arXiv:gr-qc/0009083v1 25 Sep 2000InformationErasureand
theGeneralizedSecondLawofBlackHole
Thermodynamics
DavidD.Song
CentreforQuantumComputation,ClarendonLaboratoryUniversityofOxford,ParksRoad,OxfordOX13PU,U.K.
d.song@qubit.org
ElizabethWinstanley
DepartmentofAppliedMathematics,UniversityofSheffield,HicksBuilding,HounsfieldRoad,SheffieldS37RH,U.K.
E.Winstanley@sheffield.ac.uk
Abstract
Weconsiderthegeneralizedsecondlawofblackholethermodynamicsinthelightofquantuminformationtheory,inparticularinformationerasureandLandauer’sprin-ciple(namely,thaterasureofinformationproducesatleasttheequivalentamountofentropy).AsmallquantumsystemoutsideablackholeintheHartle-Hawkingstateisstudied,andthequantumsystemcomesintothermalequilibriumwiththeradiationsurroundingtheblackhole.Forthisscenario,wepresentasimpleproofofthegeneralizedsecondlawbasedonquantumrelativeentropy.Wethenanalyzethecorrespondinginformationerasureprocess,andconfirmourproofofthegeneralizedsecondlawbyapplyingLandauer’sprinciple.PACS:04.70.Dy,03.67-a.
Thecorrespondencebetweenthelawsofthermodynamicsandblackholemechanicswasnoted,asacuriositywithoutphysicalimplications,inaseminalpaperbyBardeen,CarterandHawking[1].Ataroundthesametime,Bekenstein[2]wasadvocatingarathermoreradicalapproach.Notingtheareatheoremofblackholes,whichstatesthatthetotalareaofblackholeeventhorizonscanneverdecrease,heobservedthatthisisanalogoustotheordinarysecondlawofthermodynamics,i.e.thetotalentropyofaclosedsystemneverdecreases.Heproposedthat,multipliedbyappropriatepowersofthePlancklength,Boltzmannconstantandsomedimensionlessconstantoforderunity,theblackholeareashouldbeinterpretedasitsphysicalentropy.ThisproposalwasgivenphysicalsupportbythediscoveryofHawking[3]thatblackholesradiateatatemperature
Tbh=
κ
4
.
WheelerprovidedtheinitialmotivationforBekenstein’sblackholeentropyproposal[4].Wheelersuggestedacreature,subsequentlycalledWheeler’sdemon,whichcouldviolatetheordinarysecondlawofthermodynamicsbydroppingentropyintoablackhole,producingadecreaseintheentropyoutsidetheblackhole.ThisledBekensteintoconjecturethattheblackholeitselfhasanentropy(proportionaltotheareaoftheeventhorizon)and,furthermore,thesumoftheentropyoutsidetheblackholeandtheblackholeentropymustnotdecrease,
∆Sout+∆Sbh≥0.
(2)
Thisgeneralizedsecondlawhasbeenwidelydiscussedintheliterature,andthereareproofsduetoFrolovandPage[5],and,morerecentlyMukohyama[6].Boththeseproofsmakeuseofquantumfieldtheoryincurvedspace,andapplytoquasistationaryblackholes.FrolovandPage’sproofisapplicabletoeternalblackholespace-times,whilstMukohyamaconsidersblackholesarisingfromgravitationalcollapse.
Inthispaperwewishtoconsiderthegeneralizedsecondlawfromanotherpointofview,namelyquantuminformationtheoryand,inparticular,Landauer’sprincipleofinformationerasure[7].Wewillconsideraquantumsystemoutsideablackhole,whichthencomesintothermalequilibriumwiththeHawkingradiationsurroundingtheblackhole.Thisscenarioisdifferentfromthosethathavebeenconsideredpreviouslyinproofsofthegeneralizedsecondlaw,givingfurtherweighttoitsvalidity.
