ConsciousnessasaStateofMatterMax TegmarkDept. of Physics&MITKavli Institute, MassachusettsInstituteof Technology, Cambridge, MA02139(Dated: AcceptedforpublicationinChaos,Solitons&FractalsMarch17,2015)Weexaminethehypothesisthatconsciousnesscanbeunderstoodasastateofmatter, percep-tronium, withdistinctiveinformationprocessingabilities. Weexplorefourbasicprinciplesthatmaydistinguishconscious matter fromother physical systems suchas solids, liquids andgases:theinformation,integration,independenceanddynamicsprinciples. Ifsuchprinciplescanidentifyconsciousentities, thentheycanhelpsolvethequantumfactorizationproblem: whydoconsciousobserverslikeusperceivetheparticularHilbertspacefactorizationcorrespondingtoclassicalspace(ratherthanFourierspace, say), andmoregenerally, whydoweperceivetheworldaroundusasadynamichierarchyof objects that arestronglyintegratedandrelativelyindependent? Tensorfactorizationofmatricesisfoundtoplayacentralrole,andourtechnicalresultsincludeatheoremabout Hamiltonian separability (dened using Hilbert-Schmidt superoperators) being maximized intheenergyeigenbasis. OurapproachgeneralizesGiulioTononisintegratedinformationframeworkforneural-network-basedconsciousnesstoarbitraryquantumsystems,andwendinterestinglinkstoerror-correctingcodes, condensedmattercriticality, andtheQuantumDarwinismprogram, aswell as an interesting connection between the emergence of consciousness and the emergence of time.I. INTRODUCTIONA. ConsciousnessinphysicsAcommonlyheldviewisthat consciousnessisirrel-evant tophysics andshouldthereforenot bediscussedinphysicspapers. Oneoft-statedreasonisaperceivedlackof rigor inpast attempts tolinkconsciousness tophysics. Another argument is that physics has been man-aged just ne for hundreds of years by avoiding this sub-ject, andshouldthereforekeepdoingso. Yet thefactthatmostphysicsproblemscanbesolvedwithoutrefer-ence to consciousness does not guarantee that this appliestoallphysicsproblems. Indeed,itisstrikingthatmanyof the most hotly debated issues in physics today involvethe notions of observations and observers, and we cannotdismissthepossibilitythatpartofthereasonwhytheseissues haveresistedresolutionfor solongis our reluc-tanceasphysiciststodiscussconsciousnessandattempttorigorouslydenewhatconstitutesanobserver.Forexample, doesthenon-observabilityof spacetimeregionsbeyondhorizonsimplythattheyinsomesensedonot exist independentlyof the regions that we canobserve? Thisquestionliesattheheartof thecontro-versiessurroundingtheholographicprinciple,blackholecomplementarityandrewalls,anddependscruciallyontheroleof observers[1, 2]. Whatisthesolutiontothequantummeasurementproblem?Thisagainhingescru-ciallyontheroleofobservation: doesthewavefunctionundergoanon-unitarycollapsewhenanobservationismade, arethereEverettianparallel universes, ordoesitmakenosensetotalkaboutananobserver-independentreality, asarguedbyQBismadvocates[3]? Isourper-sistent failuretounifygeneral relativitywithquantummechanics linked to the dierent roles of observers in thetwotheories?Afterall,theidealizedobserveringeneralrelativityhas nomass, nospatial extent andnoeectonwhatisobserved, whereasthequantumobserverno-toriouslydoesappeartoaectthequantumstateoftheobservedsystem. Finally, outof all of thepossiblefac-torizations of Hilbert space,why is the particular factor-izationcorrespondingtoclassicalspacesospecial?WhydoweobserversperceiveourselvesarefairlylocalinrealspaceasopposedtoFourierspace, say, whichaccordingtotheformalismofquantumeldtheorycorrespondstoan equally valid Hilbert space factorization?This quan-tum factorization problem appears intimately related tothenatureofanobserver.Theonlyissuethereis consensus onis that thereisnoconsensus about howtodene anobserver anditsrole. Onemighthopethatadetailedobserverdenitionwill prove unnecessarybecause some simple propertiessuchastheabilitytorecordinformationmight suce;however,wewillseethatatleasttwomorepropertiesofobservers may be necessary to solve the quantum factor-izationproblem, andthatacloserexaminationof con-sciousnessmayberequiredtoidentifytheseproperties.Anothercommonlyheldviewisthatconsciousnessisunrelatedtoquantummechanicsbecausethebrainisawet, warmsystemwheredecoherencedestroysquantumsuperpositionsofneuronringmuchfasterthanwecanthink, preventingour brainfromactingas aquantumcomputer[4]. Inthispaper, Iarguethatconsciousnessandquantummechanicsarenonethelessrelated, butina dierent way: it is not so much that quantum mechan-ics is relevant tothe brain, as the other wayaround.Specically,consciousnessisrelevanttosolvinganopenproblemat theveryheart of quantummechanics: thequantumfactorizationproblem.B. ConsciousnessinphilosophyWhyareyouconscious right now? Specically, whyareyouhavingasubjectiveexperienceof readingthesewords, seeing colors and hearing sounds, while the inani-arXiv:1401.1219v3 [quant-ph] 18 Mar 20152mate objects around you are presumably not having anysubjectiveexperienceatall? Dierentpeoplemeandif-ferentthingsbyconsciousness, includingawarenessofenvironment orself. I amaskingthemorebasicques-tionofwhyyouexperienceanythingatall,whichistheessenceofwhatphilosopherDavidChalmershastermedthehardproblemofconsciousnessandwhichhaspre-occupiedphilosophersthroughouttheages(see[5] andreferences therein). Atraditional answer tothis prob-lemisdualismthatlivingentitiesdierfrominani-mateonesbecausetheycontainsomenon-physical ele-ment such as an anima or soul. Support for dualismamongscientistshasgraduallydwindledwiththereal-izationthatwearemadeofquarksandelectrons,whichasfaraswecantell moveaccordingtosimplephysicallaws. Ifyourparticlesreallymoveaccordingtothelawsof physics, then your purported soul is having no eect onyourparticles, soyourconsciousmindanditsabilitytocontrol your movements would have nothing to do with asoul. If your particles were instead found not to obey theknownlawsof physicsbecausetheywerebeingpushedaround by your soul, then we could treat the soul as justanotherphysicalentityabletoexertforcesonparticles,andstudywhatphysicallawsitobeys,justasphysicistshavestudiednewforceseldsandparticlesinthepast.The key assumption in this paper is that consciousnessis apropertyof certainphysical systems, withnose-cretsauceornon-physical elements.1, ThistransformsChalmers hardproblem. Insteadof startingwiththehardproblemof whyanarrangement of particles canfeel conscious, we will start with the hard fact that somearrangement of particles (such as your brain) do feel con-sciouswhileothers(suchasyourpillow)donot,andaskwhat properties of the particle arrangement make thedierence.Thispaperisnotacomprehensivetheoryofconcious-ness. Rather, it is aninvestigationinto the physicalpropertiesthatconscioussystemsmusthave. If weun-1More specically,we pursue an extreme Occams razor approachandexplorewhether all aspectsof realitycanbederivedfromquantummechanicswithadensitymatrixevolvingunitarilyac-cordingtoaHamiltonian. Itthisapproachshouldturnouttobesuccessful, thenall observedaspectsof realitymustemergefromthemathematical formalismalone: forexample, theBornrule for subjective randomness associated with observation wouldemerge fromthe underlyingdeterministic densitymatrixevo-lution through Everetts approach, and both a semiclassicalworld and consciousness should somehow emerge as well, perhapsthoughprocesses generalizing decoherence. Evenif quantumgravity phenomena cannot be captured with this simple quantumformalism, itisfarfromclearthatgravitational, relativisticornon-unitary eects are central to understanding consciousness orhowconsciousobserversperceivetheirimmediatesurroundings.Thereisofcoursenoaprioriguaranteethatthisapproachwillwork;this paper is motivated by the view that an Occams razorapproachisuseful if itsucceedsandveryinterestingif itfails,by giving hints as to what alternative assumptions or ingredientsareneeded.derstoodwhat thesephysical properties were, thenwecould in principle answer all of the above-mentioned openphysicsquestionsbystudyingtheequationsof physics:wecouldidentifyall conscious entities inanyphysicalsystem, andcalculatewhattheywouldperceive. How-ever, this approach is typically not pursued by physicists,with the argument that we do not understand conscious-nesswellenough.C. ConsciousnessinneuroscienceArguably, recent progressinneurosciencehasfunda-mentallychangedthis situation, so that we physicistscannolongerblameneuroscientistsforourownlackofprogress. I have long contended that consciousness is thewayinformationfeels whenbeingprocessedincertaincomplexways[6, 7], i.e., thatitcorrespondstocertaincomplexpatternsinspacetimethatobeythesamelawsofphysicsasothercomplexsystems. Intheseminalpa-perConsciousnessasIntegratedInformation: aProvi-sional Manifesto [8], Giulio Tononi made this idea morespecicanduseful, makingacompellingargumentthatforaninformationprocessingsystemtobeconscious, itneedstohavetwoseparatetraits:1. Information: Ithastohavealargerepertoireofaccessiblestates, i.e., theabilitytostorealargeamountofinformation.2. Integration: