K.M.R.Audenaert1,C.A.Fuchs2,3,C.King3,4,A.Winter5
1
arXiv:quant-ph/0308120v1 22 Aug 20032
3
4
5
UniversityofWales,BangorSchoolofInformatics
Bangor(Gwynedd)LL571UT,WalesE-mail:kauden@informatics.bangor.ac.ukBellLabs,LucentTechnologies600-700MountainAve.
MurrayHill,NJ07974,USA
E-mail:cafuchs@research.bell-labs.com
CommunicationNetworksResearchInstituteDublinInstituteofTechnologyRathminesRoadDublin6,Ireland
DepartmentofMathematicsNortheasternUniversityBoston,MA02115,USAE-mail:king@neu.edu
DepartmentofComputerScienceUniversityofBristol
MerchantVenturersBuildingWoodlandRoad
BristolBS81UB,EnglandE-mail:winter@cs.bris.ac.uk
1August2003
Abstract:Twomeasuresofsensitivitytoeavesdroppingforalphabetsofquan-tumstateswererecentlyintroducedbyFuchsandSasakiinquant-ph/0302092.Thesearetheaccessiblefidelityandquantumness.Inthispaperweproveanim-portantpropertyofbothmeasures:Theyaremultiplicativeundertensorprod-ucts.Theproofinthecaseofaccessiblefidelityshowsaconnectionbetweenthemeasureandcharacteristicsofentanglement-breakingquantumchannels.1.Introductionandstatementofresults
Thesecurityofquantumcryptographyreliesonthenotionthatanymeasure-mentonaquantumsystemcausesadisturbancetoit,therebyrevealingthepresenceofaneavesdropper.Howevertheideathat‘measurementcausesdistur-bance’mustbeappliedcarefullyinordertobeuseful.Forexample,givenastate|ψ,themeasurementwhichprojectsonto|ψanditsorthogonalcomplementcausesnodisturbancetothestate.Furthermore,ifasignalisencodedusingorthogonalstatesfordifferentlettersinanalphabet,thenaneavesdroppercangaincompleteinformationbyprojectingontothosestates,againwithoutcausinganychangesinthesignal.Soinordertobesuccessfullyexploitedforquantumcryptography(forexampleasin[4]),anencodingschememustuseanensembleofnonorthogonalsignalstatestopreventadisturbance-freemeasurement.In
2K.M.R.Audenaert,C.A.Fuchs,C.King,A.Winter
otherwords,thesendercannotuseaclassicalensembleofstatestoimplementquantumcryptography.
Thusforpurposesofimplementingquantumcryptographysomeensemblesarebetterthanothers.Thisraisesthequestionoftryingtoquantifythe‘amountofquantumness’inanensembleofstates.Wewilladdressoneaspectofthisques-tionusingtheapproachintroducedinthepaper[6].(Foradifferentapproach,see[8].)Theideaofthepresentapproachistoconsiderthetransmissionofanensembleofstatesfromasendertoareceiver,andtoseehoweasilyaneavesdrop-percanbedetectedatparticipatinginanintercept/resendstrategy.Specifically,supposethatthesenderdrawsstatesrandomlyfromanensembleE={pi,|ψi}.AftertransmissionthereceiverobtainstheensembleE′={pi,|ψi′}.Intheab-senceofnoiseoraneavesdropper,theseensemblesshouldhavefidelityequalto1.Recallthatthefidelityisgivenby
′2
.(1)piψi|ψiF=
i
Nowsupposethattheeavesdropperisallowedtomakeanymeasurementon
theinterceptedstates,thatisanyfixedPOVM{Eb}canbeapplied.Basedontheresultofthismeasurement,theeavesdroppersubstitutesanyotherstate|φbinplaceof|ψiandsendsthisontothereceiver.Thefidelitybetweentheoriginalensembleandthisnewensembleis
2′F=piψi|Eb|ψiψi|φb.(2)
i
b
Inordertominimizeherprobabilityofremainingundetected,theeavesdrop-pershoulduseaPOVMandsetofstatesthatmaximize(2).Thisleadstothe
followingdefinition:
Definition1LetE={pi,|ψi}beanensembleofstates.TheaccessiblefidelityofEisdefinedtobe
2
piψi|Eb|ψiψi|φb.(3)F(E)=supsup
{Eb}{|φb}
i
b
SinceF(E)isthepointwisemaximumoffunctionsthatarelinearinthe
weightspi,itisaconvexfunctionofthepi.Becausethesetofpossibleweights{pi}isconvex(moreprecisely,asimplex),themaximumvalueofF(E)overallweightsisachievedinanextremepointofthesimplex[11].Thesepointsarecharacterisedbyoneofthepibeing1andalltheothersbeing0.Thus
2
maxF(E)=maxsupsuppiψi|Eb|ψiψi|φb
{pi}
i
{Eb}{|φb}
b
=1.
