Quantum repeaters using coherent-state communication

Peter van Loock,1,2 Norbert Lütkenhaus,3 W. J. Munro,2,4 and Kae Nemoto2

1Optical Quantum Information Theory Group, Institute of Theoretical Physics I and Max-Planck Research Group, Institute of Optics,Information and Photonics, Universität Erlangen-Nürnberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany

2National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan3Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, N2L 3G1 Waterloo, Canada

4Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom�Received 21 July 2008; revised manuscript received 3 November 2008; published 10 December 2008�

We investigate quantum repeater protocols based upon atomic qubit-entanglement distribution through op-tical coherent-state communication. Various measurement schemes for an optical mode entangled with twospatially separated atomic qubits are considered in order to nonlocally prepare conditional two-qubit entangledstates. In particular, generalized measurements for unambiguous state discrimination enable one to completelyeliminate spin-flip errors in the resulting qubit states, as they would occur in a homodyne-based scheme due tothe finite overlap of the optical states in phase space. As a result, by using weaker coherent states, high initialfidelities can still be achieved for larger repeater spacing, at the expense of lower entanglement generationrates. In this regime, the coherent-state-based protocols start resembling single-photon-based repeater schemes.

DOI: 10.1103/PhysRevA.78.062319 PACS number�s�: 03.67.Lx, 42.50.Dv, 42.25.Hz

I. INTRODUCTION

In long-distance, classical communication networks, sig-nals that are gradually distorted during their propagationthrough a channel are repeatedly recreated via a chain ofintermediate stations along the transmission line. For in-stance, optical pulses traveling through a glass fiber and be-ing subject to photon loss can be reamplified at each repeaterstation. Such an amplification is impossible, when the signalcarries quantum information. If a quantum bit is encodedinto a single photon, its unknown quantum state cannot becopied along the line �1,2�; the photon must travel the entiredistance with an exponentially decreasing probability toreach the end of the channel.

The solution to the problem of long-distance quantumcommunication is provided by the so-called quantum re-peater �3,4�. In this case, prior to the actual quantum-statecommunication, a supply of standard entangled states is gen-erated and distributed among not too distant nodes of thechannel. If sufficiently many of these imperfect entangledstates are shared between the repeater stations, a combinationof entanglement purification and entanglement swapping ex-tends this shared entanglement over the entire channel.Through entanglement swapping �5�, the entanglement ofneighboring pairs is connected, gradually increasing the dis-tance of the shared entanglement. The entanglement purifi-cation �6,7� enables one to distill �through local operations� ahigh-fidelity entangled pair from a larger number of low-fidelity entangled pairs, as they would emerge after a fewrounds of entanglement swapping with imperfect entangledstates, or even at the very beginning after the initial, imper-fect entanglement generation and distribution.

Current implementations for quantum communication, inparticular, quantum key distribution, are limited by a dis-tance of about 200 km. In principle, one could go beyondthis distance using a quantum repeater. However, the issue ofactually realizing a quantum repeater protocol is rathersubtle, even for not too long distances. In particular, the sub-

routines of entanglement distillation and swapping requireadvanced local quantum logic including, for instance, two-qubit entangling gates; moreover, a sufficient quantummemory is needed such that local measurement results canbe communicated between the repeater stations �8�. Nonethe-less, various proposals exist, of which the most recent onesare based on the nonlocal generation of atomic �spin� en-tangled states, conditioned upon the detection of photons dis-tributed between two neighboring repeater stations. Thelight, before traveling through the communication channeland being detected, is scattered from either individual atoms,for example, in the form of solid-state single-photon emitters�9,10�, or from an atomic ensemble, i.e., a cloud of atoms ina gas �11�. In these heralded schemes, typically, the fidelitiesof the initial entanglement generation are quite high, at theexpense of rather small efficiencies. Other complications in-clude interferometric phase stabilization over large distances�12–14� and the purification of atomic ensembles. Yet someelements towards a realization of the protocol in Ref. �11�have been demonstrated already �15–17�. Further theoreticalresults were presented very recently �18�, improving thescheme of Ref. �11�.

In this paper, we will extend our previous results on theso-called hybrid quantum repeater �19,20�. This approach tolong-distance quantum communication is somewhat differentfrom those mentioned above. It relies on atom-light entangle-ment, which becomes manifest in quantum correlations be-tween a discrete spin variable and a continuous optical phasequadrature rather than a discrete single-photon occupationnumber. An optical pulse in a coherent state of about 104

photons is subject to a controlled phase rotation �achievedthrough dispersive, cavity quantum electrodynamics�CQED�-type interactions�, conditioned upon the state of theatom. After propagating to the nearest neighboring repeaterstation and a further interaction with a second spin at thatstation, the light field is measured via homodyne detectionand an imperfect entangled two-qubit state is nonlocally pre-pared between the two repeater stations through postselec-tion. Finally, the same dispersive light-matter interactions are

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exploited to achieve the local quantum gates �“qubus com-putation” �21,22�� needed for entanglement purification andswapping.

The two main advantages of the hybrid repeater protocol,distinct from the single-photon-based schemes, are the highsuccess probabilities for postselection in the entanglementgeneration step and the intrinsic phase stabilization providedthrough reference pulses propagating in the same channel asthe probes. However, these assets are at the expense of rathermodest initial fidelities of the entangled states and high sen-sitivity to photon losses and noise in the optical channel. Infact, distances between repeater stations beyond 10–20 kmturn out to be impossible with the current proposal; the de-coherence effect �a damping of the off-diagonal terms of thetwo-qubit density matrix after postselection� exponentiallygrows with distance such that only smaller mean photonnumbers lead to a sufficient degree of entanglement; how-ever, the less intense coherent states are less distinguishable,hence resulting in a further decrease of fidelity through post-selection errors. A good trade-off between these competingsources of errors is only possible for not too large distances.

