1Itcanbeviewedasapproximatelyproducingrectangleswithsigns10Section3 HKlauckandRdeWolf3protocolswithoutentanglementTheseresultsalsoimplylowerboundforquantumLasVegasorzeroerrorprotocolsie ID: 90851
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2FoolingOne-SidedQuantumProtocolsNotethatthesetwoconditionsimplythatifpairs(x;y);(x0;y0)2Faredistinct(i.e.,dierinatleastonecoordinate),thentheydierinbothcoordinates.HenceafoolingsetFformsab¼ectionbetweenjFjinputsonAlice'ssideandjFjinputsonBob'sside.Accordingly,byrenamingsomeofBob'sinputswecanalwaysassumewithoutlossofgeneralitythatFisoftheformf(x;x)g.Toillustratetheconceptofafoolingset,considerthen-bitequalityfunctionEQ,de nedonx;y2f0;1gnasEQ(x;y)=1ix=y.Thishasa1-foolingsetF=f(x;x)gofsize2n,sinceEQ(x;x)=1forallxandEQ(x;y)=0foralldistinctx;y.Thesamefoolingsetalsoworksforthen-bitgreater-thanfunction,whichisde nedasGT(x;y)=1iyx.Then-bitdisjointnessfunctionDisj,de nedasDisj(x;y)=1ijx^yj=0,alsohasa1-foolingsetofsize2n,whichcanbeseenasfollows:writeitscommunicationmatrixas1110 n,andtaketheanti-diagonalasthe1-foolingset.Allentriesontheanti-diagonalare1(givingthe rstproperty)andallentriesbelowtheanti-diagonalare0(givingthesecondproperty).Nowconsiderforsimplicityadeterministicprotocolcomputingf.Supposethelastbitoftheconversationistheoutputbit,sobothpartiesendupknowingtheoutput.Considerinputpairs(x;y);(x0;y0)2F.Forbothinputs,the rstpropertyofthefoolingsetsaysthatthecorrectoutputvalueis1.Suppose,bywayofcontradiction,thattheconversationbetweenAliceandBobisthesameonbothinputpairs.Ifweswitchinputpair(x;y)to(x;y0)thennothingchangesfromAlice'sperspective(neitherherinputnortheconversationchanges),sotheoutputwillstillbe1.Similarly,ifweswitch(x;y)to(x0;y)thentheoutputwon'tchangefromBob'sperspective.Butbythesecondpropertyoffoolingsets,foratleastoneof(x;y0)and(x0;y),thecorrectoutputis0!Hencetheconversationsoninputs(x;y)and(x0;y0)musthavebeendierent.Accordingly,thebiggerourfoolingsetFis,themoredistinctconversationswemustallowandhencethemorebitsofcommunicationareneeded.Moreprecisely,thecommunicationcomplexityislowerboundedbylogjFj+1.Aformalproofofthisfactcanbebasedonthenotionofmonochromaticrectangles.ArectangleisasetR=AB,whereAXandBY.Sucharectangleis1-monochromaticiff(x;y)=1forall(x;y)2R.Notethatarectanglecontaining1-inputs(x;y);(x0;y0)2Fcannotbe1-monochromatic,becausebytherectanglepropertyitalsocontains(x;y0)and(x0;y),atleastoneofwhichisa0-inputbyfoolingsetproperty2.Accordingly,ifwewanttoincludeFinasetof1-rectangles,weneedaseparate1-rectangleforeachelementofF,soweneedatleastjFjdierentrectangles.Itiswell-knownthatadeterministicc-bitcommunicationprotocolinducesapartitionofthesetofall1-inputsinto2c�11-monochromaticrectangles,sothepreviousargumentimplies2c�1jFj;equivalentlyclogjFj+1.InfactevennondeterministiccommunicationcomplexityislowerboundedbylogjFj+1:ac-bitnondeterministicprotocolgivesrisetoacover(ratherthanpartition)ofthesetofall1-inputsby2c�11