Theoptimumisachievedbytaking{Eb}={I}and|φb=|ψi(foranychoiceof
i).HencethemaximumofF(E)overallensemblesisnotparticularlyinteresting.Ontheotherhand,therearenontriviallowerboundsfortheaccessiblefidelityasafunctionofthe{pi}[6].Inparticular,thequantumnessofasetofstates{|ψi}providesanintrinsicandnontrivialcharacterforthesetitself:
Multiplicativityofaccessiblefidelityandquantumness3
Definition2Thequantumnessofacollectionofstates{|ψi}isdefinedtobe
Q{|ψi}=infF({pi,|ψi}).
{pi}
(4)
Thequantumnessspecifiesthebestusethatcanbemadeofasetofstatesforrevealingtheexistenceofaneavesdropper:Itisaninvertedmeasure,thesmallerthequantumness,thegreaterthedeparturefromclassicalcharacteristics(sinceintheclassicalworldanunconstrainedeavesdroppercannotbedetectedatall).Thepurposeofthispaperistoshowthatboththeaccessiblefidelityandthequantumnesssatisfyanimportantmultiplicativitypropertyforproductstruc-tures.Tobespecific,giventwoensemblesE1={pi,|ψi}andE2={qj,|θj},definetheproductensembleE1⊗E2by
E1⊗E2={piqj,|ψi⊗|θj}.
Weprovethefollowingtwotheorems:Theorem1ForanyensemblesE1andE2,
F(E1⊗E2)=F(E1)F(E2).
and
Theorem2Foranycollections{|ψi}and{|θj},
Q{|ψi⊗|θj}=Q{|ψi}Q{|θj}.
(7)(6)(5)
Thesignificanceofthesethesetheoremsisthefollowing.Inthefirstcase,imaginenotasingleshotthroughtheeavesdroppingchannel,butratherasourcethatrepeatedlygeneratesstatesfromtheensembleE.Onecouldimagineasmarteavesdropperwhosavesupmultiplesignalsbeforeperforminghermeasurementonthechancethatitwillhelpherremainundetected.Ourfirsttheoremshowsthatthismorecomplicatedstrategyprovidesnohelp.Thesecondtheoremmakesastatementaboutoptimalusesofanalphabet.Itsays,givenastatepreparationdevicethatcanonlypreparestatesfromagivencollection{|ψi},itisneverinthesender’sinteresttogeneratecorrelationsbetweenseparatetransmissions.Inthisway,quantumnessisquitedistinctfromachannelcapacity.Forincontrasttochannelcapacity—whereintroducingcorrelationisgenerallynecessaryforachievingit—eavesdroppingdetectionprefersuncorrelatedsignals.Theorem1and2togethersupportthenotionthataccessiblefidelityandquantumnessareintrinsicpropertiesofanensembleanditsunderlyingsetofstates.
Theremainderofthepaperisorganizedasfollows.InSection2welayoutthebasicingredientsrequiredforprovingthetheorems.Followingthat,inSection3weproveTheorem1,andinSection4weproveTheorem2.WeconcludeinSection5withasmalldiscussionaboutthepotentialimplicationsofthiswork.InanAppendixwegiveanewproofofthemultiplicativityofthemaximal∞-normforentanglementbreakingchannels.
4K.M.R.Audenaert,C.A.Fuchs,C.King,A.Winter
2.IngredientsoftheproofofTheorem1
Wedescribeherethetwoprincipalingredientsintheproof.Thefirstingredientisanapplicationofthedualityprincipalofconvexanalysis,whichallowstheacces-siblefidelitytoberewrittenasaninfimumoveraffinefunctionswhichmajorizeitsvalueonthepurestates.Similarideashavebeenexploitedrecentlyinotherareasofquantuminformationtheory,inparticularintheworkonequivalenceofadditivityquestions[3,13].
