Quantum information and relativity theory

QuantuminformationandrelativitytheoryAsherPeresDepartmentofPhysics,Technion–IsraelInstituteofTechnology,32000Haifa,IsraelDanielR.TernoPerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario,CanadaN2J2W9(Published6January2004)Thisarticlediscussestheintimaterelationshipbetweenquantummechanics,informationtheory,andrelativitytheory.Takentogetherthesearethefoundationsofpresent-daytheoreticalphysics,andtheirinterrelationshipisanessentialpartofthetheory.Theacquisitionofinformationfromaquantumsystembyanobserveroccursattheinterfaceofclassicalandquantumphysics.Theauthorsreviewtheessentialtoolsneededtodescribethisinterface,i.e.,Krausmatricesandpositive-operator-valuedmeasures.TheythendiscusshowspecialrelativityimposessevererestrictionsonthetransferofinformationbetweendistantsystemsandtheimplicationsofthefactthatquantumentropyisnotaLorentz-covariantconcept.ThisleadstoadiscussionofhowitcomesaboutthatLorentztransformationsofreduceddensitymatricesforentangledsystemsmaynotbecompletelypositivemaps.Quantumfieldtheoryis,ofcourse,necessaryforaconsistentdescriptionofinteractions.Itsstructureimpliesafundamentaltradeoffbetweendetectorreliabilityandlocalizability.Moreover,generalrelativityproducesnewandcounterintuitiveeffects,particularlywhenblackholes(or,moregenerally,eventhorizons)areinvolved.Inthismoregeneralcontexttheauthorsdiscusshowmostofthecurrentconceptsinquantuminformationtheorymayrequireareassessment.CONTENTSI.ThreeInseparableTheories93A.Relativityandinformation93B.Quantummechanicsandinformation94C.Relativityandquantumtheory95D.Themeaningofprobability95E.Theroleoftopology96F.Theessenceofquantuminformation96II.TheAcquisitionofInformation97A.Theambivalentquantumobserver97B.Themeasuringprocess98C.Decoherence99D.Krausmatricesandpositive-operator-valuedmeasures(POVM’s)99E.Theno-communicationtheorem100III.TheRelativisticMeasuringProcess102A.Generalproperties102B.Theroleofrelativity103C.Quantumnonlocality?104D.Classicalanalogies105IV.QuantumEntropyandSpecialRelativity105A.Reduceddensitymatrices105B.Massiveparticles105C.Photons107D.Entanglement109E.Communicationchannels110V.TheRoleofQuantumFieldTheory110A.Generaltheorems110B.Particlesandlocalization111C.Entanglementinquantumfieldtheory112D.Accelerateddetectors113VI.BeyondSpecialRelativity114A.Entanglementrevisited115B.Thethermodynamicsofblackholes116C.Openproblems118AcknowledgmentsandApologies118AppendixA:RelativisticStateTransformations119AppendixB:Black-HoleRadiation119References120I.THREEINSEPARABLETHEORIESQuantumtheoryandrelativitytheoryemergedatthebeginningofthetwentiethcenturytogiveanswerstounexplainedissuesinphysics:theblackbodyspectrum,thestructureofatomsandnuclei,theelectrodynamicsofmovingbodies.Manyyearslater,informationtheorywasdevelopedbyClaudeShannon(1948)foranalyzingtheefficiencyofcommunicationmethods.Howdotheseseeminglydisparatedisciplinesrelatetoeachother?Inthisreview,weshallshowthattheyareinseparablylinked.A.RelativityandinformationCommonpresentationsofrelativitytheoryemployfictitiousobserverswhosendandreceivesignals.These‘‘observers’’shouldnotbethoughtofashumanbeings,butratherasordinaryphysicalemittersanddetectors.Theirroleistolabelandlocateeventsinspacetime.Thespeedoftransmissionofthesesignalsisboundedbyc—thevelocityoflight—becauseinformationneedsamaterialcarrier,andthelattermustobeythelawsofphysics.Informationisphysical(Landauer,1991).However,themereexistenceofanupperboundonthespeedofpropagationofphysicaleffectsdoesnotdojusticetothefundamentallynewconceptsthatwerein-troducedbyAlbertEinstein(onecouldaswellimaginecommunicationslimitedbythespeedofsound,orthatofthepostalservice).Einsteinshowedthatsimultaneityhadnoabsolutemeaning,andthatdistanteventsmightREVIEWSOFMODERNPHYSICS,VOLUME76,JANUARY20040034-6861/2004/76(1)/93(31)/$40.00©2004TheAmericanPhysicalSociety93havedifferenttimeorderingswhenreferredtoobserversinrelativemotion.Relativistickinematicsisallaboutinformationtransferbetweenobserversinrelativemo-tion.Classicalinformationtheoryinvolvesconceptssuchastheratesofemissionanddetectionofsignals,andthenoisepowerspectrum.Thesevariableshavewell-definedrelativistictransformationproperties,indepen-dentoftheactualphysicalimplementationofthecom-municationsystem.AdetailedanalysisbyJarettandCover(1981)showedthatthetransmissionratesforob-serverswithrelativevelocityvwerealteredbyafactor(c1v)/(c2v),namely,thesquareofthefamiliarDop-plerfactorforfrequenciesofperiodicphenomena.Weshalllaterderivethesamefactorfromclassicalelectro-magnetictheory[seeEq.(36)below].Physicshasare-markablyrigidtheoreticalstructure:youcannotalteranypartofitwithouthavingtochangeeverything(Weinberg,1992).B.QuantummechanicsandinformationEinstein’stheoryelicitedstrongoppositionwhenitwasproposed,butitisgenerallyacceptedbynow.Ontheotherhand,therevolutioncausedbyquantumtheorystillproducesuneasyfeelingsamongsomephysicists.1StandardtexbooksonquantummechanicstellyouthatobservablequantitiesarerepresentedbyHermitianoperators,thattheirpossiblevaluesaretheeigenvaluesoftheseoperators,andthattheprobabilityofdetectingeigen