Information Erasure and the Generalized Second Law of Black Hole Thermodynamics

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|0󰀃and|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|ψ󰀃AB.

WeshallnowrelateourblackholeprocessandthegeneralizedsecondlawtoLan-dauer’sprincipleofinformationerasure.First,webrieflyreviewsomeofthekeyideas,andamechanismfortheerasureofinformation.Landauer’sprincipleofinformationerasure(thattheerasureofacertainamountofinformationproducesatleasttheequiv-alentamountofentropy)hasbeenusedtoexplainsomeofthefundamentalaspectsinquantuminformationtheory.Theentanglementsharedbytwopartiescanbemanipu-latedintoanotherstatebylocaloperationandclassicalcommunication(LOCC).Howeveritisknownthatlocaloperationcannotincreasetheentanglementsharedbytwosepa-ratedparties.Forexample,theconversionfroma|00󰀃AB+b|11󰀃AB,wherea,b=12,to1

(|00󰀃AB+|11󰀃AB)(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

|ψ󰀃S=

󰀅󰀁

i

λi|ai󰀃S|mi󰀃M.(10)

ThestateofthetheapparatusMcanbeobtainedbytracingoverthesystemSin(10),whichyields

󰀅

λi|mi󰀃󰀂mi|,(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.

7

References

[1]J.Bardeen,B.CarterandS.W.Hawking,Comm.Math.Phys.31,161(1973).[2]J.D.Bekenstein,Lett.NuovoCimento4,737(1972);Phys.Rev.D7,2333(1973).[3]S.W.Hawking,Nature248,30(1974);Comm.Math.Phys.43,199(1975).[4]J.D.Bekenstein,Sci.Am.,January(1980).

[5]V.P.FrolovandD.N.Page,Phys.Rev.Lett.71,3902(1993).[6]S.Mukohyama,Phys.Rev.D56,2192(1997).[7]R.Landauer,IBMJ.Rev.Develop.5,183(1961).[8]J.C.Maxwell,TheoryofHeat(Appleton,London,1871).[9]C.H.Bennett,Int.J.Theor.Phys.21,905(1982).

[10]W.H.Zurek,Maxwell’sdemon,Szilard’sengineandquantummeasurements,inG.T.

MooreandM.O.Scully,FrontiersofNon-equilibriumStatisticalPhysics,(PlenumPress,NY,1984).[11]E.Lubkin,Int.J.Theor.Phys.26,523(1987).[12]S.Lloyd,Phys.Rev.A56,3374(1996).

[13]A.Hosoya,A.CarliniandT.Shimomura,Preprintgr-qc/0008045.[14]J.B.HartleandS.W.Hawking,Phys.Rev.D13,2188(1976).[15]A.Wehrl,Rev.Mod.Phys.50,221(1978).

[16]B.SchumacherandM.D.Westmoreland,Preprintquant-ph/0004045.

[17]V.Vedral,M.B.Plenio,M.A.RippinandP.L.Knight,Phys.Rev.Lett.78,2275

(1997);

V.VedralandM.B.Plenio,Phys.Rev.A57,1619(1998).[18]V.Vedral,Proc.R.Soc.Lond.A456,969(2000).

[19]A.S.Holevo,Probl.Pereda.Inf.9,3(1973)[Probl.Inf.Transm.9,177(1973)].[20]M.Plenio,Phys.Lett.A263,281(1999).

8