ConstructionofSeveralDeterministic Quantum Channels

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1 第 9 卷第 0 期 No.0Vol.9 JOURNALOFNEIJIANG NORMALUNIVERSITY ConstructionofSeveralDeterministic Quantum Channels PENGJiaȳin MOZhi-wen * * (.SchoolofMathematicsandInformationScienceNeijiangNormalUniversityNeijiang Sichuan6499China;.SchoolofMathematicsandSoftwareScienceSichuan NormalUniversityChengdu60066China) Abstract:Teleportationschemesbasedonprobabilisticchannelsusualyrelyheavilyontheimplementationof higẖdimensionalunitaryoperationswhilehigẖdimensionalunitaryoperationsareverydifculttodirectlyimplement inphysicsexperiments.thispaperdescribeshowtoconstructdeterministicquantumchannelsviaseveralcommonly probabilisticchannels.inourschemesthesenderorreceiverneedstointroduceoneormoreauxiliaryqubitsinto theprobabilisticchannelsothatthesequbitsareincorporatedandentangledwiththeoriginalqubitsandthenexecuteaprojectivemeasurementorapositiveoperatoṟvaluedmeasure(povm)onher/hisqubitsincludingtheanciḻ lastoestablishdeterministicchannelbetweenthesenderandher/hisagentswithacertainprobability.inaquantum teleportationnetworktherearegeneralymultispatialyremoteagentswhichplaytheroleofrelaynodesbetweena senderandadistantreceiver.hencewepresentaschemeforconstructingadeterministicchannelbetweenthe senderandthedistantreceiverwiththeassistanceofrelayagentsrespectively. Keywords:probabilisticteleportation;deterministicchannel;multi-particlestate;projective measurement; POVM measurement DOI:0.3603/j.cnki.5-6/z 中图分类号 :O43. 文献标志码 :A 文章编号 :67-785(04) Introduction Quantum teleportation the disembodied transportofaquantumstatebasedonthenonlocal propertiesofanentangledstateresourcehasbeen demonstratedtobeoneofthe mostpeculiarand fascinatingaspectsofquantuminformationtheory. AfterBennetetal. [] haveproposedtheteleportationtheoryquantum teleportationhasplayedan importantroleintherapidlyevolvingquantumiṉ formationand quantum communications.it may have applications in quantum cryptography [-4] quantum computer [5-6] and quantum dense coḏ ing [7].Uptonowpeoplehaveputforwardvariouskindsofteleportationschemes [8-].Inthese protocolsdiferentquantumchannelsconsistingofvarious entangledstateshavebeenexploredinquantumtelepoṟ tationsuchas GHZstats [9] W states [0-] Graph states [] Belstates [7-8] andsoon.recently manypapersarefocusedonthepropertiesofmulti- 收稿日期 : 基金项目 : 国家自然科学基金项目 (0778) 作者简介 : 彭家寅 (96-) 男 四川资中人 内江师范学院教授 博士. 研究方向 : 量子通信. 通讯作者 : 莫智文 男 四川师范大学教授. 研究方向 : 量子通信.

