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PHYSICAL REVIEW A, VOLUME 63, 040301~R!

Bit-flip-error rejection in optical quantum communication

Dik BouwmeesterCentre for Quantum Computation, Clarendon Laboratory, University of Oxford, Parks Road, OX1 3PU Oxford, United Kingdo

~Received 23 June 2000; revised manuscript received 23 October 2000; published 16 March 2001!

An optical scheme for the error-free transfer of quantum information through a noisy quantum channel isproposed. The scheme is inspired by quantum error-correction schemes, but it avoids the currently unfeasiblerequirement for a controlled-NOT operation between single photons. The quantum communication schemepresented here rejects bit-flip errors instead of correcting them and combines quantum-measurement propertiesof three-particle entangled states with properties of the quantum teleportation protocol.

DOI: 10.1103/PhysRevA.63.040301 PACS number~s!: 03.67.2a

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I. INTRODUCTION

The possibility of detecting and correcting errors in tevolution of a quantum system has been a most remarktheoretical discovery@1–5#. This discovery, and the subsequent theoretical development of related ideas such astanglement purification@6,7#, the quantum repeater@8,9#, andfault-tolerant quantum computation@10–12#, turned the ini-tial skepticism about implementing quantum computatand long distance quantum communication into~modest! op-timism. At present, however, no practical realization of aof these ideas has been achieved in laboratories. Of particinterest would be an implementation in quantum optics, sithis would enable secure quantum cryptography and qutum communication through optical channels such as optfibers. The reason that no such implementation has beealized to date is that all theoretical schemes are basedcontrolled-NOT operations between single particles. For phtons, this operation would require a strong nonlinear intertion between individual photons, which is extremely difficuto achieve. In this paper a scheme is proposed that reerroneous transmission of photon states without uscontrolled-NOT operations.

II. QUANTUM ERROR DETECTION

In order to explain the optical scheme, we first point othe main ideas underlying classical and quantum error detion. A particularly simple classical error detection scheuses the transmission of several copies of the bits to be trferred and requires that the probability of a bit-flip error duing transmission be much smaller than unity. By comparthe copies of each initial bit after transmission, one cantermine the initial bits with high probability. Despite the fathat it is impossible to copy the state of an unknown quantstate, it is still possible to use a strategy similar to the clsical one. Consider the state of a two-level system, a qucharacterized by

uC&5au0&1bu1&.

In order to make comparison measurements after the stransmission, and thereby detect errors, we have to enthe initial qubit onto several particles. If we restrict our atention to the case in which there is a small probability tha

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bit-flip error occurs, it is sufficient to encode the initial staonto the following three-particle entangled state:

uC&12351

A2~au000&1231bu111&123). ~1!

The left-hand side of Fig. 1 indicates how this encodingobtained using two controlled-NOT operations with the initialqubit as control qubit and two auxiliary particles initiallprepared in stateu0& as target qubits.

After transmission of the three-particle entangled stthrough a ‘‘noisy’’ quantum channel, one can retrieve tinitial qubit using the comparison measurements indicatedthe right-hand side of Fig. 1. The measurements conagain of controlled-NOT operations, followed by detection othe two auxiliary particles in theu0&, u1& basis. The detectionacts as a parity check between the two particles on whichcontrolled-NOT operation acts: au0& outcome indicates thain each term of the entangled state the two particles aresame, i.e., 00 or 11; au1& outcome indicates that they aropposite, i.e., 01 or 10. If no error occurred during the tramission, both auxiliary particles should be detected instateu0&. However, if a bit-flip error occurred for the initiaparticle and not for the other two, both auxiliary particlwill be detected in the stateu1&. In the case where an errooccurred on one of the auxiliary particles, and not onremaining two, the corresponding particle will be detected

FIG. 1. Traditional scheme for the detection and correction obit-flip error. Using two controlled-NOT operations, an initial quan-tum stateuC& ~the control qubit! is entangled with two auxiliaryparticles~the target qubits!, each initially prepared in the stateu0&.After transmission of the three-particle entangled state througharea in which an error might occur, indicated by the question maeach of the two auxiliary particles becomes the target particlecontrolled-NOT operation with the initial particle as the control paticle. A final projection measurement on each of the two auxiliaparticles onto theu0&, u1& basis uniquely identifies a possibl~single! error that can then be corrected.

©2001 The American Physical Society01-1

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state u1& and the remaining auxiliary particle in stateu0&.After identification, a possible error can be corrected.

Crucial for the error detection/correction scheme isfact that the parity-check measurements project the transted entangled state onto only four possible outcomnamely, no error, or one error on one of the three particTherefore, although during the transmission throughnoisy quantum channel any qubit-rotation error can occthe final state is quantized to contain either a full bit-flerror or no error.