This information must be integratedinto a uniedwhole, i.e., it must be impossibletodecomposethesystemintonearlyindependentparts,becauseotherwisethesepartswouldsubjec-tivelyfeelliketwoseparateconsciousentities.Tononisworkhasgeneratedaurryof activityintheneuroscience community, spanning the spectrumfromtheory to experiment (see [913] for recent reviews), mak-ing it timely to investigate its implications for physics aswell. This is the goal of the present paper a goal whosepursuitmayultimatelyprovideadditional toolsfortheneurosciencecommunityaswell.Despiteitssuccesses,TononisIntegratedInformationTheory(IIT)2leavesmanyquestionsunanswered. If itis toextendour consciousness-detectionabilitytoani-mals, computers and arbitrary physical systems, then weneed to ground its principles in fundamental physics. IITtakesinformation,measuredinbits,asastartingpoint.But when we view a brain or computer through our physi-cists eyes, as myriad moving particles, then what physicalpropertiesofthesystemshouldbeinterpretedaslogical2Sinceitsinception[8], IIThasbeenfurtherdeveloped[12]. Inparticular, IIT3.0considersboththepastandthefutureof amechanisminaparticularstate(itsso-calledcause-eectreper-toire)andreplacestheKullback-Leiblermeasurewithapropermetric.3ManyStateof long-livedInformation Easily Complex?matter states? integrated? writable? dynamics?Gas N N N YLiquid N N N YSolid Y N N NMemory Y N Y NComputer Y ? Y YConsciousness Y Y Y YTABLEI: Substancesthatstoreorprocessinformationcanbe viewedas novel states of matter andinvestigatedwithtraditionalphysicstools.bitsof information? Iinterpretasabitboththepo-sitionof certainelectronsinmycomputersRAMmem-ory (determining whether the micro-capacitor is charged)and the position of certain sodium ions in your brain (de-terminingwhetheraneuronisring), butonthebasisof whatprinciple? Surelythereshouldbesomewayofidentifying consciousness from the particle motions alone,orfromthequantumstateevolution, evenwithoutthisinformationinterpretation? If so, what aspects of thebehavior of particles corresponds to conscious integratedinformation? Wewillexploredierentmeasuresofinte-gration below. Neuroscientists have successfully mappedout which brain activation patterns correspond to certaintypes of conscious experiences, and named these patternsneural correlatesof consciousness. Howcanwegeneral-izethisandlookforphysicalcorrelatesofconsciousness,denedasthepatternsofmovingparticlesthatarecon-scious?Whatparticlearrangementsareconscious?D. ConsciousnessasastateofmatterGenerations of physicists andchemists have studiedwhathappenswhenyougrouptogethervastnumbersofatoms,ndingthattheircollectivebehaviordependsonthepatterninwhichtheyarearranged: thekeydier-encebetweenasolid, aliquidandagasliesnotinthetypes of atoms, but intheir arrangement. Inthis pa-per, I conjecturethatconsciousnesscanbeunderstoodasyetanotherstateof matter. Justastherearemanytypesof liquids, therearemanytypesof consciousness.However, this shouldnot precludeus fromidentifying,quantifying,modelingandultimatelyunderstandingthecharacteristicpropertiesthatall liquidformsof matter(orallconsciousformsofmatter)share.Toclassifythetraditionallystudiedstatesof matter,weneedtomeasureonlyasmallnumberofphysicalpa-rameters: viscosity,compressibility,electricalconductiv-ityand(optionally) diusivity. Wecall asubstanceasolid if its viscosity is eectively innite (producing struc-tural stiness), andcall it auidotherwise. We callauidaliquidif itscompressibilityanddiusivityaresmallandotherwisecalliteitheragasoraplasma, de-pendingonitselectricalconductivity.Whatarethecorrespondingphysicalparametersthatcanhelpusidentifyconsciousmatter, andwhatarethekey physical features that characterize it?If such param-eters canbeidentied, understoodandmeasured, thiswill helpusidentify(oratleastruleout)consciousnessfromtheoutside, without access tosubjectiveintro-spection. Thiscouldbeimportantforreachingconsen-susonmanycurrentlycontroversialtopics,rangingfromthefutureof articial intelligencetodeterminingwhenananimal, fetus or unresponsivepatient canfeel pain.If wouldalsobeimportant forfundamental theoreticalphysics, byallowingus toidentifyconscious observersinour universe byusingthe equations of physics andthereby answer thorny observation-related questions suchasthosementionedintheintroductoryparagraph.E. MemoryAs arst warmupsteptowardconsciousness, let usrstconsiderastateofmatterthatwewouldcharacter-izeasmemory3whatphysical featuresdoesithave?Forasubstancetobeuseful forstoringinformation, itclearlyneedstohavealargerepertoireofpossiblelong-livedstates or attractors (seeTableI). Physically, thismeans that its potential energy function has a large num-berof well-separatedminima. Theinformationstoragecapacity(inbits)issimplythebase-2logarithmof thenumber of minima. This equals the entropy(inbits)ofthedegenerategroundstateifallminimaareequallydeep. Forexample, solidshavemanylong-livedstates,whereasliquidsandgasesdonot: if youengravesome-onesnameonagoldring, theinformationwill still bethere years later,but if you engrave it in the surface of apond,it will be lost within a second as the water surfacechangesitsshape. Anotherdesirabletraitofamemorysubstance, distinguishing it from generic solids, is that itisnotonlyeasytoreadfrom(asagoldring), butalsoeasytowriteto: alteringthestateofyourharddriveoryoursynapsesrequireslessenergythanengravinggold.F. ComputroniumAsasecondwarmupstep, whatpropertiesshouldweascribe to what Margolus and Tooli have termed com-3Neuroscience researchhas demonstratedthat long-termmem-oryisnotnecessaryforconsciousness. However,evenextremelymemory-impairedconscioushumanssuchasCliveWearing[14]areabletoretaininformationforseveralseconds;inthispaper,Iwill assumemerelythatinformationneedstoberememberedlongenoughtobesubjectivelyexperiencedperhaps0.1sec-ondsforahuman,andmuchlessforentitiesprocessinginforma-tionmorerapidly.4putronium [15], the most general substance that canprocess informationas acomputer? Rather thanjustremainimmobile as agoldring, it must exhibit com-plexdynamicssothatitsfuturestatedependsinsomecomplicated(andhopefullycontrollable/programmable)wayonthepresent state. Its atomarrangement mustbe less ordered than a rigid solid where nothing interest-ingchanges, butmoreorderedthanaliquidorgas. Atthe microscopic level, computroniumneednot be par-ticularlycomplicated, becausecomputerscientistshavelongknownthataslongasadevicecanperformcertainelementary logic operations, it is universal: it can be pro-grammed to perform the same computation as any othercomputer with enough time and memory. ComputervendorsoftenparametrizecomputingpowerinFLOPS,oating-pointoperationspersecondfor64-bitnumbers;moregenerically,wecanparametrizecomputroniumca-pable of universal computation by FLIPS: the numberofelementarylogical operationssuchasbitipsthatitcanperformper second. It has beenshownbyLloyd[16]thatasystemwithaverageenergyEcanperformamaximum of 4E/h elementary logical operations per sec-ond, wherehisPlancksconstant. Theperformanceoftodaysbestcomputersisabout38ordersofmagnitudelower than this,because they use huge numbers of parti-cles to store each bit and because most of their energy istiedupinacomputationallypassiveform,asrestmass.G. PerceptroniumWhat about perceptronium, themost general sub-stance that feels subjectivelyself-aware? If Tononi isright, then it should not merely be able to store and pro-cessinformationlikecomputroniumdoes, butitshouldalso satisfy the principle that its information is inte-grated,formingauniedandindivisiblewhole.Let us also conjecture another principle that conscioussystems must satisfy: that of autonomy, i.e., that in-formationcanbeprocessedwithrelativefreedomfromexternal inuence. Autonomyis thus thecombinationoftwoseparateproperties: dynamicsandindependence.Heredynamicsmeanstimedependence(henceinforma-tionprocessingcapacity)andindependencemeansthatthedynamicsisdominatedbyforcesfromwithinratherthan outside the system. Just like integration, autonomyispostulatedtobeanecessarybutnotsucientcondi-tionfor asystemtobeconscious: for example, clocksand diesel generators tend to exhibit high autonomy, butlacksubstantialinformationstoragecapacity.H. ConsciousnessandthequantumfactorizationproblemTable II summarizes the four candidate principles thatwe will explore as necessary conditions for consciousness.Ourgoal withisolatingandstudyingtheseprinciplesisPrinciple DenitionInformation Aconscioussystemhassubstantialprinciple informationstoragecapacity.Dynamics Aconscioussystemhassubstantialprinciple informationprocessingcapacity.Independence Aconscioussystemhassubstantialprinciple independencefromtherestoftheworld.Integration Aconscioussystemcannotconsistofprinciple nearlyindependentparts.Autonomy Aconscioussystemhassubstantialprinciple dynamicsandindependence.Utility Anevolvedconscioussystemrecordsmainlyprinciple informationthatisusefulforit.TABLEII: Four conjecturednecessaryconditions for con-sciousnessthatweexploreinthispaper. Thefthprinciplesimplycombines thesecondandthird. Thesixthis not anecessarycondition,butmayexplaintheevolutionaryoriginoftheothers.not merely to strengthen our understanding of conscious-ness as aphysical process, but alsotoidentifysimpletraitsof consciousmatterthatcanhelpustackleotheropen problems in physics. For example, the only propertyof consciousness that Hugh Everett needed to assume forhis work on quantum measurement was that of the infor-mationprinciple: byapplyingtheSchrodingerequationtosystemsthatcouldrecordandstoreinformation, heinferredthattheywouldperceivesubjectiverandomnessin accordance with the Born rule. In this spirit, we mighthopethataddingfurthersimplerequirementssuchasinthe integration principle, the independence principle andthedynamics principlemight sucetosolvecurrentlyopenproblemsrelatedtoobservation. Thelast princi-plelistedinTableII, theutilityprinciple, isofadier-ent character thantheothers: weconsider it not as anecessaryconditionforconsciousness,butasapotentialunifyingevolutionaryexplanationoftheothers.In this paper, we will pay particular attention to whatI will refer to as the quantumfactorization problem:whydoconsciousobserverslikeusperceivetheparticu-larHilbertspacefactorizationcorrespondingtoclassicalspace(ratherthanFourierspace, say), andmoregener-ally, whydoweperceivetheworldaroundusasady-namichierarchyof objectsthatarestronglyintegratedandrelativelyindependent? Thisfundamental problemhas receivedalmost noattentionintheliterature[18].Wewill seethatthisproblemisverycloselyrelatedtothe one Tononi confrontedfor the brain, merelyonalarger scale. Solving it would also help solve the physics-from-scratchproblem [7]: If the Hamiltonian H and thetotal densitymatrixfullyspecifyourphysical world,howdoweextract 3Dspaceandtherest of our semi-classical worldfromnothingmorethantwoHermitianmatrices, whichcomewithoutanyapriori physical in-terpretationor additional structure suchas aphysicalspace, quantumobservables, quantumelddenitions,anoutsidesystem,etc.?Cansomeofthisinformation5beextractedevenfromHalone, whichisfullyspeciedbynothingmorethanitseigenvaluespectrum? Wewillsee that ageneric Hamiltoniancannot be decomposedusingtensorproducts, whichwouldcorrespondtoade-compositionofthecosmosintonon-interactingpartsinstead,thereisanoptimalfactorizationofouruniverseintointegratedandrelativelyindependentparts. BasedonTononiswork, wemightexpectthatthisfactoriza-tion, or somegeneralizationthereof, is what consciousobserversperceive, becauseanintegratedandrelativelyautonomous information complexis fundamentally whataconsciousobserveris!Therestofthispaperisorganizedasfollows. InSec-tionII, weexploretheintegrationprinciplebyquanti-fyingintegratedinformationinphysicalsystems,ndingencouragingresultsforclassical systemsandinterestingchallenges introducedbyquantummechanics. InSec-tionIII, weexploretheindependenceprinciple, ndingthatatleastoneadditional principleisrequiredtoac-count for the observed factorization of our physical worldintoanobjecthierarchyinthree-dimensional space. InSectionIV,weexplorethedynamicsprincipleandotherpossibilities for reconcilingquantum-mechanical theorywithour observationof asemiclassical world. Wedis-cussourconclusionsinSectionV,includingapplicationsof theutilityprinciple, andcovervariousmathematicaldetailsinthethreeappendices. Throughoutthepaper,we mainly consider nite-dimensional Hilbert spaces thatcan be viewed as collections of qubits; as explained in Ap-pendixC, thisappearstocoverstandardquantumeldtheory with its innite-dimensional Hilbert space as well.II. INTEGRATIONA. Ourphysical worldasanobjecthierarchyTheproblemof identifyingconsciousness inanarbi-trarycollectionofmovingparticlesissimilartothesim-pler problem of identifying objects there. One of the moststriking features of our physical world is that we perceiveit as an object hierarchy, as illustrated in Figure 1. If youareenjoyingacolddrink,youperceiveicecubesinyourglassasseparateobjectsbecausetheyarebothfairlyin-tegrated and fairly independent, e.g., their parts are morestronglyconnectedtooneanotherthantotheoutside.Thesamecanbesaidabouteachof theirconstituents,rangingfromwatermoleculesall thewaydowntoelec-tronsandquarks. Zoomingout, yousimilarlyperceivethe macroscopic world as a dynamic hierarchy of objectsthatarestronglyintegratedandrelativelyindependent,all the way up to planets, solar systems and galaxies. Letusquantifythisbydeningtherobustnessof anobjectastheratioof theintegrationtemperature(theenergyperpartneededtoseparatethem)totheindependencetemperature (the energy per part needed to separate theparentobjectinthehierarchy). Figure1illustratesthatall ofthetentypesofobjectsshownhaverobustnessoften or more. A highly robust object preserves its identity(itsintegrationandindependence)overawiderangeoftemperatures/energies/situations. Themorerobust anobjectis,themoreusefulitisforushumanstoperceiveitasanobjectandcoinanameforit,aspertheabove-mentionedutilityprinciple.Returningtothephysics-from-scratchproblem,howcanwe identifythis object hierarchyif all we have tostartwitharetwoHermitianmatrices, thedensityma-trixencodingthestateofourworldandtheHamilto-nianHdeterminingitstime-evolution?Imaginethatweknow only these mathematical objects and H and havenoinformationwhatsoever about howtointerpret thevariousdegreesoffreedomoranythingelseaboutthem.Agoodbeginningistostudyintegration. Consider, forexample, andHfor asingledeuteriumatom, whoseHamiltonianis(ignoring spininteractions forsimplicity)H(rp, pp, rn, pn, re, pe) = (1)= H1(rp, pp, rn, pn) +H2(pe) +H3(rp, pp, rn, pn, re, pe),whererandparepositionandmomentumvectors,andthe subscripts p, n and e refer to the proton, the neutronandtheelectron. Onthesecondline, wehavedecom-posedHinto three terms: the internal energyof theproton-neutronnucleus, theinternal (kinetic)energyofthe electron, and the electromagnetic electron-nucleus in-teraction. Thisinteractionistiny, onaverageinvolvingmuchlessenergythanthosewithinthenucleus:tr H3tr H1 105, (2)which we recognize as the inverse robustness for a typicalnucleusinFigure3. Wecanthereforefruitfullyapprox-imatethenucleus andtheelectronas separateobjectsthat are almost independent, interacting only weaklywithoneanother. Thekeypointhereisthatwecouldhave performedthis object-ndingexercise of dividingthe variables into two groups to nd the greatest indepen-dence (analogous to what Tononi calls the cruelest cut)basedonthefunctional formof Halone, withoutevenhavingheardof electronsornuclei, therebyidentifyingtheirdegreesoffreedomthroughapurelymathematicalexercise.B. Integrationandmutual informationIf the interactionenergyH3were so small that wecouldneglectitaltogether,thenHwouldbedecompos-ableintotwopartsH1andH2,eachoneactingononlyoneofthetwosub-systems(inourcasethenucleusandtheelectron). Thismeansthatanythermalstatewouldbefactorizable: eH/kT= eH1/kTeH2/kT= 12, (3)6Object: Oxygen atomRobustness: 10Independence T: 1 eVIntegration T: 10 eVObject: Oxygen nucleusRobustness: 105Independence T: 10 eVIntegration T: 1 MeVObject: ProtonRobustness: 200Independence T:1 MeVIntegration T:200 MeVObject: NeutronRobustness: 200Independence T:1 MeVIntegration T:200 MeVObject: ElectronRobustness: 1022?Independence T:10 eVIntegration T:1016 GeV?Object: Down quarkRobustness: 1017?Independence T: 200 MeVIntegration T:1016 GeV?Object: Up quarkRobustness: 1017?Independence T: 200 MeVIntegration T:1016 GeV?Object: Hydrogen atomRobustness: 10Independence T: 1 eVIntegration T: 10 eVObject: Ice cubeRobustness: 105Independence T:3 mKIntegration T:300 KObject: Water moleculeRobustness: 30Independence T:300 KIntegration T:1 eV ~ 104K{mgh/kB~3mK permoleculeFIG. 