2 第 9 卷第 0 期 particleentangledstatesasquantumchannels [3-7] especialyclusterstatessuchasfouṟqubitstates [4] a genuine five-qubit states [55-6] and sixqubit states [57].Itisknownthatthen-qubit(n > 3) clusterstateismaximalyconnectedwiththebeṯ terpersistencythanthe GHZ statealsocluster statesarerobustagainstdecoherence.ontheother handpeoplealsofocus moreatentions on the noṉmaximalyentangledstatesasquantum channels in quantum teleportation schemes [8-33].Asaresultthe quantum teleportation schemescan be classfiedintotwotypes.oneworksinadetermiṉ istic.mannerwherepureentangledstatesareused; whilethe other worksin a probabilistic manner wherepartlypureentangledstatesareemployed. Intheprobabilisticschemesusualythereceiver needstointroduceoneormoreauxiliaryqubitsand perform highdimensional unitary operations to obtaintheoriginalunknownstate withacertain probability.howeveriftheteleportationfails theunknown quantum information wilbecompletelyruined.moreoverhigh dimensionalunitary operationsareverydificulttorealizeexperimeṉ taly.thecurrentschemesforthe probabilistic teleportationviaanoṉmaximalyentangledstate havebeenproposedwithonlytwoorthreeparties involved. Howeverin a realistic situationthe longdistancecommunication networkconsistsof manyspatialydistributednodes.inquantumteleportationnetworkmultispatialyremoteagents playing the role of relay nodes are necessary betweenasenderandadistantreciever.sofar thereareno knownreportsusingaprobabilistic channeltorealizethe deterministicteleportation toachievetheteleportationorprobabilistictelepoṟ tation oftwo-particle and multi-particle states. Thesepotentialbarriersbringinconvenienceinto without use of higẖdimensional unitary operations.furthermorethereisnouniformapproach theprocessoftheconstructionofaquantumtele- portationnetwork.inthispaperweproposesev- eralschemestoconstructcommonlyusedfaithful channelviaprobabilisticchannel.inourschemes wenotonly cantransform probabilisticchannel into deterministicchannelbutalso successfuly solve the problem of probabilistic teleportation involvingmultirelayagents. ConstructionoftheEPRchannelviaanon-maximalyentangledtwoparticlestate Suppose that a quantum channel between AliceandBobisestablishedasfolows φ > =α 00> +β > () whereαandβarenonzerorealmemberswith α + β = and α β.particlesandbelongtoaliceandbobrespectively. Inordertorealizethefaithfulteleportation Aliceneedstointroduceanauxiliaryparticle3with theinitialstate 0> 3.Thejointsystemofparticles and3canbewritenas φ > 0> 3= α 000> 3+ β 0> 3= [ E> 췍 G> 3+ F> 췍 H> 3 ] () where E> = ( 00>+ >) F> = ( 00>- >) G> 3= (α 0>+β >) 3 H> 3= (α 0>-β >) 3.ThenAlice saimishowtodistinguishtwo nonorthogonalstates G> 3 and H > 3.Statediscrimination measurementshavebeenstudiedand analyticalexpressionsforoptimalpovmformany quantum decision problems [8-0] have been oḇ tained.atthisstage Aliceperformsan optimal POVMtoconclusivelydistinguishbetweenthetwo nonorthogonalquantum states G> 3 and H > 3. The respective operators thatform an optimal POVMinthissubspaceare where U= β M><M U= β M><M U3=I- Mi ><Mi β i= M>= αβ ( a 0 >+ β >) 3 M>= αβ ( a 0 >- β >) 3 whileiisanidentityoperator.wecanrewritethe threeoperatorsuuandu3inthematrixforms