The analysis here will provide a possible solution to thedistance limitation. This is particularly important, as the typi-cal repeater spacing in existing classical communication net-works is of the order of 50–100 km and thus incompatiblewith the current hybrid repeater protocol. The distance limitcan be overcome by completely eliminating one source oferrors, namely, that which stems from the finite overlaps ofthe phase-rotated coherent states. This is achieved through adifferent detection scheme, where the coherent states are un-ambiguously discriminated. Such an unambiguous state dis-crimination �USD� is error-free; so for nonorthogonal states,it must include inconclusive measurement results. These willlead to lower efficiencies of the entanglement generation, inparticular, when smaller photon numbers are used in order tosuppress the decoherence effect through photon losses. Thecorresponding trade-off between success probability and fi-delity means there are ultimate quantum-mechanical boundson the accessible regimes. We will discuss these bounds andpropose suboptimal, but practical, linear optical implementa-tions.

The emphasis here is on possible measurement schemesfor the initial entanglement generation. Further, we investi-gate the hybrid entangled atom-light states before the mea-surements and potential variations of the entanglement dis-tillation and swapping steps. The latter could be performedalready on the atom-light level �“hybrid entanglement distil-lation and swapping”� rather than solely on the atomic levelafter the conditional state preparation. We do not considerissues related with the CQED part �for this, see �20��; neitherare we concerned about architecture-related issues on how tocombine the entanglement purification and swapping steps inan optimal way �for this, see �23��. Such considerations willbe needed for comparing the overall efficiencies between thehybrid approach and the single-photon-based schemes.

The plan of the paper is as follows. First, in Sec. II, wewill examine the hybrid entangled states between one atomicspin and an optical mode �Sec. II A�, the entangled states oftwo spins and an optical mode �Sec. II B�, and the measure-ments for conditional entangled-state preparation �Sec. II C�.

Secondly, in Sec. III, we will discuss the notions of hybridentanglement distillation and swapping and their potentialrealizations.

II. ENTANGLEMENT GENERATION

In the hybrid quantum repeater, the mechanism for en-tanglement distribution is based on dispersive light-matterinteractions, obtainable from the Jaynes-Cummings interac-tion Hamiltonian �g��−a†+ �+a� in the limit of large detun-ing �24�,

Hint = ���za†a . �1�

Here, a �a†� is the annihilation �creation� operator of theelectromagnetic field mode and �z= �0��0�− �1��1� is the cor-responding qubit Pauli operator for a two-level atom. Theparameter �=g2 /� describes the strength of the atom-lightcoupling; 2g is the vacuum Rabi splitting for the dipole tran-sition and � is the detuning between the dipole transition andthe light field. The Hamiltonian in Eq. �1� leads to a condi-tional phase rotation of the field mode,

Uint = exp�− i��za†a/2� . �2�

Here, �t�� /2 is an effective interaction time. The only re-quirement for a dispersive interaction resulting in a high-fidelity conditional rotation is a sufficiently large cooperativ-ity parameter in a weak or intermediate coupling regime;strong coupling is not needed �20�. We may then write theeffect of a controlled rotation on a coherent state and a qubitsuperposition state �corresponding to Eq. �2� up to an uncon-trolled phase rotation� as

Uint���0� + �1������/2 = ��0���� + �1���ei���/2. �3�

In the following, we will investigate the entangled states thatemerge from this interaction. According to the initial en-tanglement distribution procedure for the hybrid quantum re-peater, as a first step, one atomic qubit interacts with theoptical qubus mode �see Fig. 1�a��, resulting in a “hybridentangled state” between qubit 1 and the qubus, as describedby Eq. �3�. During the transmission of the qubus through the�lossy� channel, the hybrid entangled state is subject to de-coherence and becomes mixed. Then the qubus interacts withqubit 2; at this stage, the two qubits and the qubus are in atripartite �mixed� entangled state. Finally, a measurement onthe qubus mode conditionally prepares a two-qubit �mixed�entangled state. An example of a possible measurementscheme to �approximately� achieve this final step is throughhomodyne detection �see Fig. 1�b��. Let us now closely ex-amine the elements of this protocol with respect to possibleimprovements.

A. Qubit-qubus entanglement

Let us assume the qubus mode is entangled with qubit 1after the first interaction, as expressed by Eq. �3�. Photonlosses in the qubus channel are now described via a simplebeam splitter, which reflects, on average, 1−� photons intoan environment mode, initially in the vacuum state �0�E,

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�0�A����B�1 − ���E/2 + �1�A���ei��B�1 − ��ei��E/2.

�4�

The subscripts “A” and “B” denote the state of the atomicqubit and the qubus mode, respectively. We may now rewriteeach of the two pairs of pure, nonorthogonal states of thequbus mode and the loss mode in an orthogonal, two-dimensional basis, �u� , �v��,

����B = �B�u�B + B�v�B,

���ei��B = ��B�u�B − B�v�B�ei�,

�1 − ���E = �E�u�E + E�v�E,

�1 − ��ei��E = ��E�u�E − E�v�E�ei�1−��, �5�

where B=1−�B2 and E=1−�E

2 with

�B = �1 + e−��2�1−cos ���1/2/2, �6�

�E = �1 + e−�1−���2�1−cos ���1/2/2, �7�

and ��2 sin �. Here and in the following, we assume � tobe real. Now tracing over the loss mode and using the Abasis, ��0�A�ei�1�A� /2�, as a new computational basis, wecan express the “two-qubit” density matrix of A and B in avery compact way as