1: Weperceivetheexternal worldasahierarchyof objects, whosepartsaremorestronglyconnectedtooneanotherthantotheoutside. Therobustnessof anobjectisdenedastheratioof theintegrationtemperature(theenergyperpartneededtoseparatethem)totheindependencetemperature(theenergyperpartneededtoseparatetheparentobjectinthehierarchy).sothe total state canbe factoredintoaproduct ofthesubsystemstates1and2. Inthiscase,themutualinformationI S(1) +S(2) S() (4)vanishes,whereS() tr log2 (5)isthevonNeumannentropy(inbits)whichissimplytheShannonentropyofeigenvaluesof. Evenfornon-thermal states, thetime-evolutionoperatorUbecomesseparable:U eiHt/= eiH1t/eiH2t/= U1U2, (6)which(aswewilldiscussindetailinSectionIII)impliesthat the mutual information stays constant over time andno information is ever exchanged between the objects. Insummary, if aHamiltoniancanbedecomposedwithoutaninteractionterm(withH3= 0),thenitdescribestwoperfectlyindependentsystems.44Notethatinthispaper, wearegenerallyconsideringHandfor theentirecosmos, sothat thereisnooutsidecontainingobserversetc.If H3= 0,entanglementbetweenthetwosystemsthus cannot have any observable eects. This is in stark contrasttomosttextbookquantummechanicsconsiderations,whereonestudiesasmallsubsystemoftheworld.7Let us nowconsider the opposite case, whenasys-temcannotbedecomposedintoindependentparts. Letusdenetheintegratedinformationasthemutualin-formationI for thecruelest cut(thecut minimizingI)insomeclassof cutsthatsubdividethesystemintotwo(wewill discuss manydierent classes of cuts be-low). Although our -denition is slightly dierent fromTononis [8]5, it is similar in spirit, and we are reusing his-symbolforitselegantsymbolism(unifyingtheshapesofIforinformationandOforintegration).C. MaximizingintegrationWejustsawthatiftwosystemsaredynamicallyinde-pendent(H3= 0),then = 0atalltimebothforther-malstatesandforstatesthatwereindependent( = 0)at somepoint intime. Let us nowconsider theoppo-siteextreme. Howlargecantheintegratedinformationget? Aas warmupexample, let us consider thefa-miliar2DIsingmodelinFigure2wheren = 2500mag-neticdipoles (or spins) that canpoint upor downareplacedonasquarelattice,andHissuchthattheypre-fer aligning with their nearest neighbors. When T , eH/kTI, soall 2nstatesareequallylikely, allnbitsarestatisticallyindependent, and=0. WhenT0, all statesfreezeoutexceptthetwodegenerategroundstates(allspinuporallspindown),soallspinsare perfectlycorrelatedand=1bit. For interme-diatetemperatures, long-rangecorrelations areseentoexistsuchthattypicalstateshavecontiguousspin-uporspin-downpatches. Onaverage,wegetaboutonebitofmutualinformationforeachsuchpatchcrossingourcut(sinceaspinononesideknowsaboutataspinontheotherside), soforbipartitionsthatcutthesystemintotwoequallylargehalves,themutualinformationwillbeproportional to the length of the cutting curve. The cru-elest cut is therefore a vertical or horizontal straight lineof length n1/2, giving n1/2at the temperature wheretypical patches are only a few pixels wide. We would sim-ilarly get a maximum integration n1/3for a 3D Isingsystemand 1bitfora1DIsingsystem.Sinceitisthespatialcorrelationsthatprovidethein-tegration, it is interestingtospeculate about whetherthe conscious subsystem of our brain is a system near itscriticaltemperature,closetoaphasetransition. Indeed,Damasiohasarguedthattobeinhomeostasis, anum-berofphysical parametersofourbrainneedtobekeptwithinanarrowrangeof values[19] thisispreciselywhatisrequiredof anycondensedmattersystemtobenear-critical, exhibitingcorrelationsthatarelong-range5Tononis denitionof [8] applies onlyfor classical systems,whereaswewishtostudythequantumcaseaswell. Ourismeasuredinbitsandcangrowwithsystemsizelikeanextrinsicvariable, whereashisisanintrinsicvariableakinrepresentingasortofaverageintegrationperbit.(providingintegration)butnotsostrongthatthewholesystembecomescorrelatedlikeintherightpanelorinabrainexperiencinganepilepticseizure.D. Integration, codingtheoryanderrorcorrectionEvenwhenwetunedthetemperaturetothemostfa-vorablevalueinour2DIsingmodel example, theinte-gratedinformationneverexceeded n1/2bits,whichis merelyafractionn1/2of thenbits of informationthatnspinscanpotentiallystore. Socanwedobetter?Fortunately, a closely related question has been carefullystudiedinthebranchof mathematicsknownascodingtheory, with the aim of optimizing error correcting codes.Consider, forexample, thefollowingsetof m=16bitstrings,eachwrittenasacolumnvectoroflengthn = 8:M=___________00001111000011110000000011111111011010010110100101010101101010100101101001011010001100111100110000111100001111000110011010011001___________This is known as the Hamming(8,4)-code,and has Ham-mingdistanced=4, whichmeans that at least 4bitipsarerequiredtochangeonestringintoanother[20].ItiseasytoseethatforacodewithHammingdistanced,any(d 1)bitscanalwaysbereconstructedfromtheothers: You can always reconstruct b bits as long as eras-ingthemdoesnotmaketwobitstringsidentical, whichwould cause ambiguity about which the correct bit stringis. This implies that reconstruction works when the Ham-mingdistanced > b.Totranslatesuchcodes of mbit strings of lengthnintophysical systems, wesimplycreatedastatespacewithnbits(interpretableasnspinsorothertwo-statesystems)andconstructaHamiltonianwhichhasanm-folddegenerate groundstate, withone minimumcor-responding to each of the mbit strings in the code.In the low-temperature limit, all bit strings will re-ceive the same probability weight 1/m, giving an entropyS=log2m. Thecorrespondingintegratedinformationof thegroundstateis plottedinFigure3for afewexamples,asafunctionofcutsizek(thenumberofbitsassigned to the rst subsystem). To calculate for a cutsizekinpractice,wesimplyminimizethemutualinfor-mation Iover all_nk_ways of partitioning the n bits intokand(n k)bits.We see that, as advertised, the Hamming(8,4)-codegives gives =3when3bits are cut o. However,itgivesonly=2forbipartitions; the-valueforbi-partitions is not simply related to the Hamming distance,andisnotaquantitythatmostpopularbitstringcodesare optimized for. Indeed, Figure 3 shows that for bipar-titions,itunderperformsacodeconsistingof16random8MorecorrelationToo little Too much Optimum Uniform RandomLesscorrelationFIG.2: Thepanelsshowsimulationsofthe2DIsingmodelona50 50lattice,withthetemperatureprogressivelydecreasingfromleft toright. Theintegratedinformationdrops tozerobits at T (leftmost panel) andtoonebit as T0(rightmostpanel),takingamaximumatanintermediatetemperaturenearthephasetransitiontemperature.Bits cut offIntegrated informationHamming (8,4)-code(16 8-bit strings)16 random 8-bit strings128 8-bit stringswith checksum bit2 4 6 81234FIG. 3: For various 8-bit systems, the integrated informationisplottedasafunctionof thenumberof bitscutointoasub-system with the cruelest cut. The Hamming (8,4)-codeisseentogiveclassicallyoptimalintegrationexceptforabi-partitioninto4 + 4bits: anarbitrarysubsetcontainingnomore than three bits is completely determined by the remain-ingbits. Thecodeconsistingof thehalf of all 8-bitstringswhosebitsumiseven(i.e., eachofthe1287-bitstringsfol-lowed by a parity checksum bit) has Hamming distance d = 2and gives = 1 however many bits are cut o. A random setof168-bitstringsisseentooutperformtheHamming(8,4)-code for 4+4-bipartitions, but not when fewer bits are cut o.unique bit strings of the same length. A rich and diversesetof codeshavebeenpublishedintheliterature, andthestate-of-the-artintermsof maximal Hammingdis-tance for a given n is continually updated [21]. Althoughcodes witharbitrarilylargeHammingdistancedexist,thereis (just asfor our Hamming(8,4)-exampleabove)noguaranteethatwill beaslargeasd 1whenthesmaller of the two subsystems contains more than d bits.Moreover, althoughReed-Solomoncodesaresometimesbilledas classicallyoptimal erasurecodes (maximizingdfor agivenn), their fundamental unitsaregenerallyBits cut offIntegrated information256 random 16-bit words5 10 152468128 random 14-bit words64 12-bit words32 10-bit words16 8-bit words8 6-bit words4 4-bit wordsFIG. 4: Sameasforpreviousgure, butforrandomcodeswithprogressivelylongerbitstrings, consistingof arandomsubset containing2nof the 2npossible bit strings. Forbetterlegibility,theverticalaxishasbeenre-centeredfortheshortercodes.not bits but groups of bits (generallynumbers modulosome prime number), and the optimality is violated if wemake cuts that do not respect the boundaries of these bitgroups.Although further research on codes maximizing wouldbeofinterest, itisworthnotingthatsimpleran-dom codes appear to give -values within a couple of bitsofthetheoreticalmaximuminthelimitoflargen,asil-lustratedinFigure4. Whencuttingokoutofnbits,themutual informationinclassical physicsclearlycan-not exceed the number of bits in either subsystem, i.e., kandn k,sothe-curveforacodemustliewithintheshadedtriangleinthegure. (Thequantum-mechanicalcase is more complicated, and we well see in the next sec-tionthatitinasenseintegratesbothbetterandworse.)The codes for which the integrated information is plottedsimplyconsistofarandomsubsetcontaining2n/2ofthe2npossible bitstrings,so roughlyspeaking,halfthe bitsencodefreshinformationandtheotherhalfprovidethe92-logarithm of number of patterns usedIntegrated information (bits)2 4 6 8 10 12 141234567FIG. 5: The integratedinformationis shownfor randomcodes usingprogressivelylarger randomsubsets of the 214possiblestringsof 14bits. Theoptimal choiceisseentobeusingabout27bitstrings, i.e., usingabouthalf thebitstoencodeinformationandtheotherhalftointegrateit.redundancygivingnear-perfectintegration.Just as we saw for the Ising model example, these ran-domcodesshowatradeobetweenentropyandredun-dancy,asillustratedinFigure5. Whentherearenbits,howmanyof the2npossiblebit stringsshouldweusetomaximizetheintegratedinformation? Ifweusemof them, weclearlyhave log2m, sinceinclassicalphysics, cannotexceedtheentropyifthesystem(themutual informationisI =S1+ S2 S, whereS1 SandS2 SsoIS). Usingveryfewbit strings isthereforeabadidea. Ontheotherhand, if weuseall2nofthem, weloseall redundancy, thebitsbecomein-dependent, and=0, sobeinggreedyandusingtoomanybitstringsinanattempttostoremoreinforma-tionisalsoabadidea. Figure5showsthattheoptimaltradeoistouse2nofthecodewords,i.e.,tousehalfthebitstoencodeinformationandtheotherhalftoin-tegrateit. Takentogether,thelasttwoguresthereforesuggest that n physical bits can be used to provide aboutn/2bitsofintegratedinformationinthelarge-nlimit.E. Integrationinphysical systemsLet us explore the consequences of these results forphysical systems describedbyaHamiltonianHandastate. As emphasizedbyHopeld[22], anyphysicalsystemwithmultipleattractorscanbeviewedasanin-formation storage device,since its state permanently en-codes informationabout whichattractor it belongs to.Figure6showstwoexamplesof Hinterpretableaspo-tentialenergyfunctionsforaasingleparticleintwodi-mensions. They can both be used as information storagedevices, byplacingtheparticleinapotential well andkeepingthesystemcool enoughthattheparticlestaysinthe same well indenitely. The eggcrate potentialV (x, y) =sin2(x) sin2(y) (top) has 256minimaandhence agroundstate entropy(informationstorage ca-pacity)S= 8bits,whereasthelowerpotentialhasonlyFIG. 