3 04 年 0 月 PENGJia-yinMOZhi-wen:ConstructionofSeveralDeterministicQuantum Channels 3 U= β β ( αβ ) αβ α U= β β ( -αβ ) -αβ α U3 = ( α β ) ( 3) AfterperformingthePOVMoperationontheauxiliary particle 3 Alice can definitively get Ui(i =3)withtheprobability p(ui)=3<0 < φ Ui φ > 0> 3=α (i = ). AlternativelyintermsofthePOVM valuealice canpositivelyconcludethestates G> 3and H> 3of particle3.howeverwiththeprobability - α AlicemaygetU3 svalue.inthiscasealicecaṉ notinfer whichstatetheparticle3arein.once Alicedeterminesthe G> 3 and H > 3 this means shealsoknowsthestat E> and F>.Asacoṉ sequencethestateofandtransformsintoone ofthefolowingtwostates ( 00>+ >) ( 00>- >). Thefirststateisdesired. Whentheoutcomeis ( 00>- >) Alice needsto perform the unitaryoperationu = 0><0 - >< onherparticle.henceafaithfulchannelcan beestablished betweenaliceandbobwiththeprobabilityα. Itiseasytostraightforwardlygeneralizethe constructionofthe EPR channelto of multi-particle GHZ-type channel case.in the generalization supposethat Alicesharesa quantum state with m-agentsbobbob Bobm viathefolowinga multi-particleunknownquantumchannel τ>= α 0A0B >+β AB B m > wheretheparticlesof 0A>= 00 0n>and A>= n>belongto Alicewhile particlesof 0B i >= 0i 0i 0i mi >and B i >= i i i mi > belongtobobi(i = m).herebothαand βarerestrictedtotakenonzerorealvalues with α + β = and α β. InordertosharethestatesimplyAlicecan transferaltheinformationofamulti-particlehgz typestate τ>intoalice sparticleandthebob s particleswhichstateis τ >= α 00B >+β B B m >. Anditcangeneratedinthefolowing way:alice carriesoutn-controleḏnotoperationsonher nparticleswithparticleascontroledparticleand Ψ>= (α 00B >+ β B B m >) 췍 00 0> 3 n. (4) Nowwe note that altheinformation ofthe τ>havebeentransferredintothestate τ >= α 00B >+β B Bm >. (5) Soweonlyneedtoconsiderthestate τ >.After thataliceintroducesanauxiliaryparticlec with theinitialstate 0> C.Thejointsystemofthestates 0> Cand τ >canbewritenas τ > 0> C= α 00B 0C>+β B B m 0C.>= where [ E> 췍 G> C+ F> 췍 H> C (6) E>= 00B >+ B B m > E>= 00B >- B B m > G> C=(α 0>+β >) C H> C= (α 0>-β >) C. BytheabovemethodAlicecangetthestate Φ >= ( 00B >+ B B m >) (7) withtheprobabilityα.finalyaliceintroduces n-particles3 nininitialstate 00 0> 3 n thejointsystem ofthe 00 0> 3 nand the Φ >ineq.(7)is Φ > 췍 00 0> 3 n= ( 00B >+ B B m >) 췍 00 0> 3 n. (8) ThenAlicesendshernparticlesthroughn-coṉ troleḏnot gates with particle as controled particleandeach ofhern- particlesastarget eachofn-particles3 nastargetparticles.thestate τ>becomes particlesthestateineq.(8)becomesbyregroupingterms ( 0A0B >+ AB Bm >). (9) Whichmeansthatadeterministicchannelbeestaḇ lished among Alice and Bobs with the success probabilityα.if α = β thesuccessprobabilitybecomes.