-monochromaticrectangles,andwestillneedaseparaterectangleforeachelementofF.Incontrast,aquantumcommunicationprotocoldoesnotnaturallyinduceapartitionorcoverofthe1-inputsintorectangles1,sotheabovewayofreasoningfails.Infact,incontrasttotheclassicalcase,thenumberofmonochromaticrectanglesneededtopartitionthe1-inputsdoesnotprovidealowerboundonexactquantumprotocols,aswitnessedbytheexponentialseparationin[4].Nevertheless,inthispaperweshowhowfoolingsetscanstillbeusedtolowerboundquantumcommunicationcomplexity.Wedothisintwosettings:one-sided-errorquantumprotocolswithunlimitedpriorentanglementandnondeterministicquantum 1Itcanbeviewedasapproximatelyproducingrectangleswithsigns[10,Section3]. H.KlauckandR.deWolf3protocolswithoutentanglement.TheseresultsalsoimplylowerboundforquantumLasVegasorzero-errorprotocols(i.e.,quantumprotocolsthatnevererr,buthaveprobability1=2ofgivingupwithoutaresult).1.2Ourresults:foolingone-sided-errorquantumprotocolsFirst,westudyone-sided-errorprotocols:protocolsthatalwaysoutput0oninputsx;ywheref(x;y)=0,andthatoutput1withprobabilityatleast1/2oninputswheref(x;y)=1.Westartbygettinganessentiallyoptimalboundforthecaseofupper-triangularfoolingsets.Wecalla1-foolingsetF=f(x;x)gupper-triangularifthereissometotalordering`'onthex'ssuchthatx�yimpliesf(x;y)=0.Inotherwords,thematrixMwithentriesMxy=f(x;y)is0belowthediagonal.InSection2weshowthatiffhasanupper-triangular1-foolingsetofsizeN,thenQ1(f)1 2logN�1 2:Forexample,then-bitequality,disjointness,andgreater-thanfunctionsallhaveupper-triangular1-foolingsetsofsize2n,andhenceann=2�1=2lowerboundontheirone-sided-errorcomplexityQ1(f).WehaveQ1(f)n=2+1foranyBooleanfunctionwhereXf0;1gn,becausesuperdensecoding[2]allowsAlicetosend2classicalbitsusingoneEPR-pairandonequbitofcommunication.Hencetheaboveresultisessentiallytightforthefunctionsmentioned.2Wecanextendthistoaslightlyweakerresultforallfunctionsstatedintermsoftheir(notnecessarilyupper-triangular)1-fooling-setsize:Q1(f)1 4logfool1(f)�1 2:Surprisinglyforsuchbasicfunctionsasequalityanddisjointness,theseboundswerenotknownbefore.WhileitispossibletouseRazborov'stechnique[16]combinedwithresultsaboutpolynomialapproximationwithverysmallerror[5]toshowQ1(Disj)= (n),nosuper-constantlowerboundwasknownforQ1(EQ).Thisgapinourknowledgewasduetothefactthatotherexistinglowerboundmethodscannotgivegoodlowerboundsforequality,asweexplainnow.Generallowerboundmethodsforquantumcommunicationcomplexitycanbegroupedintorank-basedmethodsandmethodsbasedonapproximationnorms(inparticularbasedonthe 2-norm[14]).3ThelinearityofnormsmakesitpossibletoprovelowerboundsforquantumprotocolsinwhichAliceandBobsharepriorentanglement.Rank-basedmethods,however,donotseemtodirectlyapplytoprotocolswithentanglement:inthecaseofexactquantumprotocolsadirectsum-basedconstructionin[6]showsthatthelogarithmoftherankisalowerboundeveninthepresenceofentanglement.4Inthecaseoftwo-sidederrorandentanglement,LeeandShraibman[12]showthattheapproximationrankyieldslowerboundsbyrelatingittothe 