�E2 ��+��B����+��B�� + �1 − �E

2�� +��B��� +��B�� . �8�

The resulting density matrix is a mixture of two nonmaxi-mally entangled states,

��+��B�� = �B�0�A�u�B + 1 − �B2 �1�A�v�B, �9�

� +��B�� = �B�1�A�u�B + 1 − �B2 �0�A�v�B. �10�

The phase-rotated coherent states are now contained in theorthogonal basis of the corresponding qubus-mode subspace,

�u�B =1

2�B�����B + e−i����ei��B� ,

�v�B =1

21 − �B2

�����B − e−i����ei��B� . �11�

In the form of Eq. �8�, one can easily observe the trade-off ofthe presence of entanglement for different photon numbers�2, assuming imperfect transmission, ��1, and reasonablephase shifts, ��10−2 ,10−3. Choosing � small means thedensity matrix in Eq. �8� approaches a pure state, accordingto Eq. �7�; however, this pure state is nearly unentangled fortoo small �, according to Eqs. �6� and �9�. Conversely, forlarge �, we have �E

2 →1 /2, leading to an almost equal mix-ture of the states of Eqs. �9� and �10� in Eq. �8�; however,this time, the individual states of Eqs. �9� and �10� are nearlymaximally entangled Bell states. In other words, the qualityof the entanglement is affected either by the decoherenceeffect of the channel �for large �, when many photons trans-fer which-path information into the environment� or by thenonmaximal entanglement of the pure states in Eqs. �9� and�10� �for small �, when the phase-rotated coherent states arenearly indistinguishable and the initial atom-light entangle-ment is weak�.

Another interesting feature of the state in Eq. �8� is thatwe may consider a purification of some copies of it throughlocal operations on the qubits and the qubus modes, hencedistilling a higher degree of entanglement into a smallernumber of copies. This “prepurification” �prior to the qubusinteraction with qubit 2� or “hybrid entanglement distilla-tion” will be discussed in Sec. III A.

As the density matrix in Eq. �8� effectively describes atwo-qubit state, we can evaluate its entanglement of forma-tion using the concurrence �25�. Figure 2 shows the entangle-ment of formation as a function of the qubus amplitude ��square root of qubus photon number� for different channeltransmissions. Note that this is the maximum initial entangle-ment �prior to any entanglement distillation procedures�available in the repeater protocol. All the remaining steps ofthe initial entanglement generation, including the interactionbetween qubus and qubit 2 and the measurement of the qu-bus mode, are local; hence they can only reduce the amountof entanglement. The optimal value of the product ��, maxi-mizing the entanglement, is always of the order of ���1.This is similar to the result obtained for the two-qubit singletfidelity after the interaction with qubit 2 and homodyne mea-surement of the qubus mode, reflecting the optimal trade-offbetween distinguishability and decoherence �19�. Let us now

Coherent-State Pulse

LO PulseProbe Pulse Homodyne

Measurement

ChannelQubit 1 / Cavity 1

a)

b)

Qubit 1

Qubit 2

Qubus

Qubit 2 / Cavity 2

FIG. 1. �Color online� �a� Three steps for the generation of spinentanglement between two qubits at neighboring repeater stations:the first interaction results in an entangled state between the atomicqubit 1 and the optical qubus; after transmission, the qubus interactswith the atomic qubit 2, leading to a tripartite entangled state be-tween the two qubits and the qubus; finally, a measurement on thequbus mode conditionally creates the two-qubit entanglement. �b�Example of a possible measurement scheme for discriminating be-tween the conditionally phase-rotated coherent probe beams in thehybrid quantum repeater; the LO pulse is a sufficiently strong localoscillator used for homodyne detection �19�.

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consider the interaction of the qubus mode with qubit 2 andlook at the resulting tripartite mixed entangled state.

B. Qubit-qubus-qubit entanglement

In the hybrid quantum repeater protocol, the entangledstate of Eq. �8� is subject to a second interaction, this timebetween the qubus mode and qubit 2 �which initially is in anequal superposition state�. This interaction results in a con-trolled rotation of the qubus by an angle of −�, as describedby Eq. �3� with �→−�. The controlled rotation transformsthe two orthogonal qubus basis states of Eq. �11� togetherwith the qubit state as

�u�B � ��0�C + �1�C�/2 →1

22�����B��0�C + e−i��1�C�

+ e−i����ei��B�0�C

+ ���e−i��B�1�C�/�B, �12�

and, similarly,

�v�B � ��0�C + �1�C�/2 →1

22�����B��0�C − e−i��1�C�

− e−i����ei��B�0�C

+ ���e−i��B�1�C�/1 − �B2 .

�13�

Applying these transformations to the density matrix in Eq.�8�, a local Hadamard gate to the spin system A, and a localrotation ei��1−�z�/2 upon system C, leads to the followingtripartite density operator:

�E2 ��+���+� + �1 − �E

2���−���−� , �14�

where

��+� =12

����B��+�AC +1

2e−i����ei��B�10�AC

+1

2ei����e−i��B�01�AC,

��−� =12

����B��−�AC −1

2e−i����ei��B�10�AC

+1

2ei����e−i��B�01�AC, �14a�

with the maximally entangled Bell states ����= ��00�� �11�� /2. The state in Eq. �14� with Eqs. �14a�again illustrates the two competing sources of errors in theentanglement generation step. Bit-flip errors are caused bythe indistinguishability of the phase-rotated coherent states�if homodyne detection is used for state discrimination�;these can be reduced via sufficiently large photon numbers.Phase-flip errors occur for any imperfect transmission; thisdecoherence effect is suppressed for smaller photon num-bers.