4 4 第 9 卷第 0 期 ConstructionoftheW channelviaanon-maximalyentangledthree-particlestate Nowconsiderthisfolowingsituation:Alicewantsto teleportanunknowntwo-particlestatetobob [34].In [34] AliceandBobsharethefolowingquantumchannel φ > 3= (α 00>+β 00>+γ 00>) 3 (0) wherethecoeficientsarenonzerorealnumbersthat satisfynormalizedcondition α + β + γ = and α is smalerthan the other.the particle 3 U4= belongtothesenderalicewhiletheothertwopaṟ ticlesandbelongtothereceiverbob.inorder torealizetheteleportationbobintroducedanaṉ cilaryparticle4 withtheinitialstate 0> 4.The stateofparticles3and4 maybeexpressedas φ > 3 0> 4= (α 000>+ β 000>+γ 000>) 34. () Bobperformsthefolowingunitarytransformation ontohisparticlesandtheancilaryparticle4 æ ö 0 - α -( α β β ) ( α β ) α β α ( α () r γ ) ( α γ ) - α 0 0 γ è ø thenthestateofparticles3 and4 wilbe transformedintothefolowingform U4 φ > 3 0> 4=α( 00>+ 00>+ 00>) 3 0> 4+( β -( α β ) 000> 3+ γ -( α γ ) 00> 3 ) > 4. (3) AfterthisBob makesavon Neumann measurementontheancilaryparticle4.iftheresultis 0> 4 thestateofand3becomes 4<0 U4 φ > 3 0>4=α( 00>+ 00>+ 00>) 3 (4) whichmeansthatafaithfulchannelcanbeestaḇ lishedbetweenaliceandbob.herewedonotnoṟ malizethestatein (4)forconvenience.Ifthe resultis > 4 theestablishmentprocessfailsand weshouldstartanotherestablishment. From (3)wecanfindthatthesuccessproḇ abilitytoobtain 0> 4is3α.If α = β = γ thesuccessprobabilitybecomesandtheancilary particleisnotneeded. 3 Constructionofthefour-particlequantumchannelvia anunknownfour-particleclusterstate Supposethere are threelegitimate parties traditionalycaledalicebobandcharlie.aliceis the sender of quantum information Bob and Charliearetwoagents.Alicewantstoteleportan unknownsingle-particlestatetoanyoneofthetwo agents Bob and Charlie.Initialythe quantum channelwhichisdiferentfrom thefouṟparticle clusterstateslightly φ > 34= (α 0000>+β 00>+ γ 00>+δ >) 34 (5) sharedbyparticles (3)and4 belongingto AliceBobandCharlierespectivelyhasbeenset up.herethecoeficientsα β γandδarenonzero realnumberssuchthat α + β + γ + δ = and α issmalerthantheother. This scheme may also be applied to the famousexampleproposedbynieliandliu [35].In theirsystemalicebobandcharliesharethefoḻ

5 04 年 0 月 PENGJia-yinMOZhi-wen:ConstructionofSeveralDeterministicQuantum Channels 5 lowingfouṟparticlestate ( 0000>+ 00>+ 00>- >) i.e.thisstateasthequantumchannelaresetup amongalicebobandcharliewhichisaspecial caseof φ > 34inEq.(5). InordertoachievetheteleportationAliceiṉ troducestwoauxiliaryparticles5and6ininitial state 00> 56.Asaconsequencethejointstateof theparticles345and6is φ > 34 00> 56= (α 0000>+β 00>+ γ 00>+δ >) 34 00> 56. (6) Then AliceperformstwocontroleḏNOT operationswithand3asthecontroledparticleswhile theancilaryparticles5and6asthetargetones respectively.theabovestateistransformedinto thefolowingform by operatingtwocontroleḏ NOTtransformation φ > 3456= (α >+β 000>+ γ 000>+δ >) 3456= 4 ( R> 34 췍 S>56+ R> 34 췍 S> 56+ where R3> 34 췍 S3> 56+ R4> 34 췍 S4> 56 (7) R> 34= 0000> > > 34+ > 34 S> 56=α 00> 56+β 0> 56+γ 0> 56+δ > 56 R> 34= 0000> > 34-00> 34- > 34 S> 56=α 00> 56+β 0> 56-γ 0> 56-δ > 56 R3> 34= 0000> 34-00> > 34- > 34 S3> 56=α 00> 56-β 0> 56+γ 0> 56-δ > 56 00> 34+ > 34 S4> 56=α 00> 56-β 0> 56-γ 0> 56+δ > 56. Inordertoget Ri> 34 (i = 34)Alice adoptstoperformsanoptimalpovmontheauxiḻ iaryparticles5and6toobtain Si> 