCLUSIONWe have reported a complete architecture—and progress to-wards its instantiation—for long-distance, high-fidelity qubitteleportation that, if successfully realized, could provide thequantum communication infrastructure needed for a Quan-tum Internet.

ACKNOWLEDGMENTThis research was supported by the DoD MultidisciplinaryUniversity Research Initiative (MURI) Program under Army

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200417

Research Office Grant DAAD19-00-0177.

APPENDIXHere we shall present a bare bones introduction to the quan-tum mechanics needed to understand qubit teleportation.Because the MIT/NU architecture relies on entangled pho-tons, we also include a similarly minimal treatment for thequantum behavior of a single-mode electromagnetic field.For readers desiring a more complete introduction to thequantum mechanics of qubits, the text by Nielsen and Chuangis recommended [26]; readers seeking more information aboutquantized electromagnetic fields may consult the text byLouisell [27].

Quantum States and Quantum MeasurementsA quantum mechanical system is a physical system governedby the laws of quantum mechanics. Its state is the sumtotal of all the information that can be known about thatsystem. In Dirac notation, the state of a quantum systemis represented by a ket vector |ψ〉 in the Hilbert space Hof possible states for that system. For calculations we needthe bra (adjoint) vector, 〈ψ| associated with the state, sothat we can evaluate the inner (dot) product between twostates |ψ〉 and |ψ′〉 as the “bra-ket” 〈ψ|ψ′〉. In particular,finite-energy states all have unit length, viz., 〈ψ|ψ〉 = 1.

It is convenient to regard |ψ〉 as a column vector and 〈ψ′|as a row vector, in terms of some orthonormal basis for H.For a single qubit this is especially simple, as the underlyingHilbert space is two dimensional. Consider the single photondiscussed in the Introduction. An orthonormal basis forits Hilbert space of polarization states is provided by thehorizontal and vertical polarization states, which here willbe denoted |H〉 and |V 〉, viz., these states satisfy

〈H |H〉 = 〈V |V 〉 = 1 and 〈H |V 〉 = 〈V |H〉 = 0, (10)

and all states in H can be expressed as superpositions—with, in general, complex-number coefficients—of |H〉 and|V 〉. Thus by regarding arbitrary polarization-state kets |ψ〉and |ψ′〉 as column vectors,

|ψ〉 ↔»

αβ

–

and |ψ′〉 ↔»

α′

β′

–

, (11)

and their associated polarization-state bras 〈ψ| and 〈ψ′| asrow vectors,

〈ψ| ↔ˆ

α∗ β∗˜

and 〈ψ′| ↔ˆ

α′∗ β′∗˜

, (12)

where ∗ denotes complex conjugation, standard linear alge-bra yields

〈ψ|ψ′〉 =ˆ

α∗ β∗˜

»

α′

β′

–

= α∗α′ + β∗β′ (13)

for their inner product, whence |α|2 + |β|2 = 1 for a finite-energy state |ψ〉. For more complicated situations—likethe single-mode electromagnetic field discussed below—aninfinite-dimensional Hilbert space may be involved.

A little more linear algebra crops up in the specification ofquantum measurements. Here we shall limit our remarks tothe case of observables [27]—as opposed to the more generalcase of probability operator-valued measures [26]—becausethey suffice for the Bell-state measurements needed in qubit

teleportation. Observables are physically measurable prop-erties of a quantum system. They are represented by Hermi-tian operators on the Hilbert space of states, hence they havereal-valued eigenvalues and orthonormal eigenkets, with thelatter constituting a basis for H. If we measure an observ-able O—with eigenvalues {on} and orthonormal eigenkets{|on〉}—the outcome will be one of the eigenvalues and, as-suming that the eigenvalues are distinct (which is the onlycase we will require), the probability of getting the outcomeon, given that the quantum system was in state |ψ〉 immedi-ately before the measurement, is Pr( on | |ψ〉 ) = |〈on|ψ〉|2.Note that the absolute phase of a state vector is irrelevant,|ψ〉 and eiθ|ψ〉 have the same measurement statistics. Moreimportantly, for non-annihilative measurements, the state ofthe system immediately after on is obtained from measuringO will be the associated eigenket |on〉.

A simple example of an observable measurement is the fol-lowing. A single photon in an arbitrary polarization stateis applied at the input to a polarizing beam splitter (PBS),whose horizontally and vertically polarized outputs are de-tected by ideal photon counters. The difference between thehorizontal and vertical counts is the observable

O ≡ |H〉〈H | − |V 〉〈V |, (14)

i.e., it has eigenvalues {+1,−1} and associated eigenkets{|H〉, |V 〉}. Note that this measurement is annihilative, i.e.,the photon is destroyed in the process of the measurement,so we cannot speak of the photon’s state after the O mea-surement.

The final concept we require from basic quantum mechan-ics is the tensor product structure employed to character-ize multiple quantum systems. Suppose we have a pairof photons—denoted S and I for signal and idler, a ter-minology derived from the sources we use in the MIT/NUarchitecture—whose polarization qubits are to provide theentanglement resource needed for qubit teleportation. UsingHS and HI to denote their respective Hilbert state spaces,we define their joint state space to be the tensor productspace H ≡ HS ⊗ HI . An arbitrary state in this tensor-product space is a superposition of tensor products of thebasis states from the signal and idler state spaces, namely,

|ψ〉 = α|H〉S ⊗ |H〉I + β|H〉S ⊗ |V 〉I+ γ|V 〉S ⊗ |H〉I + δ|V 〉S ⊗ |V 〉I= α|HH〉 + β|HV 〉 + γ|V H〉 + δ|V V 〉, (15)

where the second equality provides a convenient shorthand.The state |ψ〉 is said to be a product state if it factors intothe tensor product of a state |ψ〉S on HS times a state |ψ〉Ion HI ; such will be the case in the preceding equation ifand only if

α = αSαI β = αSβI γ = βSαI δ = βSβI (16)

for some {αS , βS , αI , βI}. The inner product of two productstates |ψ〉 and |ψ′〉 is given by

〈ψ|ψ′〉 = (S〈ψ|ψ′〉S)(I〈ψ|ψ′〉I). (17)

If |ψ〉 ∈ H = HS ⊗HI is not a product state, then it is en-tangled. By linearity, inner products between two entangledstates on H can be built up from (15) using the definitionfor product states.