Thesecondingredientisanadditivityresultforaparticularclassofcom-pletelypositivemapsknownasentanglement-breakingmaps.ThispropertywasfirstestablishedbyShorfortheminimalentropyofmaps[12],andlaterextendedtothenoncommutativep-normsforallp≥1[9].
Todescribethefirstingredient,itisconvenienttodefinethefollowingcom-pletelypositivemapΦassociatedwithanensembleE={pi,|ψi}:
Φ(ρ)=
i
piΠiρΠi,(8)
whereΠi=|ψiψi|.Thentheaccessiblefidelityofanensemble(3)canberewrittenintermsofΦ:
2
piψi|Eb|ψiψi|φbF(E)=supsup
{Eb}{|φb}
=sup
{Eb}
b
ib
supφb|Φ(Eb)|φb
|φb
=sup
{Eb}
b
||Φ(Eb)||.(9)
Furthermore,givenaPOVM{Eb},weassociatetoitanensembleofstates
{αb,σb},givenby
αb=
1
dαb
Eb,
(10)
wheredisthedimensionofthestatespace.Thisdefinesa1–1correspondencebetweenPOVM’sandensembleswhoseaveragestateis1
d
Introducethefollowingfunctiononstates:
g(ρ)=||Φ(ρ)||.
I.(11)
(12)
Thisfunctionisobviouslyconvex.Theconcaveclosureofgisdefinedasfollows:
gˆ(ρ)=supαbg(σb):αbσb=ρ.(13)
{αb,σb}
b
Multiplicativityofaccessiblefidelityandquantumness5
Theconcaveclosureofafunctiongisthesmallestconcavefunctiononthesetofallstatesthatcoincideswithgonthepurestates.Comparingwith(11)wecanseethat
1
F(E)=dgˆ
dp
Shorprovedthattheminimaloutputentropyofaproductchannelisadditive,
providedthatatleastoneofthechannelsisentanglementbreaking[12].Itwaslatershownthatthemaximalp-normofsuchaproductchannelisalwaysmultiplicative,foranyp≥1[9].Infact,withaslightmodificationoftheproofof[9]onecanshowthatmultiplicativityalsoholdsforgeneralCPmaps,notnecessarilytrace-preservingones.Inthispaperwewillmakeuseofthislatter
νp(Ω)
p=1
.(21)
6K.M.R.Audenaert,C.A.Fuchs,C.King,A.Winter
resultforthecasep=∞.Theproofpresentedin[9]usesthepowerfulLieb-Thirringinequality[10]toderivetheresultforallp.Itturnsoutthatforthecasep=∞thereisasimplermethodofproofwhichdoesnotneedthislevelofsophistication.ThereforewestatethiscaseasaseparateLemmabelow,andpresentitsproofintheAppendix.
Lemma1LetΦbeanentanglement-breakingCPmap,andletΩbeanyotherCPmap.Then
ν∞(Φ⊗Ω)=ν∞(Φ)ν∞(Ω).
3.ProofofTheorem1Firstwenotethattheinequality
F(E1⊗E2)≥F(E1)F(E2)
(23)(22)
followsimmediatelyfromthedefinition(1),sincethefidelityoftheproductensembleE1⊗E2canonlydecreasebyrestrictingtoproductPOVM’sandproductstatesφb.SotheTheoremreducestoprovingtheinequality
F(E1⊗E2)≤F(E1)F(E2).
(24)
LetΦ1andΦ2denotetheCPmapsdefinedasin(8)forthetwoensemblesE1andE2.ItfollowsthatthecorrespondingCPmapfortheproductensembleE1⊗E2istheproductmapΦ1⊗Φ2.Asin(12)wedefinetheassociatedfunctions
gi(ρ)=||Φi(ρ)||,
i=1,2,
1
g12(ρ)=||(Φ1⊗Φ2)(ρ)||.
(25)
Nowrecall(14).Thisimplies
F(Ei)=digˆi
Multiplicativityofaccessiblefidelityandquantumness7
Proof:RecallthateverymatrixinthefeasiblesetofEispositivedefinite.GiventhetwomatricesXi∈F(gi),i=1,2,definetheentanglement-breakingCPmapsΩ1andΩ2by
−1/2−1/2
,i=1,2.(30)ρXiΩi(ρ)=ΦiXiThefeasibilityofXimeansthatforallpurestates|ψ:
Tr[Xi|ψψ|]≥gi(|ψψ|)=||Φi(|ψψ|)||.