6: Aparticleintheegg-cratepotential energyland-scape(toppanel) stablyencodes8bitsof informationthatarecompletelyindependentofoneanotherandthereforenotintegrated. Incontrast,aparticleinaHamming(8,4)poten-tial (bottompanel) encodesonly4bitsof information, butwith excellent integration. Qualitatively, a hard drive is morelikethetoppanel, whileaneural networkis morelikethebottompanel.16minimaandS= 4bits.Thebasinsof attractioninthetoppanel areseentobethesquaresshowninthebottompanel. If wewritethe xandycoordinates as binarynumbers withbbitseach, thentherst4bitsof xandyencodewhichsquare (x, y) is in. The informationinthe remainingbitsencodesthelocationwithinthissquare; thesebitsarenotuseful forinformationstoragebecausetheycanvaryovertime, astheparticleoscillatesaroundamini-mum. Ifthesystemisactivelycooled, theseoscillationsare gradually damped out and the particle settles towardthe attractor solution at the minimum, at the center of itsbasin. This example illustrates thatcooling is a physicalexampleoferrorcorrection: ifthermal noiseaddssmallperturbationstotheparticleposition, alteringtheleastsignicantbits,thencoolingwillremovetheseperturba-tionsandpushtheparticlebacktowardstheminimumit came from. As long as cooling keeps the perturbationssmall enough that the particle never rolls out of its basinof attraction, all the8bitsof informationencodingitsbasinnumberareperfectlypreserved. Insteadof inter-pretingourn=8databitsaspositionsintwodimen-sions, we can interpret them as positions in n dimensions,whereeachpossiblestatecorrespondstoacornerofthen-dimensional hypercube. Thiscapturestheessenceofmany computer memory devices, where each bit is storedinasystemwithtwodegenerateminima; theleastsig-nicantandredundantbitsthatcanbeerror-correctedvia cooling now get equally distributed among all the di-mensions.10How integrated is the information S?For the top panelof Figure6, not at all: Hcanbefactoredas atensorproduct of 8two-state systems, so=0, just as fortypical computer memory. In other words, if the particleis in a particular egg crate basin, knowing any one of thebitsspecifyingthebasinpositiontellsusnothingabouttheotherbits. Thepotential inthelowerpanel, ontheother hand, gives good integration. This potential retainsonly16of the256minima, correspondingtothe16bitstrings of the Hamming(8,4)-code, which as we saw gives = 3 for any 3 bits cut o and = 2 bits for symmetricbipartitions. SincetheHammingdistanced = 4forthiscode, at least 4bits must be ippedtoreachanotherminimum, which among other things implies that no twobasinscansharearoworcolumn.F. TheprosandconsofintegrationNatural selection suggests that self-reproducinginformation-processingsystemswillevolveintegrationifitisuseful tothem, regardlessofwhethertheyarecon-scious or not. Error correctioncanobviouslybe use-ful, bothtocorrecterrorscausedbythermal noiseandtoprovideredundancythatimprovesrobustnesstowardfailure of individual physical components suchas neu-rons. Indeed, suchutilityexplains the preponderanceof error correctionbuilt intohuman-developeddevices,fromRAID-storage tobar codes toforwarderror cor-rectionintelecommunications. If Tononi iscorrectandconsciousness requires integration, then this raises an in-terestingpossibility: ourhumanconsciousnessmayhaveevolvedasanaccidental by-productof errorcorrection.There is also empirical evidence that integration is usefulforproblem-solving: articiallifesimulationsofvehiclesthathavetotraversemazesandwhosebrainsevolvebynatural selection show that the more adapted they are totheir environment, the higher the integrated informationofthemaincomplexintheirbrain[23].However, integrationcomesatacost, andaswewillnowsee, near maximal integrationappears tobe pro-hibitively expensive. Let us distinguish between the max-imum amount of information that can be stored in a statedenedbyandthemaximumamountof informationthat can be stored in a physical system dened by H. TheformerissimplyS()fortheperfectlymixed(T= )state, i.e., log2 of the number of possible states (the num-ber of bits characterizingthesystem). Thelatter canbemuchlarger, correspondingtolog2of thenumberofHamiltoniansthatyoucoulddistinguishbetweengivenyourtimeandenergyavailableforexperimentation. Letusconsiderpotential energyfunctionswhosekdierentminimacanbeencodedasbitstrings(asinFigure6),andlet us limit our experimentationtondingall theminima. ThenHencodesnotasinglestringof nbits,but a subset consisting of k out of all 2nsuch strings, oneforeachminimum. Thereare_2nk_suchsubsets,sotheinformationcontainedinHisS(H) =log2_2nk_= log22n!k!(2nk)! log2(2n)kkk= k(n log2k) (7)for k 2n, where we used Stirlings approximationk!kk. Socrudelyspeaking, Hencodes not nbitsbut knbits. For the near-maximal integration givenbytherandomcodesfromtheprevioussection, wehadk= 2n/2,whichgivesS(H) 2n/2 n2bits. Forexample,if then 1011neurons inyour brainweremaximallyintegratedinthisway, thenyourneural networkwouldrequireadizzying1010000000000bits todescribe, vastlymoreinformationthancanbeencodedbyall the1089particlesinouruniversecombined.Theneuronal mechanismsofhumanmemoryarestillunclear despite intensive experimental and theoretical ex-plorations[24],butthereissignicantevidencethatthebrain uses attractor dynamics in its integration and mem-oryfunctions, wherediscreteattractorsmaybeusedtorepresent discrete items [25]. The classic implementa-tionof suchdynamicsasasimplesymmetricandasyn-chronous Hopeld neural network [22] can be conve-niently interpreted in terms of potential energy func-tions: theequationsofthecontinuousHopeldnetworkareidentical toasetofmean-eldequationsthatmini-mizeapotentialenergyfunction,sothisnetworkalwaysconvergestoabasinofattraction[26]. SuchaHopeldnetworkgivesadramaticallylowerinformationcontentS(H) of only about 0.25 bits per synapse[26], and we haveonly about 1014synapses, suggesting that our brains canstore only on the order of a few Terabytes of information.The integratedinformationof a Hopeldnetworkisevenlower. For aHopeldnetworkof nneurons withHebbian learning, the total number of attractors isboundedby0.14n[26],sothemaximuminformationca-pacityismerelyS log2 0.14n log2n 37bitsforn = 1011neurons. Even in the most favorable case wherethese bits are maximallyintegrated, our 1011neuronsthus provide a measly 37 bits of integrated informa-tion, as opposed to about 51010bits for a randomcoding.G. TheintegrationparadoxThisleavesuswithanintegrationparadox: whydoestheinformationcontentofourconsciousexperienceap-pear to be vastly larger than 37 bits?If Tononis informa-tion and integration principles from Section I are correct,the integration paradox forces us6to draw at least one of6Can we sidestep the integration paradox by simply dismissing theideathatintegrationisnecessary? Althoughitremainscontro-11thefollowingthreeconclusions:1. Our brains use some more clever scheme for encod-ingourconsciousbitsofinformation,whichallowsdramaticallylargerthanHebbianHopeldnet-works.2. These conscious bits are much fewer than we mightnaivelyhavethoughtfromintrospection, implyingthat weareonlyabletopayattentiontoaverymodestamountofinformationatanyinstant.3. Toberelevantforconsciousness, thedenitionofintegratedinformationthatwehaveusedmustbemodied or supplemented by at least one additionalprinciple.Wewillseethatthequantumresultsinthenextsectionbolsterthecaseforconclusion3. Interestingly, thereisalsosupportforconclusion2inthelargepsychophysicalliterature on the illusion of the perceptual richness of theworld. Forexample,thereisevidencesuggestingthatoftheroughly107bitsofinformationthatenterourbraineachsecondfromour sensoryorgans, we canonlybeawareofatinyfraction,withestimatesrangingfrom10to50bits[27,28].The fundamental reasonwhy a Hopeldnetwork isspecied by much less information than a near-maximallyintegratednetworkisthatitinvolvesonlypairwisecou-plings