High-Bandwidth Hybrid Quantum Repeater

W. J. Munro,1,2,* R. Van Meter,2,3 Sebastien G. R. Louis,2,4 and Kae Nemoto2

1Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom2National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan

3Keio University, 5322 Endo, Fujisawa, Kanagawa, 252-8520, Japan4Department of Informatics, School of Multidisciplinary Sciences, The Graduate University for Advanced Studies,

2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan(Received 27 October 2007; published 24 July 2008)

We present a physical- and link-level design for the creation of entangled pairs to be used in quantumrepeater applications where one can control the noise level of the initially distributed pairs. The systemcan tune dynamically, trading initial fidelity for success probability, from high fidelity pairs (F � 0:98 orabove) to moderate fidelity pairs. The same physical resources that create the long-distance entanglementare used to implement the local gates required for entanglement purification and swapping, creating ahomogeneous repeater architecture. Optimizing the noise properties of the initially distributed pairssignificantly improves the rate of generating long-distance Bell pairs. Finally, we discuss the performancetrade-off between spatial and temporal resources.

DOI: 10.1103/PhysRevLett.101.040502 PACS numbers: 03.67.Hk, 03.67.Mn, 42.50.Pq

Quantum information has reached a very interestingstage in its development, where we have seen many fun-damental experiments laying the foundation for practicalsystems [1]. Now certain applications, such as quantumkey distribution (QKD), are being readied for commercialuse [2,3], where practical distances hover around the150 km mark. Any quantum communication longer thanthis limit suffers severely from noise and exponential lossin the quantum communication channel. Hence, the quan-tum communication for either QKD over a distance beyondthe limit, or more generally, all distributed quantum infor-mation processing, requires the development of a high-bandwidth repeater which can distribute and potentiallyprocess quantum information given these constraints. In aquantum repeater system, initial imperfect Bell pairs(which we call base-level pairs) are distributed over chan-nel segments. These base pairs are then purified to highfidelity Bell pairs and connected via entanglement swap-ping, resulting in entanglement between the qubits at dis-tant stations. Iterating this procedure creates Bell pairs ateven larger distances. These pairs can be used in manydifferent applications, including QKD, quantum commu-nication, distributed quantum computation, and quantummetrology and related uses [4–8].

The many recently proposed schemes for the design of aquantum repeater fall into two categories. The majority ofthe schemes focus on the heralded creation of very highfidelity base-level pairs [9–17]. For longer segmentlengths, the generation of these high fidelity pairs comesat the expense of a very low probability of success, whichbecomes one of the major bottlenecks in the overall per-formance of a repeater system. Another significant issue isthat in the majority of these schemes, local gates betweenmultiple qubits within a single repeater station are difficult.An alternative approach has recently been proposed, in-stead creating base-level pairs of moderate fidelity and

high heralded success probability [18,19]. In this secondapproach, the physical resources used for long-distanceentanglement also efficiently implement local gates, facil-itating the purification of moderate fidelity pairs back tohigh fidelity pairs. For instance, with 16 qubits per nodewith a 10 km spacing rate, 15 pairs of fidelity F � 0:98 canbe achieved over a 1280 km repeater network; however, atlonger repeater node spacing distances (>40 km) the ratefalls to zero. This node spacing issue is one of the keylimitations for this second approach. These two approachesare radically different in the use of physical resources andin technological requirements. It is hence not trivial todirectly compare the feasibility and efficiency of schemesin different approaches. However, it has been thought thatthese two approaches are complementary to each other,trading high fidelity for high success probability orvice versa. Quantum repeaters of the latter kind typicallyuse coherent light instead of the single photons common inthe former category. It has been believed that a coherent-light quantum repeater is fundamentally unable to generatehigh fidelity Bell pairs and hence is unable to cope withsevere loss in the quantum channel. In this Letter weaddress this shortcoming by presenting the design of anew scheme for entanglement distribution utilizing coher-ent light and demonstrate that in fact such a system is moreflexible over a wide range of losses without serious over-head in physical resources. This advance will have a sig-nificant impact on the overall repeater performance.