WefirstlydiscussMaxwell’sdemon[8],whichistheanalogueinordinarythermo-dynamicsofWheeler’sdemon.Consideracontainerofgaswhichisdividedintotwohalves,leftandright,byapartition.Imaginenowthatthereisademonsittingonthe
1
partition,whoisabletomeasurethevelocitiesofindividualmoleculesinthegas.Ifthedemonletthefastmoleculesmovetotherightcontainerwhilekeepingthesloweronestotheleft,thenthiswouldcreateatemperaturedifferenceandviolatethesecondlawofthermodynamics.Bennettnoted[9]thatinordertodofreework,thedemonhastorecorditsmeasurementresult,andthenitsmemoryneedstobeerasedinordertodothenextmeasurement.Landauer’sprinciplestatesthatinordertoeraseacertainamountofinformationatleastthesameamountofentropymustbegenerated.Therefore,theerasureofthememoryofthedemongeneratesanentropygreaterthanorequaltotheamountofrecordedmemory,whichpreservesthesecondlaw.Bennett’sclassicalanalysisofMaxwell’sdemonwaslaterconfirmedquantummechanically[10,11,12].ThisprocessresemblesWheeler’sdemon,whoistryingtoeraseinformationbydroppinganobjectintoablackhole.Thisnecessarilycreatesanincreaseofblackholeentropybyatleastthesameamountasthedroppedentropy,accordingtothegeneralizedsecondlaw.Inthispaper,wegiveasimpleproofofthegeneralizedsecondlawforourmodel,usingknownresultsonquantumrelativeentropy.ThiswillbeconfirmedbyouranalysisofthecorrespondinginformationerasureprocessusingLandauer’sprinciple.Thegeneralizedsecondlawhasrecentlybeenconsideredinthecontextofquantuminformationtheory(concentratingontheentanglementofstatesinsideandoutsidetheblackholeeventhorizon)byHosoyaandcollaborators[13].
LetusconsiderablackholeinthermalequilibriumwithaheatbathattheHawkingtemperatureTbh.ThisistheHartle-Hawkingstate[14],andcanberenderedstablebyplacingtheblackholeinacavitywhosedimensionsareverymuchlargerthantheradiusoftheblackholeeventhorizon,therebyformingaclosedsystem.Weconsiderasmallquantumsystemoutsidetheblackhole,havingHamiltonianHandinitiallyinaquantumstatedescribedbythedensitymatrixρi.Wethensupposethatthesmallquantumsystemcomesintothermalequilibrium,sothatitsfinalstateisthethermalstateρf=Z−1e−βbhH,whereZ=tr[e−βbhH]andβbh=1
−1
Sinceρfisathermalstate,H=−βbhlog(Zρf),whichgives
∆Sbh=tr[ρflog(Zρf)]−tr[ρilog(Zρf)]
=−tr[(ρi−ρf)logρf]+[tr(ρf)−tr(ρi)]logZ=−tr[(ρi−ρf)logρf].
(5)
Thefinallinefollowsbyconservationofprobability.Notethatthisdoesnotassumethatthestatesevolveunitarily.
Thereforethetotalchangeinentropycanbewrittenasfollows
∆Sout+∆Sbh=tr[ρilogρi−ρilogρf].
(6)
Atthisstageitisimportanttonotethat,incommonwithotherproofsofthegeneralizedsecondlaw,wehavehadtousethefirstlaw.Herewehaveusedtheordinaryfirstlawofthermodynamics,althoughthisisdirectlyanalogoustothefirstlawofblackholemechanicsforquasistationaryblackholes,andgivesthesameresult.
Weshouldalsoemphasiseatthisstagethattheprocessweareconsideringhereisdif-ferentfromtheusualgedankenexperimentofBekenstein[2]inwhichasystemcontainingentropyisdroppeddowntheblackholehorizon.Hereweconsiderinsteadasystemwhichcomesintothermalequilibriumwiththeradiationoutsidetheblackholeeventhorizon.ThissystemwillbepertinenttooursubsequentconsiderationofinformationerasureandLandauer’sprinciple.