56 (i = 3 4).ThePOVMtakesthefolowingforms Ui= ω Mi ><Mi>(i = 34) U5=I - 4 ω Mi><Mi (8) i= where M>= ξ( α 00+ β 0+ γ 0 >+ δ >) 56 M>= ξ( α 00+ β 0- γ 0 >- δ >) 56 M3>= ξ( α 00- β 0+ γ 0 >- δ >) 56 M4>= ξ( α 00- β 0 >- γ 0 >+ δ >) 56 ξ= α + β + γ + δ Iisanidentity operatorandthecoeficientω whichrelatestoα β γandδshouldbeableto R4> 34= 0000> 34-00> 34- assureu5tobeapositiveoperator.toexactlyde- termineωwewouldliketorewritethefiveopera- torsuuu3u4andu5inthematrixforms æ ö α αβ αγ αδ U= αβ β βγ βδ ωξ αγ βγ γ γδ èαδ βδ δγ δ ø æ α αβ - αγ - ö αδ U= - αβ β βγ - βδ ωξ - αγ - βγ γ γδ - αδ - è βδ δγ δ ø æ α U3= - αβ ωξ αγ - è αδ æ U4= - αβ ωξ - αγ è - αβ αγ β -βγ - βγ βδ γ - δγ α - αβ - αγ β βγ βγ γ αδ - βδ - δγ - ö αδ βδ - γδ δ ø ö αδ - βδ - γδ δ ø

6 6 第 9 卷第 0 期 æ ωξα ωξβ U5= è ö ωξγ ωξδ ø ObviouslytoletU5 beapositiveoperatorthe parameter ω 4β γ δ α β γ +α β δ +α γ δ +β γ δ. Afterperformingtheabove POVM operation on theauxiliaryparticles5and6alicecangetui(i= 34)withtheprobability p(ui)=3456< φ Ui φ > 3456= 4 ωξ ( i = 34). AlternativelyintermsofthePOVM valuealice canpositivelyconcludethestate Si> 56 (i=34). HoweverAlicegetsU5 svaluewiththeprobability - 6 ωξ.soalicealsoknowsthestate R-i > 34 (i = 34).ByperformingtheunitarytransformationI( 0><0 - >< ) ( 0><0 - >< ) 3and ( 0><0 - >< ) 췍 ( 0><0 - >< ) 3 respectivelyalicecanobtainthefoḻ lowingstateofparticles3and4 ( 0000>+ 00>+ 00>+ >) 34 (9) which meansthatafaithfulchannelcanbesetup amongalicebobandcharliewiththeprobability 6 ωξ.if α = β = γ = δ = thesuccessprobabilitybecomesandtheancilary particlesarenotneeded. Remark (a)intheaboveanalysisprocess BobandCharliedonothaveanyactionsoifAlice sharesthequantumchannel φ > 34inEq.(5)with Bobforteleportationofatwo-particlestatewhere theparticlespairs (3)and (4)belongingto Alice and Bobrespectivelythe result ofthe faithfulconstructionofquantum channelbetween AliceandBobisthesameformasEq.(9).SymmetricalyBobcanalsoadjusttheabovefouṟpaṟ ticlestateineq.(5)toestablishfaithfulfouṟpaṟ ticlequantumchannelbyusingsimilarmethod. (b)considerthefolowing situation:alice wantstoteleportanunknownone-particlestateto Charliewhilethereisnodirectquantum channel betweenthem.howeveraliceandbobsharethe folowingquantumchannel φ > AB = (α 00>+β >) AB (0) intwodiferentplaces.aquantumchannel φ > B C= (γ 00>+δ >) B C () sharedbybobandcharliehasbeensetup.here theparametersα β γandδarenonzerorealnumbersandtheysatisfy α + β = γ + δ = α β and γ δ.particlesa BandBC belongto AliceBoband Charlie respectively.theinitialstateofthejointsystemof quantumchannelscanbewritenas φ > AB B C= (α 00>+β >) AB 췍 (γ 00+ δ > B C= (αγ 0000>+αδ 00>+ βγ 00>+βδ >) AB B C. () ItisclearthattheabovestateofEq.()iscoṉ sistentwitheq.(5).soapplyingsimilarmethod proposed beforebob can obtain the folowing state ( 0000>+ 00>+ 00>+ >) AB B C Charlieisbuiltsuccessfuly. whichhasthesameform aseq.(9).sothedeterministicquantumchannelamongaliceboband Theotheroneschemeisthedirectconstructionofthequantum channelbetweenthesender AliceandthereceiverCharliewiththehelpofBob. InthisschemeBob first performs a Beḻstate measurementon hisparticlesband Bandiṉ formsaliceandcharlieofhismeasurementresult. Thenthejointstatein Eq.()wilcolapseinto oneofthefolowingstates B B < ϕ ± φ > AB B C= (αγ 00>±βδ >) AC (3)