In this Letter we consider the design of a repeater seg-ment where one can dynamically vary the quality of thebase-level entangled pairs from very high to moderatefidelity. This tunability will allow one to trade the fidelityof the entangled pairs against the probability of theirsuccessful distribution for a given segment length. Wewill also ensure that the same physical resources can beused to implement the local gates necessary for the entan-

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glement purification and entanglement swapping [20,21].In this case, both requirements can be met via a control-lable interaction between our qubit and light field.

The core interaction between our qubit and field in ourcavity quantum electrodynamics system is the Jaynes-Cummings (JC) Hamiltonian given by HJC � @g�ay�� �a��� where 2g is the vacuum Rabi splitting for the dipoletransition. a (ay) refers to the annihilation (creation) op-erators of the electromagnetic field mode in a cavity and�� (��) the raising (lowering) operators of the qubit withground state j0i and excited state j1i. Our qubit is a solid-state electronic spin which is also coupled to a nuclear spinqubit, allowing for a coherent transfer of quantum infor-mation to the long-lived nuclear spin qubit. Physically, theelectronic and nuclear spin systems may be achieved, forexample, by single electrons trapped in quantum dots,nitrogen vacancy centers in diamond, or neutral donorimpurities in semiconductors.

Our basic JC Hamiltonian can be used to implement acontrolled displacement D���z� � e�z��a

y���a� operation[21,22] between the qubit and field, where �z � j0ih0j �j1ih1j. This operation displaces the field mode by anamount � conditioned on the state of the qubit. There area number of ways this interaction can be achieved rangingfrom shaped pulse sequences modulating the qubit or field[23] to sequences of controlled rotations and unconditionaldisplacement operations [22].

These controlled displacement operations now form thebasis of an efficient and tunable entanglement distributionscheme, as depicted in Fig. 1. The scheme works as fol-lows: our first qubit is prepared in an equal superposition ofboth basis states �j0i � j1i�=

���2p

with the probe mode in thecavity initially being the vacuum. The qubit interacts withthe probe beam via the controlled displacement operationD���z� resulting in the combined qubit-probe state�j0ij�i � j1ij � �i�=

���2p

. The probe beam is eitherswitched out or leaks out of the cavity and is transmittedover the noisy loss channel to the second cavity. Here thesecond qubit and probe mode interact via a controlleddisplacement operation. The probe beam then leaks outof this second cavity and is measured, projecting our qubitsinto an appropriate entangled state. In more detail, if our

two distributed qubits are prepared in a state �j00i �j01i � j10i � j11i�=2, then after both controlled displace-ments and the noisy channel our two qubit-light field statecan be represented by

� �1� e� ��=2

2jZ�ihZ�j �

1� e� ��=2

2jZ�ihZ�j; (1)

where jZ�i��������N �

p

2 j��ijc�i��������N

p

2 j��ijci�1��2p j��ij0i,

j��i� j00�11��

2p i, j��i�j

01�10��2p i, jc�i � �j2�e

�l=2l0i�

j�2�e�l=2l0i�=�������������2N �

pwith N��1�exp�8j�j2e�l=l0�

and �� � 2j�j2�1� e�l=l0�. Here l=l0 represents the attenu-ation of the probe in the channel. Now the probe beam inEq. (1) is in one of three possible states, jc�i or the vacuumj0i. The odd cat state jc�i is orthogonal to both jc�i andj0i. However, jc�i and j0i are nonorthogonal with anoverlap 2 exp�4j�j2e�l=l0�=�1� exp�8j�j2e�l=l0��. Ofcourse, if �e�l=2l0 � 1, these two states are effectivelyorthogonal, and so one could distinguish all probe beamstates. However, the channel loss has had two significanteffects. First, it has mixed jZ�i with jZ�i with a mixingparameter 1�e� ��=2

2 . This mixing parameter is small onlywhen j�j2�1� e�l=l0� 1, which is in conflict with thedesire to have all probe beam states nearly orthogonal forchannels of moderate length. Second, the channel has alsoattenuated the probe beam’s amplitude and so the secondcontrolled displacement must be by D��e�l=2l0�z2