Thequantity(6)isknownasquantumrelativeentropywhichisdefinedasS(σ||ρ)=tr[σlogσ−σlogρ].QuantumrelativeentropyS(σ||ρ)hasbeenshown[15]tobealwaysnon-negativeandiszeroifandonlyifσ=ρ.Therefore(6)isnon-negativeandthisprovesthegeneralizedsecondlaw.
Quantumrelativeentropyhasbeenshowntohavevariousapplicationsinquantuminformationtheory(see[16],forexample)includingentanglementquantification.Wenowgiveasimpleexampletoillustratethisconcept.Theunitofquantuminformationiscalledaquantumbitorqubit.Aqubitisasuperpositionof|0and|1,anorthonormalbasisinatwo-dimensionalHilbertspace.Forexample,aspin-1
Inordertogiveacorrectmeasureofentanglement,ρABinS(σAB||ρAB)satisfiesthe
i
followingconditions:(1)itisdisentangled(i.e.ρAB=ipiρiA⊗ρB)and(2)S(σAB||ρAB)isminimal.Forpurestates,aρABsatisfyingboththeseconditionscanalwaysbefoundasσABwithoff-diagonaltermssettozero,i.e.
ρAB=
a0
2
0b2
.(8)
WithσABandρABin(7)and(8),wecouldcalculatetr[σABlogσAB−σABlogρAB]wheretr[σABlogσAB]iszerosinceσABisapurestate.Thesecondterm,tr[−σABlogρAB]yields−a2loga2−b2logb2whichissameasthevonNeumannentropyof|ψAB.
WeshallnowrelateourblackholeprocessandthegeneralizedsecondlawtoLan-dauer’sprincipleofinformationerasure.First,webrieflyreviewsomeofthekeyideas,andamechanismfortheerasureofinformation.Landauer’sprincipleofinformationerasure(thattheerasureofacertainamountofinformationproducesatleasttheequiv-alentamountofentropy)hasbeenusedtoexplainsomeofthefundamentalaspectsinquantuminformationtheory.Theentanglementsharedbytwopartiescanbemanipu-latedintoanotherstatebylocaloperationandclassicalcommunication(LOCC).Howeveritisknownthatlocaloperationcannotincreasetheentanglementsharedbytwosepa-ratedparties.Forexample,theconversionfroma|00AB+b|11AB,wherea,b=12,to1
(|00AB+|11AB)(knownasentanglementpurification)cannotbedonewithprobabil-2
ity1byLOCCbecausetheentanglementhasincreasedfrom−a2loga2−b2logb2tolog2.Vedral[18]hasshownthatLandauer’sprincipleyieldsanupperboundforentanglementpurification,linkingnolocalincreaseofentanglementtothesecondlawofthermodynam-ics.AnotherfundamentalideainquantuminformationtheoryistheHolevobound[19]whichlimitstheamountofclassicalinformationencodedinquantummixedstatesthatcanberecovered.Plenioshowed[20]howthisHolevoboundmaybeillustratedusingLandauer’sprinciple.Inthefollowing,weshowhowinformationerasuremayberealizedphysically,followingLubkin’smethod[11],aspresentedin[18,20].Wereferthereaderto[20]fordetailsofthemethod.
LetusconsideraquantumstateofasystemS
|ψS=
i
λi|aiS|miM.(10)
ThestateofthetheapparatusMcanbeobtainedbytracingoverthesystemSin(10),whichyields
λi|mimi|,(11)ρ=
i
4
Reservoir
'
Tbh
Apparatus
M
%
Figure1:TheapparatusMisthrownintoareservoirwithblackholetemperatureTbhandthenreachesthermalequilibrium.
i.e.withprobabilityλitheapparatusisinstate|mi.Afterthemeasurement,theapparatuswillthereforebeinoneofthesepurestates,withtheassociatedprobability.Thegeneralwaytoerasetheinformationofapparatusistoputtheapparatusintoathermalreservoir.Theapparatusreachesthermalequilibriumwiththereservoirandwethenbringinanotherpurestatetoperformthenextmeasurement.Thentheerasureentropyhastwoparts:oneistheentropychangeofapparatusduetoitschangeofstatefromoneofthepurestatesin(11)toastatewhichisinthermalequilibrium,andtheotheristheentropychangeofthereservoirduetotheapparatus.