ACM SIGCOMM Computer Communications Review Volume 34, Number 5: October 200418

Quantization of the Single-ModeElectromagnetic FieldThe throughput and fidelity analyses reported in Sec. II de-pend on a full quantum treatment for the electromagneticfield. Even though we did not include the details of thatdevelopment, it is still of value to supply some of the foun-dational concepts here. A propagating electromagnetic wavemay be decomposed into a collection of modes, e.g., by se-lecting particular values for the frequency, direction of prop-agation, and polarization. Quantization of each mode thenleads, in turn, to a quantum description for the full field.Any particular mode has equations of motion that mimicthose of the harmonic oscillator. Its state space is spannedby orthonormal eigenkets { |n〉 : n = 0, 1, 2, . . . , } that areknown as photon-number states, because |n〉 is a state con-taining exactly n photons. An ideal photon-counting mea-surement made on this single mode field is thus representedby the observable

N =

∞X

n=0

n|n〉〈n|. (18)

Standard optical sources do not produce photon-numberstates. In particular, the single-mode field from a laser canapproximate the coherent state,

|α〉 ≡∞X

n=0

αn

√n!e−|α|2/2|n〉; (19)

so that a photon-counting measurement on this state thenyields the familiar Poisson distribution that classical physicsassociates with shot noise:

Pr(n | |α〉 ) =|α|2n

n!e−|α|2 , for n = 0, 1, 2, . . . (20)

Nonlinear optics in second-order (χ(2)) or third-order (χ(3))media can produce signal and idler modes in the entangledstate

|ψ〉 =∞X

n=0

s

Nn

(N + 1)n+1|n〉S ⊗ |n〉I , (21)

where N is the average photon number in each mode. WhenN ≪ 1, this state reduces to the simpler form,

|ψ〉 ≈r

1

N + 1|00〉 +

s

N

(N + 1)2(|01〉 + |10〉). (22)

Generalizing this state—to two polarizations modes (H andV ) each for the signal and the idler—then provides theentangled-photon pair needed for the MIT/NU architecture,because the trapped-atom quantum memories permit thevacuum state (zero-photon) component |00〉 to be ignored.

7. REFERENCES[1] A. Einstein, B. Podolsky, and N. Rosen, “Can

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[9] For ease of calculation, without appreciable loss ofgenerality, our architectural analysis relies on asymmetric arrangement, wherein the entanglementsource is equidistant from the two trapped-atommemories. In actual implementation the source will beco-located with one of these quantum memories.

[10] J. H. Shapiro and N. C. Wong, “An ultrabrightnarrowband source of polarization-entangled photonpairs,” J. Opt. B: Quantum Semiclass. Opt. 2, L1–L4(2000).

[11] S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R.Hemmer, “Long-distance, unconditional teleportationof atomic states via complete Bell statemeasurements,” Phys. Rev. Lett. 87, 167903 (2001).

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[16] T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller,“Decoherence, continuous observation, and quantumcomputing: a cavity-QED model,” Phys. Rev. Lett. 75,3788–3791 (1995).

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[17] M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C.Wong, and J. H. Shapiro, “Generation of ultrabrighttunable polarization entanglement without spatial,spectral, or temporal constraints,” Phys. Rev. A 69,041801(R) (2004).

[18] E. J. Mason, M. A. Albota, F. Konig, and F. N. C.Wong, “A frequency-nondegenerate entanglementsource using periodically poled lithium niobate,” in J.H. Shapiro and O. Hirota, eds., Proceedings of the Sixth

International Conference on Quantum Communication,

Measurement and Computing (Rinton Press, Princeton2003), pp. 107–110.

[19] E. J. Mason, M. A. Albota, F. Konig, and F. N. C.Wong, “Efficient generation of tunable photon pairs at0.8 and 1.6µm,” Opt. Lett. 27, 2115 (2002).

[20] M. A. Albota and F. N. C. Wong, “Efficientsingle-photon counting at 1.55 µm by means offrequency upconversion,” Opt. Lett. 29, 1449 (2004).

[21] F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli,G. Di Giuseppe, and A. V. Sergienko, “Effectivefiber-coupling of entangled photons for quantumcommunication,” Opt. Commun. 227, 343–348 (2003).

[22] M. Fiorentino, P. L. Voss, J. E. Sharping, and P.Kumar, “All-fiber photon-pair source for quantumcommunication,” IEEE Photon. Technol. Lett. 27,491–493 (2002).

[23] X. Li, P. L. Voss, J. E. Sharping, and P. Kumar,“Optical-fiber source of polarization-entangled photonpairs in the 1550 nm telecom band,” submitted to Phys.Rev. Lett., e-print quant-ph/040219.

[24] X. Li, J. Chen, P. L. Voss, J. E. Sharping, and P.Kumar, “All-fiber photon-pair source for quantumcommunications: Improved generation of correlatedphotons,” Opt. Express 12, 3737–3744 (2004).

[25] X. Li, P. L. Voss, J. Chen, J. E. Sharping, and P.Kumar, “Storage and long-distance distribution ofpolarization entanglement generated in the telecomband of standard optical fiber,” in preparation forsubmission to Optics Letters.

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