2-norm.SincethecommunicationmatrixofEQis 2Whilethefoolingsetmethodgivesverygoodboundsforthesefunctions,itdoesnotgivegoodboundsforallfunctions.Forexample,arandomfunctionwillwithhighprobabilityhavelinearquantumcommunicationcomplexity(whichcanbeshownforinstanceusingthediscrepancymethod),butonlysmallfoolingsets.Innerproductmod2isanexampleofanexplicitfunctionwiththisproperty[11,Example4.16].3Information-theoreticmethods[8]havealsobeenusedtolowerboundquantumcommunicationcomplex-ity.However,thenotionisde nedthereforinternalinformationcost,andinthiscasetheinformationcostforequalityisO(1),evenforclassicalprotocolswithouterror[3,Proposition3.21].4Footnote2of[6]claimssuchaboundforzero-errorquantumprotocolsforequalityanddisjointnesswithoutproof,butinretrospecttheydidn'tseemtohaveaproofofthis. 4FoolingOne-SidedQuantumProtocolstheidentitymatrixI,and 2(I)=O(1)forIofanysize,thereisnohopetouseaconnectionbetweenaone-sided-errorversionofapproximationrankandthe 2-normtoestablishalargelowerboundonQ1(EQ).Whetheraone-sided-errorversionofapproximationrankgiveslowerboundsforQ1remainsopen,butwenotethattheconstructionin[12]cannotbeadaptedtotheone-sided-errorscenario.SoneitherofthetwomainapproachestoquantumcommunicationcomplexitylowerboundsprovidesuswithagoodlowerboundforQ1(EQ).Henceinthispaperwetakeadierentapproach.We rstsimulateaquantumprotocolwithentanglementbyagamewithoutcommunication,inwhichAliceandBobshareentanglement,andtheyneedtocomputeafunctionfconditionedonpostselectionontheirlocalmeasurements.Thisapproachitselfisnotnew,andcanforinstancebeusedtoshowthatthe 2-normisalowerbound,see[13].WethenanalyzetheimpactofAliceandBob'smeasurementsonthesingleentangledstateusedinthegame.Theone-sided-errorrequirementplacesstrongconstraintsonthosemeasurements,whichweexploittoderiveourlowerboundintermsoffoolingsets.InaquantumLasVegasprotocolAliceandBobcomputeafunctionfwithouterror,buttheyareallowedtogiveupwithoutaresultwithprobability1/2.ThequantumLasVegascommunicationcomplexitywithentanglementQ0(f)istheminimumworst-casecommunicationofanyprotocolthatcomputesfundertheserequirements.5QuantumLasVegasprotocolswereinvestigatedin[5,9,19]inthecasewherenopriorentanglementisavailable.SinceQ0(f)maxfQ1(f);Q1(:f)gweimmediatelygetlargelowerboundsonthequantumLasVegascomplexityofDisjandEQ,andalsothefollowinggenerallowerbound:Q0(f)1 4logfool(f)�1 2;wherefool(f)isthestandardmaximumfoolingsetsize,i.e.,themaximumoverthelargest1-foolingsetand0-foolingset.1.3Ourresults:foolingnondeterministicquantumprotocolsAsasecondmainresult,justlikeintheclassicalworldfoolingsetslowerboundnonde-terministicprotocols,weshowherethattheyalsolowerboundnondeterministicquantumprotocols.Forourpurposes,wecande