If more than one error occurred, the error-correctischeme is not useful. Therefore, it is crucial that the prability for an error on each particle is much smaller thunity (Perror!1). Under this condition, it is reasonableconsider, for optical quantum communication purposessimplified scheme that rejects transmissions that containerror instead of identifying a specific error and correctingit. Such a simplified scheme requires only one auxiliary pticle as shown in Fig. 2. If the parity check measuremyields theu0& result, no error took place, or, with the versmall probabilityPerror

2 , a fatal double error took place. If thmeasurement yields theu1& result, a single error occurred foone of the two particles and the transmission is invalidat

III. AVOIDING CNOT OPERATIONS

To present our error-free optical quantum communicatscheme, we note that the controlled-NOT operation in thepreparation step of the schemes shown in Figs. 1 andused in order to encode anarbitrary initial quantum stateonto a multiparticle entangled state. It is, however, not nessary to be able to encode an arbitrary input state. Accing to the teleportation scheme@13#, illustrated in Fig. 3, thetransmission of an arbitrary quantum state can be decposed into the transmission of aknown entangled state, alocal Bell-state measurement, and the transmission of clacal information. Therefore, in order to establish error-frquantum communication, it is sufficient to be able to excluerroneous transmission of one of the particles of a fixedtangled state.

Consider a pair of entangled photons in the state

uC&2351

A2~ u0&2u0&31u1&2u1&3). ~2!

To be able to detect errors on the transmission of, say, pton 2, the preparation scheme shown on the left-hand sidFig. 2 would produce the state

FIG. 2. Scheme for bit-flip error rejection. One auxiliary particis sufficient in order to detect an error, without revealing on whparticle the error occurred.

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A2~ u0&2u00&341u1&2u11&34). ~3!

Since state~3! is a well-defined state, the use of thcontrolled-NOT operation is no longer necessary, as shoon the left-hand side of Fig. 4.

The right-hand side of Fig. 4 illustrates how thcontrolled-NOT operation for parity checking can also bavoided by using a polarizing beam splitter and a coindence detection measurement in an appropriate basis.bit-flip error occurred for one of the two transmitted photonboth photons will exit the polarizing beam splitter in th

FIG. 3. Schematic drawing of the quantum teleportation procol. The transmission of the unknown quantum stateuC& of particle1 is broken down into the distribution of an auxiliary pair of etangled particles~2 and 3!, a Bell-state measurement on particlesand 2 ~i.e., a projection onto a complete basis of maximally etangled particles!, and the transfer of classical information~the out-come of the Bell-state measurement!. After receiving the classicainformation, the relation of the state of particle 3 to the initial stauC& is fully determined. The initial state can therefore be recoveby a well-defined unitary transformationU on particle 3.

FIG. 4. Scheme for error-free quantum-state transmission wout controlled-NOT operations. In order to transfer a quantum statis sufficient to restrict the use of a quantum channel for the tramission of one of an entangled pair of particles~see Fig. 3!. In orderto reject erroneous transmissions, a three-particle entangled stused. Two of the three entangled particles are sent through‘‘noisy’’ quantum channel. A parity check measurement on pticles 2 and 3 identifies an error-free transmission and is obtaby using a polarizing beam splitter followed by a coincidencetection of one particle in arma and the other in armb. The mea-surement in armb must be such that the remaining two particles aprojected onto a well defined two-particle entangled state. Thiachieved by performing the measurement in the linear basis rot45° with respect to theu0&, u1& basis. After the result of the measurement on the particle in armb is known, the remaining particles~one to be detected in arma and particle 1! are guaranteed to be ina well defined entangled state and can be used for error-free qtum teleportation or quantum cryptography.

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BIT-FLIP-ERROR REJECTION IN . . . PHYSICAL REVIEW A63 040301~R!

same output arm. Therefore, no coincidence will be obserbetween the detectors in armsa andb, and the transmissionwill be invalidated.

If no error occurred, the state after the polarizing besplitter will have one photon in each output arm, indicatithat the two outgoing photons have the same polarizarelation as when initially prepared, i.e., the polarizationsparallel in each term of the entangled state~see note added inproof!. The detection scheme proceeds by detecting theticle in armb in the basis

u08&51

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1

A2~ u0&2u1&). ~4!

The specific measurement outcome corresponds to a prtion of the remaining particles, provided a particle is presin arm a, onto one out of two well defined pure two-particentangled states:

u08&b→1

A2~ u0&2u0&a1u1&2u1&a), ~5!

u18&b→1

A2~ u0&2u0&a2u1&2u1&a). ~6!

The teleportation procedure can now be completed by a Bstate measurement and the transfer of classical informaas illustrated in Fig. 4.