7 04 年 0 月 PENGJia-yinMOZhi-wen:ConstructionofSeveralDeterministicQuantum Channels 7 B B < ψ ± φ > AB B C= (αδ 00>±βγ >) AC. (4) Herethe probabilities to obtain the Bel state ϕ ± > B B and ψ ± > B B are α γ + β δ and α δ + β γ respectively.accordingto themeasurementoutcomes ϕ + > B B ϕ-> B B ψ+> B B and ψ - > B B AliceperformstheIσz σx orσxσzoperationontheparticleawhereσzand σxaretheusualpaulimatricesandiistheidentity matrix.thenthechannelbetweenaliceandchaṟ liewilbesetupasoneofthefolowingforms AC (αγ 00>+βδ >) (αδ 00>+βγ >) AC. (5) EvidentlytheabovestatesofEq.(5)areconsistent witheq.().hencealiceandcharliecanusethe methodproposedinsectiontoadjustthechannel whentheyreceive Bob sresult.andthetotal probabilityofsuccessis ( α γ + β δ ) α γ + α γ + β δ ( α δ + β γ ) α δ α δ + β γ = α (( α γ + β δ ) α γ + α γ + β δ ( α δ + β γ ) β γ α δ + β γ = γ if αδ βγ ( αδ > βγ ). (c)nowconsidertogeneralizetheconstructionof fouṟparticlechannelin (b)involvingthreerelay agentsto multiagents.supposethatthereisa quantum teleportation network with n agents wherethe sender Alicereceiver Bob and n - intermediatespatialyseparatedagentsareinvolved. Alice wantstoteleportan unknown oneparticle statetobob withassistanceofm-intermediate agentswhereagenti (i = n-)shares thefolowingchannelwiththenextadjacentagent i + (αi 00>+βi >) B i B i. (6) Agenti(i = 3 n - )possessestheparticles pair (Bi-B i).particlesb andbn- belongto Aliceand Bobrespectively.Theinitialstateof thejointsystemcomposedofparticlesb iandbi(i= n-)canbewritenas φ > B BB B B n-bn-= 췍i=(αi 00)+βi >) n- B B. (7) i Agents makesabelstate measurementonhis particlesbandb andtheninformshisnextaḏ jacentnodeofhis measurementresult.thestate ofwholesystem becolapsedintooneofthefoḻ lowingforms B B < ϕ ± φ > B B B B B n- B n- = (αα 00>± ββ >) B B 췍i=3(αi 00)+βi >) n- B i B i B B < ψ ± φ > B B B B B n- B n- = (αβ 0>± αβ 0>) B B 췍i=3(αi 00)+βi >) n- B i B. i According to received outcomes ϕ + > B B ϕ - > B B ψ + > B B and ψ - > B B Aliceperforms theiσzσx orσzσx operationontheparticleb the whole system wil becomeinto one ofthe folowingstates (αα 00>+ββ >) B B 췍 n- i=3 (αi 00)+βi >) B (αβ 0>+αβ 0>) B B 췍 n- i=3 (αi 00)+βi >) B i B i i B i. Afterthatagenti(i = 34 n-)performsa similaroperationsuchasoperatinganappropriate unitarytransformationonhisparticlebi-makinga Belstate measurementon hisparticlesbi- and B i andinforming hisnextadjacentagentofhis measurementresult.finalythestateparticlesof B andbn-is n- (α n- i=λi 00>+β n- i=λi >) B B n- (8) whereλi=αiandλi=βi(resp.λi=βiandλi=αi) if measurement result is ϕ ± > Bi- B i ( resp. ψ ± > Bi- B i ) i=3 n -.ThestateofEq.(8) isalsoconsistentwith Eq.()so AliceandBob canusethemethodproposedinsectiontoadjust