In the following, we will consider unambiguous state dis-crimination �USD� �26–28� of the corresponding phase-rotated coherent states in Eq. �14�. This enables us to com-pletely eliminate bit-flip errors, at the expense of a reducedefficiency coming from inconclusive measurement results.The relevant, quantum-mechanical USD problem providesultimate performance bounds. These shall be approached us-ing practical linear-optics solutions for the required general-ized measurements.

C. Conditional state preparation

As illustrated in Fig. 1�a�, after the interaction of the qu-bus with qubit 2, the final step is to prepare conditionally anentangled two-qubit state through measurements on the qu-bus mode including postselection. The postselection proce-dure may either filter out approximate, mixed entangled two-qubit states, still containing some errors from the finiteoverlap of the phase-rotated coherent states �19�; or it may, ata lower success rate, perfectly rule out those contributionsbelonging to the “wrong” coherent states �see Fig. 3� andproject onto an entangled two-qubit state whose imperfectionoriginates solely from the losses in the communication chan-nel. The latter scenario can be achieved via a scheme basedon USD measurements, providing the ultimate, distance-dependent limits on the quality of the initially generated en-tangled states. At the same time, it yields an alternative ap-proach to feasible implementations of the entanglementgeneration step. Let us first briefly recall the homodyne-based entanglement generation scheme.

1. Homodyne-based state preparation

A very efficient and practical way to discriminate thephase-rotated coherent states in Fig. 3 is through homodynedetection �19�. While an x measurement could, in principle,project onto both the even and the odd qubit subspace�spanned by �00�, �11�� and �10�, �01��, respectively�, a p

EoF

α0 50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8

4km

10km17km

30km

100km

FIG. 2. �Color online� The entanglement of formation of thequbit-1-qubus states as a function of the qubus amplitude � �squareroot of the qubus photon number� for different channel transmis-sions, i.e., different distances of qubus propagation; photon loss isassumed to be 0.18 dB per km. The phase � is always 0.01.

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measurement only leads to an entangled state in the evensubspace and those results consistent with either the �10� orthe �01� state must be discarded. Nonetheless, for the quan-tum repeater protocol, the p measurement is preferred to thex measurement, as the errors in the former scale as �� andthose of the latter as ��2, resulting in fast decoherence forsmall � and � sufficiently large. Let us now discuss the ul-timate bounds on the performance of the entanglement gen-eration step using an error-free, USD-based measurementscheme.

2. Ultimate bounds

In order to derive some bounds on the attainableentangled-state fidelities at a certain rate for a given distance,let us consider the binary USD problem of discriminating thestate ���� versus the set of states ���ei�� , ���e−i���. Wemay rephrase this problem as the USD of the two densityoperators

�1 = �������� ,

�2 =1

2����ei�����ei�� + ���e−i�����e−i��� . �15�

Any measurement scheme that is intended to filter out unam-biguously an entangled Bell state from the two individualstates in Eqs. �14a� of the mixture in Eq. �14� must be asolution to the above USD problem; thus, the best possibleentanglement generation scheme cannot outperform the op-timal USD scheme. However, note that the converse is nottrue. A generalized measurement, even for optimal USD,does not necessarily result in an entangled state. In particular,unambiguously identifying the state �2 in Eq. �15� may alsomean that the sign of the phase rotation is determined. In thiscase, the two qubits end up in a separable state, according toEq. �14� with Eqs. �14a�. In fact, in order to coherentlyproject onto the odd qubit subspace �as it could be doneerroneously via x homodyne detection �see Fig. 3�� throughUSD, the measurement scheme becomes less practical in-volving photon-number resolving detectors �29�.

The USD of mixed quantum states is a much more subtleissue than that of pure states �30–33�. Nonetheless, for equala priori probabilities �which is the case we are interested in�,

the failure probability �the probability for obtaining an in-conclusive measurement outcome� is bounded from belowby the square root of the fidelity of the two density operators�32�. For the USD problem of Eq. �15�, this means

P? � �����2���� . �16�

Using Eq. �15�, this leads to the optimal �minimal� failureprobability

P?opt = e−��2�1−cos ��. �17�

This bound can be inserted into the fidelity �this time for thequbit states� �E

2 �F of the desired ��+� state in the mixtureof Eq. �14�. Using Eq. �7�, we obtain

P?opt�F� = �2F − 1��/�1−��. �18�

The optimal failure probability as a function of the fidelity isshown in Fig. 4 for different distances. The regions beloweach curve are quantum mechanically inaccessible. Note thathere P?

opt�F� depends implicitly on � and � through the fidel-ity �E

2 �F, according to Eq. �7�; any point of the curves inFig. 4 for a given transmission � �distance� corresponds tocertain combinations of � and �. In particular, for any �, wehave P?

opt�F�→1 for perfect purity F→1 �vanishing deco-herence� with either �→0 or �→0, because the probe statesapproach identical coherent �or vacuum� states. Conversely,for ��0 and �→�, the probe states become orthogonal,leading to perfect distinguishability, P?

opt�F�→0, but maxi-mum decoherence, F→1 /2.

The larger the distances, the larger the failure probabilitiesbecome at a given fidelity. Reasonably high fidelities areonly achievable at the expense of small success probabilities.However, the bound for the USD problem does allow forfidelities much greater than 1 /2 at distances of 50 km andmore. We will now investigate whether there are practicalimplementations of the corresponding USD measurementthat approach the quantum-mechanical bounds and hence areno longer limited by distances of 20 km and below.