IV. POSTSELECTION

One might be alarmed by the fact that the photon in armahas still to be detected in order to complete the error-ftransmission scheme. This will in practical applications iply the destruction of the photon, although absorption-fdetection of single photons has been demonstrated exmentally @14#. The fear of losing the photon before beinable to use it is unjustified, at least for applications in qutum cryptography and other quantum communication procols, since the detection of the photon is an integral parall such applications. In fact, any realistic single-photcommunication scheme needs a final verification stepguarantee that the fragile photon survived the transmissThe detection of the photon, therefore, plays the doubleof enabling a projection onto a pure entangled state for pton 2 and the photon in arma, as well as exploring thisentanglement for quantum cryptography or for quantcommunication purposes. The same arguments hold forof Bell’s inequalities based on postselected entangled ptons @15#, for the Innsbruck quantum teleportation and etanglement swapping experiments@16,17#, and for the recentGreenberger-Horne-Zeilinger~GHZ! entanglement experiments@18,19#.

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V. ERROR-FREE QUANTUM CRYPTOGRAPHYTHROUGH HOSTILE QUANTUM CHANNELS

We will now briefly consider the importance of thpresent scheme for the security of quantum cryptogra@5,20,21#. A major threat to the security of quantum cryptoraphy based on entangled state distribution is that thetanglement distribution established between the two usein general not perfect. The users have to agree on a minimlevel of security corresponding to a certain purity of the dtributed entanglement. There are many technical reasonsthe accidental loss of entanglement~transmission through anoisy quantum channel!, but one should also take into account the possibility that a third party is deliberately intrducing small errors on what could otherwise be a perfquantum channel~transmission through ahostile quantumchannel!. Since the users cannot distinguish between a noand a hostile quantum channel, they are forced to use imfect entangled photon pairs for the distribution of a seckey at the risk of leaking some of the secret key to a thparty. The scheme presented in this paper resolves both‘‘noisy’’ and the ‘‘hostile’’ quantum-channel problems. Thusers will obtain entangled photon pairs with the same puas the initially created three-particle entanglement despitefact that the quantum channel is imperfect. Any accidentadeliberately induced errors on the transmission will be stematically rejected. Note that a quantum key-distributiprotocol based on the proposed scheme for error-freetanglement distribution will involve one more classical communication step in addition to the familiar classical commnication steps, namely, the publication of the measuremresult on the photon in armb.

VI. DISCUSSIONS AND CONCLUSIONS

We have restricted our attention to the error-free transmsion of half of an entangled pair. The scheme can easilyextended to error-free distribution of both particles of an etangled pair as illustrated in Fig. 5. Furthermore, we ondiscussed bit-flip errors. In addition, there could be pherrors. Error detection/correction schemes have been deoped for correcting general errors consisting of both bit-fland phase errors. Such schemes involve at least four aiary particles and more elaborate preparation and detecprocedures. It remains to be seen whether an all-optscheme is possible to reject both bit-flip and phase error

Currently we are working towards an experimental reization of the scheme presented in this paper. At first glan

FIG. 5. Generalization of the error-free transmission of onean entangled pair of particles to the error-free distribution of bentangled particles. Starting with a four-particle entangled stateerror-free entangled pair of particles can be obtained.

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it seems that previous experiments on quantum teleporta@16# and three-photon entanglement@18# provide all the nec-essary techniques to implement the scheme. Unfortunathe source for three-particle entanglement reported in R@18,19# is based on a postselection detection methodfilters out the appropriate three-photon entanglement frovariety of other photon states. Since in the proposed schtwo of the three entangled photons are recombined obeam splitter before detection, the postselection cannoapplied. What seems to be needed is a three-photon sothat produces three spatially separated outgoing photonsgenuine three-photon GHZ state. Methods of encoding inmation on more than one degree of freedom of single ptons ~polarization and momentum! might also be employedto achieve an experimental realization@22–24#.

Finally, we point out the generality of the two main ideof this paper. First, errors on the evolution of entang

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states can be detected by starting with higher-order entanstates. Second, it appears that an experimental implemetion of the controlled-NOT operation, or any other universatwo-qubit quantum gate, is not crucial for an experimendemonstration of the essence of a variety of quantum cmunication protocols.

Note added in proof:Recently research avoidingCNOT

operations in an optical entanglement purification schehas been reported@25#.

ACKNOWLEDGMENT

I am very grateful to John Rarity, Sandu Popescu, AnLamas Linares, Christian Mikkelsen, and Micheal Seevinfor stimulating discussions and useful suggestions. Twork was supported by the EPSRC GR/M88976 and theropean QuComm~ISI-1999-10033! projects.

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