betweenneurons, thus requiringonly n2cou-plingparameterstobespeciedasopposedto2npa-rameters givingtheenergyfor eachof the2npossiblestates. It is strikinghowHis similarlysimplefor thestandardmodel of particlephysics, withtheenergyin-volving only sums of pairwise interactions between parti-clessupplementedwithoccasional3-wayand4-waycou-plings. HforthebrainandHforfundamental physicsthusbothappeartobelongtoanextremelysimplesub-class of all Hamiltonians, that require an unusually smallamountofinformationtodescribe. Justasasystemim-plementingnear-maximalintegrationvia randomcodingistoocomplicatedtotinsidethebrain, itisalsotoocomplicatedtoworkinfundamental physics: SincetheinformationstoragecapacitySof aphysical systemisapproximately bounded by its number of particles [16] orbyitsareainPlanckunitsbytheHolographicprinciple[17], itcannotbeintegratedbyphysical dynamicsthatitself requires storage of the exponentially larger informa-tionquantityS(H) 2S/2 S2unlesstheStandardModelHamiltonian is replaced by something dramatically morecomplicated.versial whether integratedinformationisasucient conditionforconsciousnessasassertedbyIIT, itappearsratherobviousthat it is anecessary conditionif the conscious experience isunied: if therewerenointegration, theconsciousmindwouldconsistof twoseparatepartsthatwereindependentof onean-otherandhenceunawareofeachother.An interesting theoretical direction for further research(pursuing resolution 1 to the integration paradox) istherefore toinvestigate what maximumamount of in-tegrated information can be feasibly stored in a physi-cal system using codes that are algorithmic (such as RS-codes)ratherthanrandom. Aninterestingexperimentaldirectionwouldbe tosearchfor concrete implementa-tionsoferror-correctionalgorithmsinthebrain.Insummary, we have exploredthe integrationprin-ciplebyquantifyingintegratedinformationinphysicalsystems. We have found that although excellent integra-tionispossibleinprinciple, itismoredicultinprac-tice. Intheory, randomcodes providenearlymaximalintegration,withabouthalfofallnbitscodingfordataandtheotherhalfproviding nbitsofintegration),but inpractice, the dynamics requiredfor implement-ingthemistoocomplexforourbrainorouruniverse.Mostofourexplorationhasfocusedonclassicalphysics,wherecutsintosubsystemshavecorrespondedtoparti-tionsofclassicalbits. Aswewillseeinthenextsection,nding systems encoding large amounts of integrated in-formationisevenmorechallengingwhenweturntothequantum-mechanicalcase.III. INDEPENDENCEA. Classical versusquantumindependenceHowcrueliswhatTononicallsthecruelestcut, di-vidingasystemintotwoparts that aremaximallyin-dependent? Thesituationisquitedierentinclassicalphysicsandquantumphysics, asFigure7illustratesforasimple2-bitsystem. Inclassical physics, thestateisspecied by a 22 matrix giving the probabilities for thefourstates00,01,10and11,whichdeneanentropySandmutual informationI. Sincethereisonlyonepos-siblecut, theintegratedinformation=I. Thepointdenedbythepair(S, )canlieanywhereinthepyra-midinthegure, whos topat (S, ) =(1, 1) (blackstar) gives maximum integration, and corresponds to per-fect correlation between the two bits: 50% probability for00and11. Perfectanti-correlationgivesthesamepoint.Theothertwoverticesof theclassicallyallowedregionare seen to be (S, ) = (0, 0) (100% probability for a sin-gle outcome) and (S, ) = (2, 0) (equal probability for allfouroutcomes).Inquantummechanics, wherethe2-qubitstateisde-nedbya4 4densitymatrix,theavailableareainthe(S, I)-planedoublestoincludetheentireshadedtrian-gle, withtheclassicallyunattainableregionopenedupbecause of entanglement. The extreme case is a Bell pairstatesuchas[ =12([[ +[[) , (8)whichgives(S, I)=(0, 2). However, whereastherewasonly one possible cut for 2 classical bits, there are now in-120.5 1.0 1.5 2.00.51.01.52.0Entropy SMutual information IPossible onlyquantum-mechanically(entanglement)PossibleclassicallyBell pair().4.4.2.0().91.03.03.03()1/31/31/30()1000().7.1.1.1()1/2001/2()1/21/200()1/41/41/41/4().3.3.3.1Quantum integratedUnitary transformationUnitary transformationFIG. 7: Mutual informationversusentropyforvarious2-bitsystems. The dierent dots, squares andstars correspondtodierent states, whichinthe classical cases are denedbytheprobabilities for thefour basis states 00, 0110and11. Classical states canlie onlyinthepyramidbelowtheupperblackstarwith(S, I)=(1, 1), whereasentanglementallowsquantumstatestoextendallthewayuptotheupperblack square at (0, 2). However,the integrated information foraquantumstatecannotlieabovetheshadedgreen/greyregion, intowhichanyotherquantumstatecanbebroughtbyaunitarytransformation. Alongtheupperboundaryofthisregion,eitherthreeofthefourprobabilitiesareequal,ortotwoofthemareequalwhileonevanishes.nitelymanypossiblecutsbecauseinquantummechan-ics,allHilbertspacebasesareequallyvalid,andwecanchoose to perform the factorization in any of them. SinceisdenedasIafterthecruelestcut, itistheI-valueminimized over all possible factorizations. For simplicity,we use the notation where denotes factorization in thecoordinatebasis,sotheintegratedinformationis = minUI(UU), (9)i.e., the mutual informationminimizedover all possi-ble unitarytransformations U. Since the Bell pair ofequation(8)isapurestate = [[,wecanunitarilytransformitintoabasiswheretherstbasisvectoris[,makingitfactorizable:U____12001200000000120012____U=___1000000000000000___ =_1000__1000_.(10)Thismeansthat=0, soinquantummechanics, thecruelest cut can be very cruel indeed: the most entangledstates possible in quantum mechanics have no integratedinformationatall!The same cruel fate awaits the most integrated 2-bitstatefromclassical physics: theperfectlycorrelatedmixedstate=12[ [ +12[ [. It gave=1bitclassicallyabove(upperblackstarinthegure), butaunitarytransformationpermutingitsdiagonal elementsmakesitfactorable:U____120000000000000012____U=____12000012000 0000 000____=_1000__120012_,(11)so = 0quantum-mechanically(lowerblackstarinthegure).B. Canonical transformations, independenceandrelativityThe fundamental reason that these states are more sep-arable quantum-mechanicallyis clearlythat more cutsareavailable, makingthecruelestonecrueler. Interest-ingly, the same thing can happen also in classical physics.Consider, for example, our example of the deuteriumatomfromequation(1). Whenwerestrictedourcutstosimplyseparatingdierentdegreesoffreedom,wefoundthatthegroup(rp, pp, rn, pn)wasquite(butnotcom-pletely) independent of the group (re, pe), and that therewas nocut splittingthings intoperfectlyindependentpieces. Inotherwords, thenucleuswasfairlyindepen-dentoftheelectron,butnoneofthethreeparticleswascompletely independent of the other two. However, if weallowour degrees of freedomtobetransformedbeforethecut, thenthingscanbesplitintotwoperfectlyin-dependent parts! Theclassical equivalent of aunitarytransformationis of course acanonical transformation(onethatpreservesphase-spacevolume). Ifweperformthecanonicaltransformationwherethenewcoordinatesarethecenter-of-masspositionrMandtherelativedis-placements r

p rp rMandr

e re rM, andcor-respondinglydenepMas thetotal momentumof thewholesystem, etc., thenwendthat(rM, pM)iscom-pletelyindependentoftherest. Inotherwords, theav-eragemotionoftheentiredeuteriumatomiscompletelydecoupledfromtheinternal motionsarounditscenter-of-mass.Interestingly,thiswell-knownpossibilityofdecompos-ing any isolated system into average and relative motions(the average-relative decomposition, for short) is equiv-alent to relativitytheoryinthe following sense. Thecoreof relativitytheoryis that all laws of physics (in-cludingthespeedof light) arethesameinall inertialframes. Thisimpliestheaverage-relativedecomposition,sincethelawsof physicsgoverningtherelativemotionsof the systemare the same inall inertial frames andhenceindependentof the(uniform)center-of-massmo-tion. Conversely,wecanviewrelativityasaspecialcase13of the average-relative decomposition. If two systems arecompletelyindependent, thentheycangainnoknowl-edgeof eachother, soaconscious observer inonewillbeunawareoftheother. Theaverage-relativedecompo-sitionthereforeimplies that anobserver inanisolatedsystemhasnowayofknowingwhethersheisatrestorin uniform motion, because these are simply two dierentallowed states for the center-of-mass subsystem, which iscompletely independent from (and hence inaccessible to)theinternal-motionssubsystemof whichherconscious-nessisapart.C. Howintegratedcanquantumstatesbe?We saw in Figure 7 that some seemingly inte-grated states, such as a Bell pair or a pair of clas-sically perfectly correlated bits, are in fact not inte-grated at all. But the gure also shows that somestates aretrulyintegratedevenquantum-mechanically,with I > 0 even for the cruelest cut. How inte-gratedcana quantumstate be? The following theo-rem, provedby Jevtic, Jennings &Rudolph[29], en-ables the answer to be straightforwardly calculated7:-DiagonalityTheorem(DC):Themutualinformationalwaystakesitsmin-imuminabasiswhereisdiagonalThe