Substituting
|ψ=Xi
itfollowsthatforanypurestate|φ
||Ωi(|φφ|)||≤1,
andhencethat
ν∞(Ωi)≤1.
HencefromLemma1weget
||(Ω1⊗Ω2)(|ψ12ψ12|)||≤1=Tr|ψ12ψ12|
foranypurestate|ψ12.Thisimpliesinturnthat
||(Φ1⊗Φ2)(ρ12)||≤Tr[(X1⊗X2)ρ12]
foranybipartitestateρ12.HenceX1⊗X2isinthefeasiblesetforg12.⊓⊔4.ProofofTheorem2
First,byrestrictingtoproductdistributionsitfollowsimmediatelyfromTheo-rem1that
Q{|ψi⊗|θj}≤Q{|ψi}Q{|θj}.(37)Soitsufficienttoprovetheboundintheotherdirection.
Weneedtoprovethatforanyjointdistribution{pij}onthecollectionofproductstates{|ψi⊗|θj},wehave
F({pij,|ψi⊗|θj})≥Q{|ψi}Q{|θj}.(38)Indeed,takingtheinfimumoveralldistributions{pij}in(38)givestheinequalityQ{|ψi⊗|θj}≥Q{|ψi}Q{|θj},(39)whichtogetherwith(37)yields(7).
(36)(35)(34)
i=1,2
(33)
−1/2
(31)
|φ,(32)
8K.M.R.Audenaert,C.A.Fuchs,C.King,A.Winter
SincetheaccessiblefidelityisasupremumoverPOVMs,toprove(38),itisenoughtofindaparticularPOVM{Mb,c}suchthat
˜j≥Q{|ψi}Q{|θj},(40)˜jMb,cΠi⊗ΠpijΠi⊗Π
b,c
i,j
˜j=|θjθj|.ItwillbecomeclearfurtheronwhywewithΠi=|ψiψi|andΠ
haveequippedMb,cwithtwoindices.
ThePOVM{Mb,c}isconstructedintwosteps.First,definethemarginaldistribution
pi=pij.(41)
j
Let{Eb}beanoptimalPOVMrealisingF({pi,|ψi}),sothat
piΠiEbΠi.F({pi,|ψi})=
b
i
(42)
Foreachb,let|φbbethedominatingeigenvectorof
φb|
i
piΠiEbΠi.piΠiEbΠi|φb=
i
i
piΠiEbΠisothat
(43)
Defineforeverybanewdistribution{qb,j}jby
qb,j=
1
Multiplicativityofaccessiblefidelityandquantumness9
Hence,
F({qb,j,|θj})
˜j|χb,c˜jFb,cΠqb,jΠ=χb,c|
c
=
1
c
j
Nb≤
1
c
φb⊗χb,c|
ij
˜j)Eb⊗Fb,c(Πi⊗Π˜j)|φb⊗χb,cpij(Πi⊗Π
a
Na
˜j)Eb⊗Fb,c(Πi⊗Π˜j)pij(Πi⊗Π.
b,c
ij
(49)
ThePOVM{Mb,c}isnowdefinedby
Mb,c=Eb⊗Fb,c.
Fromcombiningalltheaboveitfollowsthat
F({pi,|ψi})
rbF({qb,j,|θj})
(51)(50)
b˜˜pij(Πi⊗Πj)Mb,c(Πi⊗Πj).≤
b,c
ij
Nowthedefinitionofquantumnessimpliesthat
F({pi,|ψi})≥Q({|ψi})
and
F({qb,j,|θj})≥Q({|θj}),
sothat
b
(52)
(53)
rbF({qb,j,|θj})≥Q({|θj}),(54)
andtogetherwith(51)thisgives(40).⊓⊔
10K.M.R.Audenaert,C.A.Fuchs,C.King,A.Winter
5.Discussion
Thepresentworkclarifiestwothings.First,thatbothaccessiblefidelityandquantumnessshouldbewrittenin“single-letterized”expressions,astheywereoriginallyproposed.Second,Theorem1maylendsomeevidencetotheideathatcollectiveeavesdroppingstrategiesneednotbeconsideredinafullquan-tumeavesdroppinganalysisafterall—anideathathasbeentoyedwithinthepast[7].Iftrue,thiswouldsignificantlyrelievethetechnologicalrequirementsforoperationalsystemsinwhichunconditionalsecurityissought.