√η αeiθ-

-

-√η α

√η αe-iθ

p

x

FIG. 3. �Color online� Phase-rotated coherent states to be dis-criminated for entangled-state preparation; the unrotated state be-longs to the preferred qubit subspace �even parity�; the two rotatedstates are correlated with the odd subspace of the two qubits.

fidelity

0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

failureprobability

fidelity

4km

10km

17km

30km

50km

FIG. 4. �Color online� The optimal failure probability for unam-biguous state discrimination of the two density operators in Eqs.�15� as a function of the fidelity of the desired entangled two-qubitstate in Eq. �14�. The regions below each curve are quantum me-chanically inaccessible. Note that for a transmission �=1 /2�17 km�, the functional dependence is linear. Photon loss is againassumed to be 0.18 dB per km.

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3. Unambiguous state preparation

Apart from an initial entanglement generation over poten-tially larger distances, there are other advantages of usingUSD for the conditional entangled-state preparation. In par-ticular, the resulting imperfect entangled states will be mix-tures of just two Bell states �rank two mixtures�, in the formof Eq. �14� after ruling out the odd parity terms in Eqs. �14a�.For some copies of this type of mixed-entangled states, en-tanglement distillation is more efficient �34� and the so-called entanglement pumping is no longer bounded by somefidelity threshold below unity �as for higher rank Bell-diagonal mixtures� �4�; entanglement pumping means thatspatial resources in the repeater protocol can be turned intotemporal resources by always distilling the same entangledpair with the help of freshly prepared elementary pairs.

A scheme for unambiguously discriminating the phase-rotated coherent states in Fig. 3, and hence realizing USD ofthe density matrices in Eq. �15�, based upon linear optics andphoton detection, is shown in Fig. 5. The qubus mode is sentthrough a linear three-port device, together with two ancillavacuum modes, and subsequently, the three output modes aredisplaced in phase space before being detected. The three-port device acts upon a coherent state ��� as

��,0,0� → ���,��,1 − 2�2�� , �19�

choosing � real. It can be realized via a sequence of twobeam splitters,

U�3� = 1 0 0

0�

1 − �2 1 − 2�2

1 − �2

0 1 − 2�2

1 − �2

− �

1 − �2�

� � 1 − �2 0

1 − �2 − � 0

0 0 1�

= � 1 − �2 0

�− �2

1 − �2 1 − 2�2

1 − �2

1 − 2�2 − �1 − 2�2

1 − �2

− �

1 − �2� .

�20�

The subsequent phase-space displacements are

D�− ���ei�� � D�− ���e−i�� � D�− 1 − 2�2��� .

�21�

Via the three-port device and the displacements, the threedifferent qubus input states to be discriminated are trans-formed as

���,0,0� → �����1 − ei��,����1 − e−i��,0� ,

���ei�,0,0� → �0,���2i sin �,1 − 2�2���ei� − 1�� ,

���e−i�,0,0� → �− ���2i sin �,0,1 − 2�2���e−i� − 1�� .

�22�

There are now six out of eight possible detection patterns,considering detectors which do not resolve photon numbers�going either “click” or “no click”�. These patterns unam-biguously identify the corresponding coherent states of theinput,

�click,click,no click� → ���� ,

�no click,no click,click� → ���e�i�� ,

�no click,click,click� → ���ei�� ,

�click,no click,click� → ���e−i�� ,

�no click,click,no click� → ����or���ei�� ,

�click,no click,no click� → ����or���e−i�� . �23�

The pattern �no click, no click, no click� is inconclusive �cor-responding to the vacuum contributions from all threemodes�, whereas the pattern �click, click, click� does not oc-cur at all.

Among the remaining six detection patterns, there are twopatterns that are inconclusive, in addition to the vacuum-based, inconclusive pattern �no click, no click, no click�. Theextra inconclusive patterns are �no click, click, no click� and�click, no click, no click� �the last two patterns of Eq. �23��.As they rule out only ���e−i�� and ���ei��, respectively,they are conclusive results neither for the USD problem norfor the entanglement generation in the repeater protocol.However, the conditional states that emerge after the detec-tion of these patterns �i.e., after obtaining just one click ineither the first or the second mode� are, in principle, stillusable for both USD and entanglement generation. Here wewill not consider such conditional dynamics.

Focusing on the remaining four patterns, we observe thefollowing. Any one of the first four patterns of Eq. �23� con-clusively identifies the quantum state in the USD problem ofEq. �15�. Among these four, only the first pattern, �click,click, no click�, identifies the state ����. The other threepatterns are only consistent with the state �2 in Eq. �15�,ruling out �1.

U(3) (−λ√η αe )-iθD

(−λ√η αe )iθD

[− (1−2λ ) √η α]2 ½D

FIG. 5. �Color online� Unambiguous state discrimination of azero-phase coherent state from two phase-rotated coherent states vialinear optics, phase-space displacements, and photon detection. Thecircuit U�3� represents a linear three-port device, corresponding to asequence of two beam splitters �see Eq. �20��.

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If we now look at the entanglement generation step of therepeater protocol, then even the patterns �no click, click,click� and �click, no click, click� must count as failure, be-cause conclusively identifying either the state ���ei�� or thestate ���e−i�� means that the two atomic spins will end upin a separable state, according to Eq. �14� with Eqs. �14a�.

Eventually, only the two patterns �click, click, no click�and �no click, no click, click� are useful for the entanglementgeneration. The former one, conclusively identifying thestate ����, projects the two qubits onto the even parity sub-space. The latter one, ruling out ���� and being consistentwith both ���ei�� and ���e−i��, leads to the odd subspace.However, in this case, just obtaining a click for mode 3 is notenough to project the two qubits onto a maximally entangledstate, not even in the ideal case without losses �see Eq. �14��.Such a measurement would result in a superposition of thetwo “odd” Bell states ����= ��10�� �01�� /2 with an evennumber �without the vacuum� and an odd number coherent-state superposition, ���i��� �−�i��, respectively. Thus, onlythrough detection of the photon-number parity, a maximallyentangled Bell state of the two qubits can emerge. This couldbe achieved via photon-number resolving detectors �29�.