neanondeterministicprotocol(quantumaswellasclassical)foraBooleanfunctionfasonethathaspositiveacceptanceprobabilityoninputx;yif(x;y)=1.Inotherwords,thisistheunbounded-errorversionoftheone-sided-errormodel:therequirementofacceptanceprobability0on0-inputsremains,buttherequirementoflargeacceptanceprobabilityon1-inputsisrelaxedtopositiveacceptanceprobabilityon1-inputs.6Thequantumversionofthismodelwasintroducedin[19],whichalsoexhibitsatotalfunctionwithanexponentialseparationbetweenquantumandclassicalnondeterministiccommunicationcomplexities.Notethatallowingunlimitedpriorentanglementtrivializesthenondeterministicmodel,forthesamereasonthatunlimitedsharedrandomnesstrivializesitintheclassicalcase:AliceandBobcansharearandomvariableruniformlydistributedoverthesetXofAlice'sinputs;Alicesendsabitindicatingwhetherx=r;if`yes'thenBoboutputsf(r;y)=f(x;y),andif 5Itispossibletode neLasVegasprotocolsasprotocolsthatnevererrandplaceboundsonexpectedcommunication.Thecorrespondingcomplexitymeasureisalwayslargerorequaltotheoneconsideredhere,andissmallerthan2timesourmeasure.6Nondeterministiccommunicationcomplexity(classicalaswellasquantum)canbeexponentiallylessthanone-sided-errorcommunicationcomplexity,evenifthelatterisassistedbyunlimitedpriorentanglement.Thenegationofthedisjointnessfunctionisanexampleofthis. H.KlauckandR.deWolf7ApplyingClaim1withtheactualentangledstatej iusedbyprotocolQ,weobtainN2�2q�1Xx2[N]Pr[outcomeAx Bxwhenmeasuringj i]=Xx2[N]h jAx Bxj ik k2=1:Rearranginggivesthetheorem.JICorollary3.Then-bitequality,disjointnessandgreater-thanfunctionshaveQ1(f)n=2�1=2.Proof.Thesethreefunctionsallhaveupper-triangular1-foolingsetsofsize2n.JNowweuseatrickofcombiningtwocopiesofthefunctiontoextendtheresultfromupper-triangularfoolingsetstoallfoolingsets,attheexpenseofafactorof2inthelowerbound(wedonotknowifthislossisnecessary).Thisissimilartotheproofthatfoolingsetsizeisatmostquadraticallybiggerthanrank[11,Lemma4.15]:ICorollary4.Forallf:XY!f0;1gwehaveQ1(f)1 4logfool1(f)�1 2.Proof.De neanewfunctiong:X2Y2!f0;1gbyg(xx0;yy0)=f(x;y)f(y0;x0).Notethereversedroleofthetwoinputsinthesecondf.AliceandBobcancomputegwithone-sidederrorp=1=4byseparatelycomputingf(x;y)andfT(x0;y0)=f(y0;x0)withone-sidederror1=2each,andoutputtingtheproductofthetwooutputbits.ThistakesQ1(f)qubitsofcommunicationforeachcomputation,soatmost2Q1(f)intotal.Letf(x;x)gbea1-foolingsetforfofsizeN=fool1(f).Thenitiseasytoseethatf(xx;xx)gisa1-foolingsetforg,withtheadditionalpropertythatg(xx;yy)=f(x;y)f(y;x)=0wheneverx6=y.HencethecommunicationmatrixforgcontainstheNNidentityasasubmatrix(i.e.,theequalityfunction).Thesameproofasabovegivesalowerboundof1 2logN�1forone-sided-errorprotocolsforequalitythataccept1-inputswithprobabilityatleast1=4(insteadofatleast1=2asabove).Hencewehave1 2logN�12Q1(f);whichimpliesthestatement.J 3LowerboundfornondeterministicquantumprotocolsInthissectionwestudynondeterministicquantumprotocols.Thefollowingalgebraiccharacterizationofnondeterministicquantumcommunicationcomplexityoffisknown.ThecommunicationmatrixMfforfisthejXjjYjBooleanmatrixMf(x;y)=f(x;y).AnondeterministicmatrixforfisanyrealorcomplexmatrixMwiththesamesupportasMf,i.e.,suchthatMx;y=0if(x;y)=0.Thenondeterministicrankoff(abbreviatedtonrank(f))offistheminimalrank(overthereals)amongallsuchmatrices.