8 8 第 9 卷第 0 期 thechannelwiththeprobability n- (( α n- i= + β n- ) λi i= λi χ= min( α n- i= λ χ (i) β n- i= λ χ (i) ) α n- i= λi + β n- i= λi )= n- min( α n- λ i= χ β (i) n- λ i= χ ). (i) χ= 5 Construction offive-particlequantum channel viaanunknownfive-particleclusterstate SupposethatthesenderAlicewantstoteleportan arbitraryunknowntwo-particlestatetooneofthe twoagentsboband Charlieandthereceivercan reconstructthestateonlywhenhe/sheobtainsthe help ofthe other agent. The quantum channel whichissharedbyalicebobandcharlieisunknown five-particleclusterstate φ > 345= (α 00000>+β 00>+ γ 0>+δ 00>) 345 (9) whereparticlespair(5)belongtoaliceparticles and3 belongto Boband Charlieandparticle4 belongsto eitherone ofthereceiver. Herethe coeficientsα β γandδarenonzerorealnumbers suchthat α + β + γ + δ = and α is smalerthantheother.supposethataliceintends torestoretheoriginalstateinbob splace. Thisschemecanalsobeappliedtothefamous exampleproposedbyliandhou [36].Intheirsystem Alicewantstoteleportanarbitraryunknowntwoparticlestatetowiththecooperationfrom Charlie usingthestate Ψ>= ( 00000>+ 00>+ 0>+ 00>) (30) asthequantumchannel.soinordertorealizeour teleportationweneedtoconstructthequantum channelstate Ψ>inEq.(30)viathestate φ > 345 ineq.(9). FirstAliceintroducestwoauxiliaryparticles6 and7ininitialstate 00> 67.Asaconsequencethe jointstateoftheparticles3456and7is φ > > 67= (α 00000>+β 00>+ γ 0>+δ 00>) > 67. (3) Then AliceperformstwocontroleḏNOT operationswithand5asthecontroledparticleswhile theancilaryparticles6and7asthetargetones respectively.theabovestateistransformedinto thefolowingform by operatingtwocontroleḏ NOTtransformation φ > 34567= (α >+β 000>+ γ 0>+δ 000>) = 4 ( K> 345 췍 Q> 67+ K> 345 췍 Q> 67+ K3> 345 췍 Q3> 67+ K4> 345 췍 Q4> 67 (3) where K> 345= 00000> > > > 345 Q> 67=α 00> 67+β 0> 67+γ > 67+δ 0> 67 K> 345= 00000> > 345-0> > 345 Q> 67=α 00> 67+β 0> 67-γ > 67-δ 0> 67 K3> 345= 00000> > > > 345 Q3> 67=α 00> 67-β 0> 67+γ > 67-δ 0> 67 K4> 345= 00000> > 345-0> > 345 Q4> 67=α 00> 67-β 0> 67-γ > 67+δ 0> 67. Inordertoget Ki> 345 (i = 34)Alice adoptstoperformsanoptimalpovmontheauxiḻ iaryparticles6and7toobtain Qi> 67 (i = 3 4).ThePOVMtakesthefolowingforms Vi= ><Mi (i ω Mi = 34) V5=I - 4 ω Mi><Mi (33) i= where M>= ( ξ α 00 >+ β 0 >+ γ >+ δ 0 >) 56 M>= ( ξ α 00 >+ β 0 >- γ >- δ 0 >) 56

9 04 年 0 月 PENGJia-yinMOZhi-wen:ConstructionofSeveralDeterministicQuantum Channels 9 M3>= ξ ( α 00 >- β 0 >+ γ >- δ 0 >) 56 M4>= ξ ( α 00 >- β 0 >- γ >+ δ 0 >) 56 ξ= α - β - γ + δ Iisanidentity operatorandthecoeficientω whichrelatestoα β γandδshouldbeabletoassurev5tobeapositiveoperator.toexactlydeteṟ mineωwewouldliketorewritethefiveoperators VVV3V4andV5inthematrixforms æ ö α αβ αδ αγ V= αβ β βδ βγ ωξ αδ βδ δ δγ èαγ βγ γδ γ ø æ α αβ - αδ - ö αγ V= - αβ β βδ - βγ ωξ - αδ - βδ δ δγ - αγ - è βγ γδ γ ø æ - α αβ - αδ V3= - αβ β βδ ωξ - αδ βδ δ è αγ - βγ - γδ æ α V4= - αβ ωξ αδ - è αγ - αβ β - βδ βγ αδ - βδ δ - γδ ö αγ - βγ - δγ γ ø - ö αγ βγ - δγ γ ø æ ωξα ωξβ V5= è ö ωξδ ωξγ ø ObviouslytoletV5 beapositiveoperatorthe parameter ω 4β γ δ α β γ +α β δ +α γ δ +β γ δ. Afterperformingtheabove POVM operation on theauxiliaryparticles6and7alicecangetvi(i = 34)withtheprobability p(vi)=34567< φ Vi φ > 34567= 4 ωξ ( i = 34). AlternativelyintermsofthePOVM valuealice