As a result, we obtain a highly practical solution for en-tanglement generation, based upon two detectors firing at thesame time, �click, click, no click�. We denote the successprobability for this event to occur as Peven. Similarly, theprobability for projecting onto the odd subspace shall bePodd,USD and Podd,ent, where Podd,USD includes those patternswhich may or may not resolve the two states ���ei�� and���e−i�� and hence are only partly useful for entanglementgeneration �but still entirely for USD�. The probabilityPodd,ent only includes the pattern �no click, no click, click�,which, using photon-number resolving detectors, leads to anentangled state. For these probabilities, we obtain,

Peven =1

2�1 − e−�2��22�1−cos ���2, �24�

Podd,USD =1

2�1 − e−�1−2�2���22�1−cos ��� ,

Podd,ent = Podd,USDe−�2��24 sin2 �. �25�

Finally, we use Ptotal,USD= Peven+ Podd,USD and Ptotal,ent

= Peven+ Podd,ent to describe the corresponding total successprobabilities.

Compared to the ultimate bounds derived in Sec. II C 2,we may now consider three different scenarios. First, themost practical scheme for entanglement generation, namely,by unambiguously identifying the state ����, projecting thetwo qubits onto a mixture of even-parity entangled Bellstates. This scheme works with a probability of Peven �whichis always smaller than 1 /2�, and does not require photon-number resolving detectors. Secondly, we consider the oddqubit subspace for entanglement generation, resulting in aslightly less practical scheme with a need for photon-numberresolving detectors; the probability here is Podd,ent. Finally,we add those patterns what resolve the states ���ei�� and���e−i�� to consider the total probability for a reasonably

practical USD scheme, Ptotal,USD. This comparison is shownin Fig. 6, where the success probabilities are replaced byfailure probabilities, as functions of the fidelity �E

2 �F in Eq.�14� with Eq. �7�. Again, moving along these curves meanschanging the parameter pair �� ,��, and hence F, for a given�. The linear-optics parameter � can be used to tune betweenthe even and the odd subspaces.

In Fig. 6, the regions below “USD bound” are quantummechanically inaccessible. The curves “USD” correspond tothose linear-optics implementations in which all conclusivepatterns for both even and odd subspaces are combined�choosing a beam-splitter parameter �=0.4�; the failureprobabilities shown correspond to 1− Ptotal,USD. The plots for“even” ��=0.7� and “odd” ��=0.01� describe those measure-ment schemes where only a single detection pattern is usedin order to project onto the respective two-qubit subspaces��click, click, no click� for even and �no click, no click, click�for odd�; the failure probabilities shown correspond to 1− Peven and 1− Podd,ent, respectively.

Here, even is less efficient, but more practical than odd, asit does not require photon-number resolving detectors. Notethat for entanglement generation, tuning the beam-splitterparameter � in order to project onto both even and odd sub-spaces at the same time �as for USD with �=0.4� does notlead to better performances. For larger distances, also for thecase of USD, beam-splitter tuning no longer helps; eitherprojecting onto the even ��=0.7� or the odd ��=0.01� sub-space is optimal in this case as well. Therefore, USD per-forms worse than odd for 30 km and beyond.

Note that the three patterns of Eq. �23� with clicks inevery mode except one unambiguously identify each indi-vidual state of the set ���� , ���ei�� , ���e−i���. In otherwords, for different �, one obtains a family of solutionsto the corresponding trinary USD problem. As the threecoherent states here are not symmetrically distributed

0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.0

0.5 0.6 0.7 0.8 0.9 1.00.50.60.70.80.91.0

0.5 0.6 0.7 0.8 0.9 1.00.900.920.940.960.981.00

0.5 0.6 0.7 0.8 0.9 1.00.99900.99920.99940.99960.99981.0000

failureprobability

fidelity

evenodd

USD USD bound

10 km 30 km

50 km 100 km

FIG. 6. �Color online� Failure probabilities as functions of thefinal two-qubit maximally entangled-state fidelities for different dis-tances. The regions below USD bound are quantum mechanicallyinaccessible. The curves USD correspond to those linear-opticsimplementations in which all conclusive patterns for both even andodd subspaces are combined �choosing a beam-splitter parameter�=0.4�. The plots for even ��=0.7� and odd ��=0.01� describethose measurement schemes where only a single detection pattern isused in order to project onto the respective two-qubit subspaces��click, click, no click� for even and �no click, no click, click� forodd�. Photon loss is again 0.18 dB per km.

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�exp�i�a†a����ei��� ���e−i�� for ��2� /3�, the successprobability of the quantum-mechanically optimal USD is ac-tually not known. For realizing optimal USD of N symmetri-cally distributed coherent states, protocols have been pro-posed by van Enk �35�, similar to the scheme here,approximately implementing the optimal N-state USD.

III. ENTANGLEMENT PURIFICATION AND SWAPPING

In the original hybrid quantum repeater proposal �19�, thefirst step is to distribute two-qubit entanglement betweennearest-neighbor stations at a high rate, but with rather mod-est fidelities. In order to achieve high-fidelity quantum com-munication over the entire repeater channel, the imperfectlyentangled qubit states must be purified; the resulting high-fidelity pairs can be used to connect the segments of thechannel through entanglement swapping. Further rounds ofentanglement purification and swapping will eventually pro-duce a high-fidelity entangled pair between the remote endsof the channel �3,4�.