[19,Theorem3.3]showsthatNQ(f)=dlognrank(f)e+1.Thekeytousingfoolingsetsfornondeterministicquantumlowerboundsisthefollowingsimplelemma:ILemma5.Foreveryfunctionf:XY!f0;1gwehavenrank(f)2fool1(f).Proof.LetN=fool1(f).LikeintheproofofCorollary4,de neg(xx0;yy0)=f(x;y)f(y0;x0)andobservethatthecommunicationmatrixofgcontainstheNNidentitymatrixINas 8FoolingOne-SidedQuantumProtocolsasubmatrix.IfMisanondeterministicmatrixforf,thenM MTisanondeterministicmatrixforg.Hence,choosingMofminimalrank,wehavenrank(f)2=rank(M)2=rank(M MT)nrank(g)nrank(IN)=N:JTakinglogarithmsandusingthatNQ(f)=dlognrank(f)e+1,wegetICorollary6.NQ(f)1 2logfool1(f)+1.Forexamplefortheequalityfunction,thisshowsNQ(f)n=2+1.However,fortheequalityfunctionwealreadyknewNQ(f)=n+1sinceobviouslynrank(f)=2n[19].Henceitisnaturaltoaskwhethertheconstant1/2intheabovecorollaryisneeded.Wedon'tknow,butatleastwecanshowthatitneedstobelessthan1.Speci cally,wegiveanexamplewhereNQ(f)log3 log6logfool1(f)+1,wherelog3 log60:613.Considerthefollowing66matrix:0BBBBBBB@110001010�1�10�1110�10�1011001001110111011CCCCCCCA:Itiseasytoseethatthishasrank3.TheBooleanmatrixobtainedbydroppingtheminussignscorrespondstoacommunicationcomplexityfunctiong:[6][6]!f0;1gwitha1-foolingsetofsize6(justtakethediagonal).Nowletf:XY!f0;1gbetheANDofkindependentinstancesofg(sojXj=jYj=6k).Because1-foolingsetsizeismultiplicativeundertakingANDs,wehavefool1(f)=6k.Ontheotherhand,takingthek-foldtensorproductoftheaboverank-3matrixgivesanondeterministicmatrixforfofrank3k.HenceNQ(f)=dlognrank(f)e+1log3 log6logfool1(f)+10:613logfool1(f).Asimplerbutslightlyweakerseparationcanbeobtainedfromthe3-inputnon-equalityfunction,whereX=Y=[3]andthefunctiontakevalue0whentheinputsxandyareequal.Thishasnrank=2vsfool1=3,hencetakingak-foldANDofthisgivesafunctionf:XY!f0;1gwithjXj=jYj=3kandnrank(f)=2kvsfool1(f)=3k.Takinglogarithms,wehaveNQ(f)0:63logfool1(f). 4ConclusionandopenproblemsEqualityanddisjointnessaretwoofthemostimportantfunctionsconsideredincommuni-cationcomplexity.Priortothispapernolargelowerboundontheone-sidederrororLasVegasquantumcommunicationcomplexityofthesefunctionswasknownforthecaseofprotocolswithpriorentanglement.Inparticular,forEQpreviouslowerboundmethodswerenotapplicable.Wehaveshownthatthefoolingsetmethodisapplicabletoone-sided-errorprotocolswithentanglement,obtaininglinearlowerboundsforbothfunctions.Itisinterestingtonotethatforclassicalprotocolsthereisessentiallynoneedtoconsiderfoolingsetsatall:themethodiscompletelysubsumedbytherectanglebound(i.e.,boundingthesizeofthelargestmonochromaticrectangleundersomedistribution).However,therectanglebounddoesnotapplytoquantumprotocolswithone-sidederrorandentanglement,nortoquantumnondeterministiccommunicationcomplexity.Weconcludewithsomeopenproblems: 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