canpositivelyconcludethestate Qi> 67 (i = 34).HoweverAlicegets V5 svalue withthe probability - 6 ωξ.soalicealsoknowsthestate Ki> 34 (i = 34).Byperformingtheunitary transformationi( 0><0 - >< ) ( 0><0 >< ) 5 respectivelyalicecanobtainthefoḻ lowingstateofparticles3and4 ( 00000>+ 00>+ 0>+ 00>) 345 (34) which meansthatafaithfulchannelcanbesetup amongalicebobandcharliewiththeprobability 6 ωξ.if α = β = γ = δ = thesuccess probabilitybecomes andtheancilary particles arenotneeded. Remark Wecanthestate Ψ >in Eq.(30) - >< ) 췍 ( 0><0 - >< ) 5and( 0><0 - usingthemethodinsection3.infactaliceintroducesanancilaryparticle6 withtheinitialstate 0> 6 thenstateofparticles345and6 maybewritenas φ > 345 0> 6= (α >+β 000>+ γ 00>+δ 000>) (35) Alicethenperformsthefolowingunitarytransfoṟ

10 0 第 9 卷第 0 期 mationontohisparticles5andtheancilarypaṟ U56= ticle6: æ ö 0 - α -( α β β ) ( α β ) α β α -( α ) β β (36) ( α ) α β δ α -( α γ γ ) ( α α γ ) 0 γ è ø thenthestateofparticles3 45and6wil betransformedintothefolowingform U56 φ > 345 0> 6=α( 00000>+ 00>+ 0>+ 00>) 345 0> 6 +( β -( α β ) 000>+γ -( α β ) 00>+ δ -( α δ ) 00>) 345 >6. (37) AfterthisAlicemakesavon Neumann measurementontheancilaryparticle6.iftheresultis 0> 6 thestateof34and5becomes 6<0 U56 φ > 345 0> 6=α( 00000>+ 00>+ 0>+ 00>) 345 (38) whichmeansthatafaithfulchannelcanbeestaḇ lishedbetweenalicebobandcharle.herewedo notnormalizethestatein (38)forconvenience.If theresultis > 6 theestablishmentprocessfails and weshould startanotherestablishment.from (37)wecanfindthatthesuccessprobabilityto obtain 0> 6is4α.If α = β = γ = δ thesuccessprobabilitybecomesandtheancilary particleisnotneeded. 6 Conclusion Entanglementplaysanimportantroleinquaṉ tum informationprocessingtasks [37-48].Quantum teleportationoriginatesfrom thenocloningtheoremresultingfrom quantum mechanicswhichfoṟ bidsthecreationofidenticalcopies withanarbitraryunknownquantum state [37].Thistopichas atracted muchinterestintherecentdecades.so far.mostsuchresearchhasfocusedonprobabilis- sionalunitary operationsto obtainthe unknown state withacertain probability [8-33].However higẖdimensional unitary operations are usualy lishdeterministicteleportationchannelsbetweena senderandher/hisagents.thepresentedschemes needsonlythesenderorreceivertooferoneor moreauxiliaryparticlesfolowedbyimplementing aprojective measurementorapositiveoperatoṟ valuedmeasureonher/hisqubitsincludingtheaṉ cilas.inadditionmultispatialyremoteagents playtheroleofrelaynodesbetweenasenderanda distantreceiverin a quantum teleportation neṯ ticteleportationofanunknownstatebyintrodu- cinganancilaryqubitandexecutinghigẖdimeṉ dificulttoimplementbyexperiment.inthispaperwehavepresentedseveralschemestoestaḇ work.henceweproposeaprotocolforconstructinga deterministicchannelbetweenthesender andthedistantreceiverwiththeassistanceofrelay agentsrespectively. References: []BennetCHBrassardGCrepeauCetal.Teleportingan unknownquantumstateviadualclassicalandeinsteiṉ Podolsky-Rosenchannels [J].PhysicalReview Leter 99370(3): []EkertAK.QuantumcryptographybasedonBel stheorem

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