Instead of using standard entanglement distillation andswapping procedures on the level of the qubits �4–7,9�, wemay also consider a purification of the imperfectly entangled,light-matter hybrid pairs, as described by Eq. �8�. In thiscase, local operations would partly act upon the qubits andpartly on the light field �see Fig. 7�. This potentially reducesthe number of qubit resources and, moreover, only high-fidelity entanglement would be transferred from the lightmodes to the qubits. Similarly, we could employ a hybridversion of entanglement swapping, where the Bell-state mea-surements are performed on the light-matter hybrid systems�Fig. 8�.

A. Hybrid entanglement distillation

Entanglement purification of optical, nonhybrid, en-tangled coherent states has been considered in Ref. �36�. Pro-

vided the initial copies of mixed entangled states are of aspecific form �rank two mixtures of a certain pair of en-tangled coherent, quasi-Bell states�, simple linear optics andphoton detectors suffice to enhance the fidelity of the en-tangled states. However, for realistic, dissipative environ-ments, the decohered states would not end up in the desiredform; additional local Hadamard-type gates �transformingGaussian coherent states into non-Gaussian superpositions ofcoherent states� would be needed in order to accomplish theentanglement purification protocol.

The situation turns out to be similar for the present hybridprotocol. As can be inferred from Eq. �8� with Eqs. �9� and�10�, the local Hadamard gates in an entanglement purifica-tion scheme must act upon the qubit states of system A andthe coherent-state superposition basis states of the qubus sys-tem B, Eq. �11�. Even though there are recent proposals toachieve such logical gates for coherent-state superpositions�“coherent-state quantum computing” �37,38��, using off-lineprepared coherent-state superpositions, the need for extranon-Gaussian, optical resources �or, possibly, extra CQED-based, optical resources� may be just as expensive as usingadditional cavity-based qubit resources. Therefore we con-clude that hybrid entanglement distillation does not appear tobe a more practical alternative to the standard distillationprocedures solely on the level of the two-qubit states.

B. Hybrid entanglement swapping

In a hybrid version of entanglement swapping, the Bell-state measurements are performed on the light-matter hybridsystems �Fig. 8�. For this hybrid Bell measurement, we canjust use the same CQED interactions as for the initial en-tanglement distribution, described by Eq. �3�; that interactionprovides the entangling gate needed for a projection onto thehybrid “Bell basis.” A subsequent Hadamard gate can beapplied to the qubit system before measuring both the qubitand the optical qubus in the “computational basis.”

More precisely, the following “Bell states” are to be dis-criminated:

��0���� � �1���ei���/2,

��0���ei�� � �1�����/2. �26�

In order to distinguish these states, an interaction gate similarto Eq. �3� is applied, where �0����→ �0���� and �1����→ �1���e−i��. The first pair of Bell states in Eq. �26� is trans-formed into ��0�� �1����� /2; in this case, measuring in theHadamard-gate-rotated qubit basis reveals the phase of theinitial Bell state. In order to additionally identify the secondpair in Eq. �26�, which will be transformed into��0���ei��� �1���e−i��� /2, apart from the qubit Hadamardgate and qubit detection, a measurement on the optical qubusmode must discriminate the unrotated coherent state from thetwo rotated ones in phase space �see Fig. 3�.

For a nearly complete Bell measurement �approximatelyidentifying any one of the four hybrid Bell states�, the tworotated coherent states must not be distinguished by the mea-surement. This could be achieved via x homodyne detection.However, as discussed previously, the distinguishability in

Qubits Qubus Modes

FIG. 7. �Color online� Purifying two copies of a light-matterhybrid entangled pair into one copy with higher fidelity �purity�through local operations on the qubits and the light modes.

Qubits Qubus Modes

FIG. 8. �Color online� Hybrid entanglement swapping throughunambiguous Bell-state measurements on the joint systems of qubitand light mode.

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phase space scales badly with distance along the x axis.Therefore, for a partial Bell measurement identifying onlyhalf of the Bell states, either p homodyne detection can beused, or, alternatively, the USD-based scheme for unambigu-ously detecting the unrotated coherent state, as introduced inthe preceding sections. In either case, p homodyne or USDmeasurement, the efficiency of the Bell measurement wouldbe limited by 1 /2.

Now using the hybrid Bell measurement for entanglementswapping �Fig. 8� means projecting subsystems 2 �the firstqubus mode� and 3 �the second qubit� of the initial pair ofentangled qubit-qubus states,

��0����12 + �1���ei��12� � ��0����34 + �1���ei��34�/2,

onto the Bell basis in Eq. �26�. According to the methoddescribed in the preceding paragraph, the interaction be-tween the qubit �system 3� and the qubus �system 2� leads to

��0,�,0,�� + �1,�,1,�ei�� + �0,�e−i�,1,�ei��

+ �1,�ei�,0,���/2. �27�

When the first qubus mode �system 2� is unambiguously de-termined to be in the state ���, a Hadamard gate on the sec-ond qubit �system 3� plus measurement in the computationalbasis yields one of two possible hybrid Bell states for thefirst qubit �system 1� and the second qubus mode �system 4�,with the phase depending on the measurement result. In or-der to obtain any one of the four hybrid Bell states of Eq.�26�, in addition, the first qubus mode �system 2� must becoherently projected onto the subspace corresponding to��ei�� , ��e−i��� �for example, via USD and photon-numberresolving detectors�.