rst step in computing the integrated information()isthustodiagonalizethenndensitymatrix.If all neigenvaluesaredierent, thentherearen! pos-sibleways of doingthis, correspondingtothen! waysof permutingtheeigenvalues, sotheDCsimpliesthecontinuous minimization problem of equation (9) to a dis-creteminimizationproblemoverthesen! permutations.Supposethatn = l m,andthatwewishtofactorthen-dimensionalHilbertspaceintofactorspacesofdimen-sionalityl andm, sothat=0. Itiseasytoseethatthisispossibleiftheneigenvaluesofcanbearrangedintoanlmmatrixthat is multiplicativelyseparable(rank1),i.e.,theproductofacolumnvectorandarowvector. Extractingtheeigenvaluesforourexamplefromequation(11)wherel = m = 2andn = 4,weseethat_12120 0_isseparable,but_120012_isnot,andthattheonlydierenceisthattheorderofthefournumbers has been permuted. More generally, we see thattondthecruelestcutthatdenestheintegratedin-formation , we want to nd the permutation that makesthematrixof eigenvaluesasseparableaspossible. Itis7Theconverseof theDCisstraightforwardtoprove: if =0(which is equivalent to the state being factorizable; = 12),thenitisfactorizablealsoinitseigenbasiswhereboth1and2arediagonal.easytosee that whenseekingthe permutationgivingmaximumseparability,wecanwithoutlossofgeneralityplace the largest eigenvalue rst (in the upper left corner)andthesmallestonelast(inthelowerrightcorner). Ifthereareonly4eigenvalues(asintheaboveexample),theorderingoftheremainingtwohasnoeectonI.D. ThequantumintegrationparadoxWe now have the tools in hand to answer the key ques-tionfromthelastsection: whichstatemaximizestheintegrated information ? Numerical search suggeststhat the most integratedstate is arescaledprojectionmatrixsatisfying2. Thismeansthatsomenum-ber k of the neigenvalues equal 1/k andthe remain-ing ones vanish.8For the n =4 example fromFig-ure7, k=3is seentogivethebest integration, witheigenvalues (probabilities) 1/3, 1/3, 1/3and0, giving = log(27/16)/ log(8) 0.2516.For classical physics, we sawthat the maximal at-tainable grows roughly linearly withn. Quantum-mechanically,however,itdecreasesasnincreases!9Insummary, nomatterhowlargeaquantumsystemwecreate,itsstatecannevercontainmorethanaboutaquarterof abitof integratedinformation! Thisexacer-bates the integration paradox from Section II G, eliminat-ing both of the rst two resolutions: you are clearly awareof more than 0.25 bits of information right now, and thisquarter-bit maximum applies not merely to states of Hop-eld networks, but to any quantum states of anysystem.Let us therefore begin exploring the third resolution: thatour denition of integrated information must be modiedorsupplementedbyatleastoneadditionalprinciple.8Aheuristicwayofunderstandingwhyhavingmanyequaleigen-valuesisadvantageousisthatithelpseliminatetheeectoftheeigenvaluepermutationsthatweareminimizingover. Iftheop-timal statehastwodistincteigenvalues, thenif swappingthemchanges I, it must by denition increase Iby some nite amount.Thissuggeststhatwecanincreasetheintegrationbybringingthe eigenvalues innitesimally closer or further apart, and repeat-ingthisprocedureletsusfurtherincreaseuntilalleigenvaluesareeitherzeroorequaltothesamepositiveconstant.9One nds that is maximizedwhenthe k identical nonzeroeigenvalues are arranged in a Young Tableau, which corre-sponds to a partition of k as a sum of positive integersk1+k2+..., giving = S(p) +S(p) log2 k, wherethe probability vectors pand pare dened by pi=ki/kand pi= ki /k. Here kidenotes the conjugate partition.For example, if we cut an even number of qubits into twoparts with n/2 qubits each, then n = 2, 4, 6, ..., 20 gives 0.252, 0.171, 0.128, 0.085, 0.085, 0.073, 0.056, 0.056, 0.051and0.042bits,respectively.14E. HowintegratedistheHamiltonian?Anobvious waytobeginthis explorationis tocon-siderthestatenotmerelyatasinglexedtimet,butas a function of time. After all, it is widely assumed thatconsciousness is relatedto informationprocessing, notmereinformationstorage. Indeed, Tononisoriginal -denition[8](whichappliestoclassicalneuralnetworksrather than general quantum systems) involves time,de-pendingontheextenttowhichcurrenteventsaectfu-tureones.Because the time-evolution of the state is determinedbytheHamiltonianHviatheSchrodingerequation = i[H, ], (12)whosesolutionis(t) = eiHteiHt, (13)weneedtoinvestigatetheextenttowhichthecruelestcutcandecomposenotmerelybutthepair(, H)intoindependentparts. (Hereandthroughout, weoftenuseunitswhere= 1forsimplicity.)F. EvolutionwithseparableHamiltonianAswesawabove,thekeyquestionforiswhetherititisfactorizable(expressibleasproduct =1 2ofmatrices acting on the two subsystems), whereas the keyquestionfor His whether it is what wewill call addi-tivelyseparable, beingasumof matricesactingonthetwosubsystems,i.e.,expressibleintheformH = H1I +I H2(14)forsomematricesH1andH2. Forbrevity,wewilloftenwrite simply separable instead of additively separable. AsmentionedinSectionII B, aseparable HamiltonianHimplies that boththe thermal state eH/kTandthetime-evolutionoperatorU eiHt/arefactorizable.Animportant propertyof densitymatrices whichwaspointedoutalreadybyvonNeumannwhenheinventedthem[30]isthatifHisseparable,then 1= i[H1, 1], (15)i.e., the time-evolution of the state of the rst subsystem,1 tr2,isindependentoftheothersubsystemandofanyentanglement withitthatmayexist. This is easytoprove: Usingtheidentities(A12)and(A14)showsthattr2[H1I, ] =tr2(H1I) tr2(H1I)=H111H2= [H1, 1]. (16)Usingtheidentity(A10)showsthattr2[I H2, ] = 0. (17)Summingequations(16)and(17)completestheproof.G. ThecruelestcutasthemaximizationofseparabilitySinceageneral HamiltonianHcannotbewritteninthe separable form of equation (14), it will also include athirdtermH3thatisnon-separable. TheindependenceprinciplefromSectionI thereforesuggests aninterest-ingmathematical approachtothephysics-from-scratchproblemof analyzingthe total HamiltonianHfor ourphysicalworld:1. Find the Hilbert space factorization giving thecruelestcut,decomposingHintopartswiththesmallestinteractionHamiltonianH3possible.2. Keep repeating this subdivision procedure for eachpart until onlyrelativelyintegratedparts remainthat cannot be further decomposedwithasmallinteractionHamiltonian.ThehopewouldbethatapplyingthisproceduretotheHamiltonian of our standard model would reproduce thefull observed object hierarchy from Figure 1, with the fac-torizationcorrespondingtotheobjects, andthevariousnon-separabletermsH3describingtheinteractionsbe-tweentheseobjects. AnydecompositionwithH3=0would correspond to two parallel universes unable tocommunicatewithoneanother.Wewill nowformulatethisasarigorousmathemat-icsproblem,solveit,andderivetheobservationalconse-quences. We will nd that this approach fails catastroph-icallywhenconfrontedwithobservation,givinginterest-ing hints regarding further physical principles needed forunderstandingwhyweperceiveourworldasanobjecthierarchy.H. TheHilbert-SchmidtvectorspaceToenablearigorousformulationof ourproblem, letus rst briey review the Hilbert-Schmidt vector space, aconvenient inner-product space where the vectors are notwavefunctions [butmatricessuchasHand. Foranytwomatrices AandB, theHilbert-Schmidt innerproductisdenedby(A, B) tr AB. (18)For example, the trace operator canbe writtenas aninnerproductwiththeidentitymatrix:tr A = (I, A). (19)This inner product denes the Hilbert-Schmidt norm(alsoknownastheFrobeniusnorm)[[A[[ (A, A)12= (tr AA)12=__

ij[Aij[2__12. (20)15If AisHermitian(A=A), then [[A[[2issimplythesumofthesquaresofitseigenvalues.Real symmetricandantisymmetricmatricesformor-thogonal subspaces under the Hilbert-Schmidt innerproduct, since(S, A) =0for anysymmetricmatrixS(satisfying St=S) andanyantisymmetric matrixA(satisfying At= A). Because a Hermitian matrix (sat-isfyingH=H) canbewritteninterms of real sym-metricandantisymmetricmatricesasH=S + iA, wehave(H1, H2) = (S1, S2) + (A1, A2),whichmeans that the inner product of twoHermitianmatricesispurelyreal.I. SeparatingHwithorthogonal projectorsByviewingHasavectorintheHilbert-Schmidtvec-torspace, wecanrigorouslydeneandecompositionofit into orthogonal components,two of which are the sep-arable terms from equation (14). Given a factorization ofthe Hilbert space where the matrix H operates, we denefourlinearsuperoperators10iasfollows:0H1n(tr H) I (21)1H_1n2tr2H_I20H (22)2HI1_1n1tr1H_0H (23)3H(I 123)H (24)Itisstraightforwardtoshowthatthesefourlinearop-eratorsiformacompletesetoforthogonalprojectors,i.e.,that3