Inordertomakethisinformationerasureprocessapplicabletoblackholes,wechooseareservoirwhichisattheblackholetemperature,Tbh.Then,asshowninFigure1,theapparatusMinstateρisthrownintothereservoirwithtemperatureTbh.Aftertheapparatusisthrownintothereservoir,itreachesthermalequilibriumandthestatebecomes
ω=Z−1e−βbhH(12)whereβbh=
1
give
∆Sres=−β{tr[ωH]−tr[ρH]}
=tr[ωlog(ωZ)]−tr[ρlog(ωZ)]=tr[(ω−ρ)logω],
(14)
wherewehaveagainusedconservationofprobability.Theentropyoferasureisthen
∆Sera=∆Sapp+∆Sres
=−tr[ρlogω].
(15)(16)
Thereforeifweconsidertheentropyofthelostinformationas∆Sinf=0−(−tr[ρlogρ]),then
∆Sinf+∆Sera=tr[ρlogρ−ρlogω].(17)Withtheidentificationofρandωasρiandρfintheblackholecase,respectively,(17)
yieldsthesameresultasin(6).
ApplicationofLandauer’sprinciplethentellsusthat(17)mustbepositive,confirmingourproofofthegeneralizedsecondlaw.Howevernotethat∆Seraisnotequalto∆Sbh,contrarytointuition.Thereasonforthisliesinthedetailsoftheerasureprocess[20].Asdescribedabove,theapparatusisinapurestatebeforetheerasureprocesstakesplace,sothatthechangeinentropyoftheapparatusduetotheerasureprocess(13)involvesonlythechangebetweenthispurestateandthethermalstateω,ratherthanbetweenthethermalstateandtheinitialstateρ.
Ourtwoapproachesinthispaperarethereforecomplementary.Inthefirstmethod,wechangetheentropyofthesystemoutsidetheblackhole,andthereisacorrespondingchangeintheentropyoftheblackhole.Inthesecondscenario,weworkouthowmuchinformationwearelosingintermsofdestroyingtheinitialstate,andthencalculatetheamountofentropyrequiredinordertoerasethisamountofinformation.Notethatintheblackholesituation,theinformationiseffectivelydestroyedbecausethefinalstateofthequantumsystemisthesamethermalstateasthesurroundingradiation.Wealsoemphasisethatinthefirstscenarioalltheentropylostgoesdowntheblackholeeventhorizon,andtheentropyofthethermalradiationsurroundingtheblackholedoesnotchange.
Inconclusion,inthispaperwehaveconsideredthegeneralizedsecondlawfromthepointofviewofquantuminformationtheory,especiallyinformationerasureandLan-dauer’sprinciple.Wehaveconsideredaquantumsystemoutsideablackhole,whichcomesintothermalequilibriumwiththeradiationsurroundingtheeventhorizon.Forthissituation,wehavebeenabletogiveasimpleproofofthegeneralizedsecondlawofblackholethermodynamicsbyappealingtoknownresultsonquantumrelativeentropy.Thisresultisconfirmedbyananalysisofthecorrespondinginformationerasureprocess
6
usingLandauer’sprinciple.Thisillustratesthepowerofquantuminformationtheoreticideaswhenappliedtoblackholeprocesses.
AcknowledgmentDSisgratefultoSougatoBose,LucienHardy,VlatkoVedralandErnestoGalv˜aoforvaluablediscussions.
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