In the above entanglement swapping scheme, clearly themost practical choice is either p homodyne measurement orUSD measurement of the unrotated coherent state. A successprobability below 1 /2 does not automatically render thisscheme inferior to the conventional, deterministic entangle-ment swapping with the two-qubit entangled states, becausein the latter case, first the entanglement needs to be distrib-uted probabilistically with a success probability of at most1 /2 using either p homodyne or USD measurements. Thisprobabilistic element is now simply incorporated into thehybrid entanglement swapping protocol. In other words, us-ing the hybrid Bell-state analysis, two initial entanglementdistributions and subsequent qubit entanglement swappingcan be done almost in one go �when the final hybrid Bellstate is again converted into a two-qubit entangled statethrough another CQED interaction and selective measure-ments with probability 1 /2�. However, the Bell-state analysisfor the two-qubit entangled states in the conventional proto-col relies upon complicated two-qubit quantum logic gates;realizable, for instance, using another four CQED-based dis-persive interactions �19�. The hybrid Bell-state analysis herewould not require any extra dispersive interactions in addi-tion to those for the initial entanglement distributions. How-ever, a drawback is that we cannot efficiently purify the hy-brid entangled states �as discussed in the preceding section�in order to combine sequences of hybrid entanglement swap-ping steps with hybrid entanglement purification steps.

IV. DISCUSSION AND CONCLUSION

In summary, we investigated the protocol for a hybridquantum repeater, based upon dispersive light-matter interac-tions between electronic spins and bright coherent light, withrespect to the different kinds of entangled states at the inter-mediate steps of the protocol and with regard to the finaloptical measurements for conditionally preparing two-qubitentangled states. As an alternative detection scheme, we pro-pose to apply USD-based measurements on the optical qubusmodes.

Compared to the homodyne-based scheme, there are vari-ous advantages of the USD-based protocol. First of all, onesource of errors can be completely eliminated from the pro-tocol, namely, those errors arising from the inability of per-fectly discriminating phase-rotated coherent states in phasespace. In the USD scheme, this imperfection only leads tosmaller efficiencies for the entanglement generation, but thefidelities are unaffected. As a result, the fidelities are solelydegraded through the decoherence effect caused by photonlosses in the communication channel. By choosing weakercoherent states, the decoherence effect can be suppressedand, in principle, repeater spacings of far beyond 10 km arepossible at the expense of smaller entanglement distributionrates. For example, initial fidelities of about 0.7 are achiev-able over 50 and 100 km with success probabilities of about1% and 0.01%, respectively, using simple on-off detectors�discriminating between vacuum and nonvacuum states�. Thefinal two-qubit entangled states here, being, in principle,ideal rank two mixtures, can be purified very efficiently. Forthe USD-based protocol, we also derived ultimate, distance-dependent bounds on the performance of the entanglementgeneration step in terms of success probabilities and fideli-ties.

The advantages of the USD-based scheme compared tothe homodyne-based scheme become apparent, as we dem-onstrated, for ideal operations, i.e., with an ideal three-portdevice and ideal phase-space displacements, and for pre-cisely known phase rotation angles. However, for the morerealistic case, including all kinds of imperfections, these ben-efits may gradually disappear. In particular, if the qubus am-plitudes are extremely weak �with only a few photons left�, itis well known that the corresponding linear-optics imple-mentation of USD is affected by dark counts; but choosingthe amplitudes small �for small phase angles, as large rota-tions are hard to achieve� means the probe states becomehighly nonorthogonal, and hence the efficiency of even theideal USD scheme drops quickly.

Conversely, in the more practical regime of the hybridrepeater, where the weak nonlinear phase rotations are effec-tively enhanced via larger probe amplitudes, stray light fromimperfect beam-splitter transformations may trigger photonclicks and lead to errors in the USD. In general, qubus am-plitudes should be chosen such that both the success prob-abilities �efficiencies� and the fidelities are reasonable at re-peater spacings potentially larger than 10 km. Only then canone decide whether experimental imperfections render thepresent USD scheme inferior to the original homodyne-basedscheme or whether it is indeed beneficial to employ USD forentanglement distribution. The overall performance of the

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repeater will then also depend on the efficiencies and thefidelities of entanglement purification and swapping. In orderto assess such more complete figures of merit and to com-pare quantum repeaters using coherent-state communicationwith those based upon single-photon transmission, furthertheoretical work is needed.

In the current work, we examined the entanglement puri-fication and swapping steps for the hybrid repeater protocolfrom a different perspective. Instead of performing thesesteps solely on the level of the two-qubit entangled states, weconsidered purification and swapping with the hybrid en-tangled states of the atomic qubit and the optical qubusmode. It turns out that entanglement purification is difficultto achieve, unless optical, non-Gaussian gates �such as Had-amard gates acting upon coherent-state superposition states�are available. Hybrid entanglement swapping, however, canbe accomplished easily with exactly the same resources asused for the initial entanglement distribution. In fact, theprobabilistic entanglement distribution steps can be incorpo-

rated into the hybrid entanglement swapping step, leading tothe same overall efficiencies as for the deterministic, qubitentanglement swapping requiring complicated, less feasiblequantum logic gates. However, a combination of hybrid en-tanglement swapping with hybrid entanglement purificationin a nested repeater protocol would again require optical,non-Gaussian gates.

ACKNOWLEDGMENTS

P.v.L. acknowledges support from NICT in Japan and theEmmy Noether program of the DFG in Germany. W.J.M.acknowledges the EU program QAP. K.N. acknowledgessupport in part from NICT and MEXT through a Grant-in-Aid for Scientific Research on Priority Area “Deepening andExpansion of Statistical Mechanical Informatics �DEX-SMI�,” Grant No. 18079014. N.L. acknowledges QAP,QuantumWorks, and the Ontario Centres of Excellence.

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