� ratherthan D���z2

� to minimize the effect of the loss.We now turn our attention to the measurement of the

probe and the resulting conditioning it causes on the dis-tributed qubits. There are various measurement strategies,ranging from highly idealized cat state projectors (CSP)jc�ihc�j to single-photon detection (SPD). Using suchdetection strategies, our qubits are conditioned to

��F� � Fj��ih��j � �1� F�j��ih��j (2)

with heralded success probabilities

PCSPF� �1

4f1� 2F� 1�8e

�l=l0=�1�e�l=l0 �g (3)

PSPDF� �1

2

dd�2F� 1�f4�

2e�l=l0=1��7�8�2�e�l=l0 �g�

����������1;

(4)

respectively. The single-photon detector is assumed tohave a nonunit quantum efficiency �2. Equations (2)–(4)have been expressed in terms of the fidelity F of the j��istate generated, the attenuation parameters l=l0, and thedetection efficiency rather than �. The initial displacement� can be expressed in terms of F, l=l0, P, and �2. We alsoneed to point out that Eq. (2) is a mixture of only two Bellstates, j��i, which has important advantages in entangle-ment purification. Such states are much more efficient topurify.

In Fig. 2 we plot these probabilities PF� versus fidelityfor both measurement strategies (with �2 � 0:9; 1) for an

FIG. 1 (color online). Schematic of an entanglement distribu-tion scheme based on two qubits in individual cavities interact-ing indirectly via a shared probe beam and controlled displace-ment operations. An optical circulator before the second qubitroutes the probe field into the cavity and the resulting beamleaking out to the detector. A phase reference is sent along thesame lossy channel.

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attenuation length l=l0 � 0:8. Our idealized cat projector(CSP) allows a nice range of fidelities to be achieved,ranging from F�1=2 to F � 1. For higher fidelities, lowersuccess probabilities are achieved. The major issue withthe cat projector is that it is difficult to implement in prac-tice, but for the moderate to high fidelity regimes, thesingle-photon detection scheme closely follows the catprojector results and so is an excellent compromise. It alsohighlights that single photons (and hence single-photonsources) are not needed for creating high fidelity distrib-uted Bell pairs. The sending of coherent states (whetherweak or strong) can achieve the same goal. High fidelitypairs can be generated over long distances but at the ex-pense of the success probability. Other measurement tech-niques (such as homodyne, bucket, or vacuum detection)also result in entangled states, but these states tend be com-posed of more than two Bell states, and so are more dif-ficult to purify. However, they can provide success proba-bilities greater than one-half but result in low fidelity finalstates.

We now have a highly tunable segment where one candynamically vary the quality of the base-level entangledpair from low to high fidelity utilizing nonunit efficiencysingle-photon detection. These pairs can be used in a re-peater protocol to create long-distance entangled pairs. Thebasic repeater protocol works as follows: multiple copiesof lower fidelity base-level pairs are purified to create highfidelity pairs, then entanglement swapping of the highfidelity pairs between adjacent repeater nodes creates lon-ger distance but lower fidelity pairs. These resulting pairscan then be purified to high fidelity pairs and entanglementswapping gives even longer range pairs. The procedure isiterated until pairs over the desired length are obtained.The purification and entanglement swapping protocolsrequire efficient local two-qubit C-Z (or CNOT) gates. Inour architecture these can be achieved using another se-quence of controlled displacement operations D�i�2�z2��D��1�z1�D��i�2�z2�D���1�z1� � exp2i�1�2�z1 ��z2� with �1 and �2 satisfying �1�2 � �=8 [21,22].These quantum bus (qubus) based local gates are needed

throughout the protocol, and so the more efficient androbust they are the better our overall performance.

One of the major issues for performance becomes thechosen quality of base-level entangled pairs for our lowest-level segment. Conventional quantum repeater wisdomgenerally suggests that before one performs entanglementswapping to create longer pairs, one should purify thenoisy base-level pairs as best as possible. For our discus-sions here we will set a working fidelity of F � 0:98 beforeattempting entanglement swapping. There are now a num-ber of ways we can use our tunable segment to achieve thisrequired fidelity. We could just directly create a pair of thatfidelity, or we could create lower fidelity pairs and purifythem using standard protocols [24–27]. Which approach isbest will depend on both the probability of generating theentangled pairs Pg and the purification probability Ppur.For a one-round purification protocol with no limitation onthe physical resources, the effective probability of generat-ing the final fidelity pair (per channel) is given by Peff �

PgPpur=2. The purification probability depends criticallyon the form of the initial entangled pairs, and so ourengineering of Eq. (2) above is very advantageous. Amixture of two Bell states is much easier to purify thanmore general mixed states. In this situation, two copies of��F� can be purified to a new entangled �F0 �F2=�1�2F�2F2�� with a Ppur�F��F2��1�F�2 [27]. This givesan overall probability of success for generating a F � 0:98pair from two F�0:9 pairs of Peff�0:0547. In compari-son, we have generation probabilities of 0.033 74 for di-rectly manufacturing the F�0:98 pair. Thus, using lowerfidelity pairs and purification improves our overall proba-bility of generating the final pair, assuming efficient localgates.

Does a multiple-round purification protocol improve ourresults? Moving to multiple rounds, there are a number ofchoices for how to implement the protocol [24,26]. As anillustration, consider a symmetric purification protocol[26] where four F � 0:75 pairs are purified to the F �0:98 pair. In this case, we have an effective generationprobability of Pg � 0:0236 which is lower than the proba-bility achieved by the single-round protocol. The differ-ence is primarily due to the lower probability of success-fully purifying lower fidelity pairs.

We now need to turn our attention to a more detaileddiscussion of the physical resources required for creatingour long-distance pairs. It is important to consider both thespatial and temporal resources necessary. The tunability ofour source allows us significant flexibility in how we useresources. We can implement the minimum physical re-source strategies (two qubits per station) developed byChildress et al. [11], as well as the modest physical re-source 16 qubit per station schemes of van Loock et al. [18]and Ladd et al. [19]. Because of our low to moderateprobabilities of successfully creating the base pairs be-tween stations, the minimum physical resource approachwill have significant performance issues due to this distri-

ββ

FIG. 2. Plot of the probability PF� of successfully establish-ing an entangled pair at an attenuation length of l=l0 � 0:8(20 km in commercial fiber) versus fidelity for the differentmeasurement strategies: (a) odd state cat projector CSP, (b) idealSPD, and (c) SPD with �2 � 0:9.

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bution bottleneck. Significant time will be spent idle wait-ing for the base pairs to be successfully created between allstations. However, by allowing moderate physical resour-ces, we can simultaneously attempt to create multiple basepairs per segment, and so reduce the time waiting for thenecessary resources to become available. This approachwill dramatically increase the throughput of the entire re-peater chain. To quantify this degree of the improvement,we have performed a Monte Carlo simulation of a nestedentanglement protocol over 51:2l=l0 (1280 km) for varyingnumbers of qubits per station with dynamical resourceallocation [28]. The results are presented in Table I. First,they show that, of the three segment lengths considered, thebest results were obtained from the 0:8l=l0 (20 km) situ-ation. For longer distances, the initial success probabilitydrops dramatically, and for shorter segment lengths, errorsin local gates have a significant impact. We also found thatthe protocol could run with local gate error rates exceeding1%, though in this situation the generation rate falls to onlya few pairs per second. Raising the number of qubits ineach half-station also gives a slight improvement comparedto stacking smaller repeater nodes in parallel. Still, ourresults show a good generation rate with 8–16 qubits perhalf node. In the scheme of van Loock, with 16 qubits perhalf node with 0:4l=l0 segment spacing (4096 total qubits),a rate of 15 F � 0:98 Bell pairs per second was achieved.Using our new scheme (for the same total number of qu-bits), we achieve a rate of 3190 pairs per second, an im-provement of over 2 orders of magnitude. Our new schemealso produces relatively high throughput of 437 pairs at alink distance of 40 km compared with zero for the originalvan Loock case. In the new scheme, the fidelity remainshigh over long distances, but the probability of successdeclines. Finally, some of our improvements in the proto-col are obtained by tuning the base-level fidelity to opti-mize the number of purification rounds before entangle-ment swapping.