Beyondthis,oneoftheauthors(CAF)ishopefulthatthesemeasures—particularlyquantumness—willbeusefultoacertainlineofattackinquantumfoundations[5].Inthatapproach,aquantumstaterepresentsnotanintrinsicpropertyofasystem,butratheranobserver’sinformation—namely,thebestinformationthatcanbehadgiventhatthecomponentsoftheworldhaveacertainfundamentalsensitivitytothetouch.
Acknowledgement.KAandAWthankCNRIinDublinforitshospitality,wherepartofthisworkwasperformed.CFandCKweresupportedinpartbyScienceFoundationIrelandundertheNationalDevelopmentPlan.CKwasalsosupportedinpartbyNationalScienceFounda-tionGrantDMS-0101205.AWwassupportedbytheU.K.EngineeringandPhysicalSciencesResearchCouncil.
6.Appendix:ProofofLemma1
Theprooffollowsthelineofargumentpresentedin[9],butreplacingtheLieb-Thirringinequalitywithasimplerboundfortheoperatornorm.
Weshowherethatentanglement-breakingCPmapssatisfymultiplicativityofthemaximal∞-norm.Themaximal∞-normofaCPmapΩisdefinedas
ν∞(Ω)=sup||Ω(ρ)||,
ρ
(55)
wherethesuprunsoveralldensitymatricesinthedomainofΩ.Itistrivialtoshowthat
ν∞(Ψ⊗Ω)≥ν∞(Ψ)ν∞(Ω).
Simplyletρ1andρ2bestatesthatachieveν∞(Ψ)andν∞(Ω),respectively.Thenρ1⊗ρ2isnotnecessarilyoptimalforν∞(Ψ⊗Ω),sothat
ν∞(Ψ⊗Ω)≥||(Ψ⊗Ω)(ρ1⊗ρ2)||
=||Ψ(ρ1)||||Ω(ρ2)||=ν∞(Ψ)ν∞(Ω).
Therefore,toprovetheLemma,weonlyneedtoshowthat
ν∞(Ψ⊗Ω)≤ν∞(Ψ)ν∞(Ω).
Tosetupthenotation,considertheactionofthemap(17)onabipartitestateρ12:
(Ψ⊗I)(ρ12)=
K
Rk⊗Tr1[(Xk⊗I)ρ12](56)
k=1
Multiplicativityofaccessiblefidelityandquantumness11
andlet
ρ12=(I⊗Ω)(τ12).
Then
(Ψ⊗I)(ρ12)=(Ψ⊗Ω)(τ12).
Define
xk=Tr[(Xk⊗I)τ12]G′k=Tr1[(Xk⊗I)τ12]/xkGk=Ω(G′k)=Tr1[(Xk⊗I)ρ12]/xk.
Then(56)reads
(Ψ⊗I)(ρ12)=(Ψ⊗I)(τ12)=
K
(57)
(58)
(59)
xkRk⊗GkxkRk⊗G′k,
(60)(61)
k=1K
k=1
wherenow{Rk,Gk}areallpositivematrices,G′kisadensitymatrixandxk≥0.
Writingτ1=Tr2(τ12)forthereduceddensitymatrixitfollowsfrom(61)that
Ψ(τ1)=
K
xkRk.(62)
k=1
NotingthatforanyHermitianmatrixX,X≤||X||I,wehave
(Ψ⊗Ω)(τ12)=(Ψ⊗I)(ρ12)
=≤
KK
xkRk⊗GkxkRk⊗||Gk||I
K
k=1
k=1
≤(max||Gk||)
kk
xkRk⊗I
(63)
k=1
=(max||Gk||)Ψ(τ1)⊗I.
′
NowrecollectthatGk=Ω(G′k)andthatGkisadensitymatrix.Therefore(55)impliesthat
||Gk||≤ν∞(Ω)
foranyk.Togetherwith(63)andthefactthattensoringintheidentitydoesnotchangetheoperatornorm,thisimplies
||(Ψ⊗Ω)(τ12)||≤ν∞(Ω)||Ψ(τ1)||.
(64)
12K.M.R.Audenaert,C.A.Fuchs,C.King,A.Winter
Usingagain(55)weget
||(Ψ⊗Ω)(τ12)||≤ν∞(Ω)ν∞(Ψ).
Sincethisboundholdsforallτ12itfollowsthat
ν∞(Ψ⊗Ω)≤ν∞(Ψ)ν∞(Ω).
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