In summary, we have shown how to implement a high-bandwidth quantum repeater using the fundamental atom-light interactions in quantum optics, through a qubus-

mediated entangling operation. This new approach refutesthe criticism that hybrid repeaters cannot adapt to a widerange of losses and offers flexibility on fidelity and entan-glement success probability. In this hybrid scheme, theonly required interactions are controlled displacement op-erations between the light field and qubit. These controlleddisplacement operations result in the distribution of mod-erate to high fidelity entanglement between the qubits inthe repeater stations, conditioned on single-photon detec-tion of the qubus mode. They also can implement a deter-ministic C-Z gate for use in all the purification andentanglement swapping steps. Such tools allow for thenatural design of a scalable and homogeneous quantumrepeater network and thus a distributed computation withmany concurrent steps happening in different locations.Using high-efficiency single-photon detection, we haveshown that long-distance communication rates over 1000pairs per second with a fidelity above 98% are possiblewith only modest resources.

We thank T. Ladd for the use of the base simulation codeupon which our simulations are based, and we also thankhim, L. Jiang, P. van Loock, T. P. Spiller, and J. M. Taylorfor valuable discussions. This work was supported in partby MEXT and NICT in Japan and the EU project QAP.

*bill.munro@hp.com[1] T. P. Spiller et al., Contemp. Phys. 46, 407 (2005).[2] N. Gisin et al., Rev. Mod. Phys. 74, 145 (2002).[3] J. L. Duligall et al., New J. Phys. 8, 249 (2006).[4] R. Van Meter et al., ACM J. Emerging Technol. Comput.

Syst. 3, 1 (2008).[5] J. I. Cirac et al., Phys. Rev. A 59, 4249 (1999).[6] L. K. Grover, arXiv:quant-ph/9704012.[7] A. Serafini et al., Phys. Rev. Lett. 96, 010503 (2006).[8] I. L. Chuang, Phys. Rev. Lett. 85, 2006 (2000).[9] H.-J. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998).

[10] W. Dur et al., Phys. Rev. A 59, 169 (1999).[11] L. Childress et al., Phys. Rev. Lett. 96, 070504 (2006).[12] A. Klein et al., Phys. Rev. A 73, 012332 (2006).[13] S. J. Enk et al., Science 279, 205 (1998).[14] L.-M. Duan et al., Nature (London) 414, 413 (2001);[15] Z.-B. Chen et al., Phys. Rev. A 76, 022329 (2007).[16] B. Zhao et al., Phys. Rev. Lett. 98, 240502 (2007).[17] L. Jiang et al., Phys. Rev. A 76, 012301 (2007).[18] P. van Loock et al., Phys. Rev. Lett. 96, 240501 (2006).[19] T. D. Ladd et al., New J. Phys. 8, 184 (2006).[20] K. Nemoto and W. J. Munro, Phys. Rev. Lett. 93, 250502

(2004).[21] T. P. Spiller et al., New J. Phys. 8, 30 (2006).[22] P. van Loock et al., arXiv:quant-ph/0701057.[23] D. F. Walls and G. J. Milburn, Quantum Optics (Springer,

New York, 2008), 2nd ed.[24] W.Dur and H. J.Briegel, Rep. Prog. Phys. 70, 1381 (2007).[25] C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).[26] W. Dur et al., Phys. Rev. A 59, 169 (1999).[27] J. Pan et al., Nature (London) 410, 1067 (2001).[28] O. A. Collins et al., Phys. Rev. Lett. 98, 060502 (2007).

TABLE I. Rate of final generation of distant 51:2l=l0(1280 km) entangled pairs of minimum fidelity F � 0:98resulting from a nested entanglement protocol with 8, 16, and32 qubits per half-station. The stations are separated by either0:4l=l0, 0:8l=l0, or 1:6l=l0 attenuation lengths. The number inthe brackets indicates the total numbers of qubits over allrepeater stations. We have assumed an initial fidelity reductionin the base pairs due to loss in the fiber (assumed to be0:17 dB=km) and distortion of 0.1% due to local losses duringthe measurement-free displacement-based C-Z gate.

Qubits half-station 0:4l=l0 0:8l=l0 1:6l=l0

8 520 (2048) 693 (1024) 193 (512)16 1097 (4096) 1528 (2048) 437 (1024)32 2297 (8192) 3190 (4096) 987 (2048)

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