i=0i=I, (25)ij=iij, (26)(iH, jH) =[[iH[[2ij. (27)This means that anyHermitianmatrixHcanbe de-composed as a sum of four orthogonal components Hi iH, sothatitssquaredHilbert-Schmidtnormcanbedecomposed as a sum of contributions from the four com-10Operators ontheHilbert-Schmidt spaceareusuallycalledsu-peroperators in the literature,to avoid confusions with operatorsontheunderlyingHilbertspace, whicharemerevectorsintheHilbert-Schmidtspace.ponents:H=H0 +H1 +H2 +H3, (28)HiiH, (29)(Hi, Hj) =[[Hi[[2ij, (30)[[H[[2=[[H0[[2+[[H1[[2+[[H2[[2+[[H3[[2. (31)We see that H0 I picks out the trace of H, whereas theotherthreematricesaretrace-free. Thistracetermisofcourse physically uninteresting, since it can be eliminatedby simply adding an unobservable constant zero-point en-ergytotheHamiltonian. H1andH2correspondstothetwoseparabletermsinequation(14)(withoutthetraceterm, whichcouldhavebeenarbitrarilyassignedtoei-ther), and H3corresponds to the non-separable residual.A Hermitian matrix His therefore separable if and only if3H = 0. Just as it is customary to write the norm or avector r by r [r[ (without boldface), we will denote theHilbert-SchmidtnormofamatrixHbyH [[H[[. Forexample, with this notation we can rewrite equation (31)assimplyH2= H20+H21+H22+H23.Geometrically, we can think of nn Hermitian matri-cesHaspointsintheN-dimensional vectorspaceRN,where N= nn (Hermiteal matrices have n real numbersonthe diagonalandn(n 1)/2complexnumbers othediagonal,constitutingatotalofn +2 n(n 1)/2 = n2real parameters). Diagonal matricesformahyperplaneof dimensionninthisspace. Theprojectionoperators0,1,2and3projectontohyperplanesofdimen-sion1, (n 1), (n 1) and(n 1)2, respectively, soseparable matrices form a hyperplane in this space of di-mension2n 1. Forexample,ageneral4 4Hermitianmatrix can be parametrized by 10 numbers (4 real for thediagonal partand6complexfortheo-diagonal part),and its decomposition from equation (28) can be writtenasfollows:H=___t+a+b+v d+w c+x yd+wt+abv z cxc+xzta+bv dwycxdwtab+v___ ==___t 0000t 0000t 0000t___+___a 0 c 00 a 0 cc0a 00 c0 a___+___b d 0 0d b 0 00 0 b d0 0 d b___++___v w x ywv z xxzv wyx wv___ (32)Weseethat t contributes tothetrace(andH0) whilethe other three components Hiare traceless. We also seethattr1H2=tr2H1=0, andthatbothpartial tracesvanishforH3.16J. MaximizingseparabilityWenowhaveallthetoolsweneedtorigorouslymax-imizeseparabilityandtestthephysics-from-scratchap-proachdescribedinSectionIII G. GivenaHamiltonianH, we simply wishto minimize the normof its non-separablecomponentH3overall possibleHilbertspacefactorizations, i.e., overall possibleunitarytransforma-tions. Inotherwords,wewishtocomputeE minU[[3H[[, (33)wherewehavedenedtheintegrationenergyEbyanal-ogywiththeintegratedinformation. IfE=0, thenthere is a basis where our system separates into two paral-lel universes, otherwiseE quanties the coupling betweenthetwopartsofthesystemunderthecruelestcut.TheHilbert-Schmidtspaceallowsustointerprettheminimizationproblemofequation(33)geometrically,asillustratedinFigure8. LetHdenotetheHamiltonianinsomegivenbasis, andconsideritsorbitH=UHUunder all unitary transformations U. This is a curved hy-persurfacewhosedimensionalityisgenericallyn(n 1),i.e.,n lower than that of the full space of Hermitian ma-trices, sinceunitarytransformationleaveall neigenval-uesinvariant.11Wewill refer tothiscurvedhypersur-faceasasubsphere,becauseitisasubsetofthefulln2-dimensional sphere: theradiusH(theHilbert-Schmidtnorm [[H[[)isinvariantunderunitarytransformations,but the subsphere may have a more complicated topologythanahypersphere; forexample, the3-sphereisknownto topologically be the double cover of SO(3), the matrixgroupof3 3orthonormaltransformations.We are interested in nding the most separable point Hon this subsphere, i.e., the point on the subsphere that isclosest to the (2n1)-dimensional separable hyperplane.In our notation, this means that we want to nd the pointH on the subsphere that minimizes [[3H[[, the Hilbert-Schmidt normof the non-separable component. If weperform innitesimal displacements along the subsphere,[[3H[[ thus remains constant to rst order (the gradientvanishesattheminimum), soall tangentvectorsofthesubsphereareorthogonal to3H, thevectorfromtheseparablehyperplanetothesubsphere.Unitary transformations are generated by anti-Hermitianmatrices, sothemostgeneral tangentvectorHisoftheformH = [A, H] AHHA (34)for some anti-Hermitian nn matrix A (any matrix sat-isfyingA= A). Wethusobtainthefollowingsimple11nn-dimensional Unitary matrices Uare known to form an nn-dimensional manifold: theycanalwaysbewrittenasU=eiHforsomeHermitianmatrixH, sotheyareparametrizedbythesamenumberofrealparameters(n n)asHermitianmatrices.conditionformaximalseparability:(3H, [A, H]) = 0 (35)for any anti-Hermitian matrix A. Because the most gen-eral anti-Hermitian matrix can be written as A = iB fora Hermitian matrix B,equation (35) is equivalent to thecondition(3H, [B, H])=0forall HermitianmatricesB. Since there are n2anti-Hermitianmatrices, equa-tion(35)isasystemof n2coupledquadraticequationsthatthecomponentsofHmustobey.Tangent vectorH=[A,H]Integrationenergy E=||3||Non-separablecomponent3Separable hyperplane: 3=0SubsphereH=UH*U {FIG. 8: Geometrically, wecanviewtheintegrationenergyastheshortestdistance(inHilbert-Schmidtnorm)betweenthe hyperplane of separable Hamiltonians andasubsphereof Hamiltonians that canbeunitarilytransformedintooneanother. The most separable Hamiltonian Hon the subsphereis suchthat its non-separablecomponent 3 is orthogonaltoall subsphere tangent vectors [A, H] generatedbyanti-HermitianmatricesA.K. TheHamiltoniandiagonalitytheoremAnalogously to the above-mentioned -diagonality the-orem,we will now prove that maximal separability is at-tainedintheeigenbasis.H-DiagonalityTheorem(HDT):TheHamiltonianisalwaysmaximallysepara-ble(minimizing [[H3[[)intheenergyeigenba-siswhereitisdiagonal.Asapreliminary,letusrstprovethefollowing:Lemma1: For anyHermitianpositivesemidenitematrixH,thereisadiagonalmatrixHgivingthesamesubsystemeigenvalue spectra, (1H) = (1H),(2H) = (2H),andwhoseeigenvaluespectrumismajorizedbythatofH,i.e.,(H) ~ (H).17Proof: DenethematrixH

UHU, whereU U1U2, and U1and U2are unitary matrices diagonal-izingthepartial tracematricestr2Handtr1H, respec-tively. Thisimpliesthattr1H

andtr2H

arediagonal,and(H

) = (H). NowdenethematrixHtobeH

with all o-diagonal elements set to zero. Then tr1H=tr1H

andtr2H= tr2H

,so(1H) = (1H)and(2H) = (2H). Moreover, since the eigenvalues ofanyHermitianpositivesemidenitematrixmajorizeitsdiagonal elements[31], (H) (H

)=(H), whichcompletestheproof.Lemma 2: The set S(H) of all diagonal matriceswhose diagonal elements are majorized by the vector(H)isaconvexsubsetofthesubsphere,withboundarypointsonthesurfaceof thesubspherethat arediagonalmatriceswithall permutationsof(H).Proof: AnymatrixHS(H) must lie either onthesubspheresurfaceor inits interior, becauseof thewell-knownresultthatforanytwopositivesemideniteHermitianmatricesofequaltrace,themajorizationcon-dition(H) (H)isequivalenttotheformerlyingintheconvexhulloftheunitaryorbitofthelatter[32]:H=

ipiUiHUi,pi 0,

ipi= 1,UiUi= I. S(H)containstheabove-mentionedboundarypoints,becausetheycanbewrittenas UHUfor all unitarymatricesUthat diagonalize H, andfor adiagonal matrix, thecorrespondingHis simplythematrixitself. ThesetS(H) is convex, because the convexityconditionthatp1+ (1 p)2 ~if 1 ~, 2 ~, 0 p 1followsstraightfromthedenitionof ~.Lemma3: Thefunctionf(H) [[1H[[2+[[2H[[2is convex, i.e., satises f(paHa+ pbHb) paf(Ha) +pbf(Hb) for any constants satisfying pa0, pb0,pa +pb= 1.Proof: If wearrangetheelements of Hintoavec-tor handdenote the actionof the superoperators ionhbymatricesPi, thenf(H) = [P1h[2+ [P2h[2=h(P1P1+ P2P2)h. Sincethematrixinparenthesisissymmetricandpositivesemidenite, thefunctionfisapositivesemidenitequadraticformandhenceconvex.We are now ready to prove the H-diagonality theorem.Thisisequivalenttoprovingthatf(H)takesitsmaxi-mum value on the subsphere in Figure 8 for a diagonal H:sinceboth [[H[[and [[H0[[areunitarilyinvariant, mini-mizing [[H3[[2= [[H[[2[[H0[[2f(H)isequivalenttomaximizingf(H).LetO(H)denotethesubphere, i.e., theunitaryorbitof H. By Lemma 1, for every H O(H), there is an H S(H)suchthatf(H) = f(H). Ifftakesitsmaximumover S(H) at apoint Hwhichalsobelongs toO(H),thenthisisthereforealsothemaximumoffoverO(H).Since the function fis convex (by Lemma 3) and the setS(H)isconvex(byLemma2), fcannothaveanylocalmaxima within the set and must take its maximum valueatatleastonepointontheboundaryoftheset. AsperLemma2, theseboundarypointsarediagonal matriceswithall permutations of theeigenvalues of H, sotheyalsobelongtoO(H)andthereforeconstitutemaximaoff overthesubsphere. Inotherwords, theHamiltonianis always maximallyseparableinits energyeigenbasis,q.e.d.ThisresultholdsalsoforHamiltonianswithnegativeeigenvalues, sincewecanmakeall eigenvalues positivebyaddinganH0-componentwithoutalteringtheopti-mizationproblem. Inadditiontothediagonaloptimum,therewill generallybeotherbaseswithidentical valuesof [[H3[[, corresponding to separable unitary transforma-tionsofthediagonaloptimum.Wehavethusprovedthatseparabilityisalwaysmax-imizedintheenergyeigenbasis,wherethen nmatrixHis diagonal andtheprojectionoperators idenedbyequations (21)-(24) greatlysimplify. If we arrangethen = lmdiagonalelementsofHintoanl mmatrixH,thenthe actionofthe linear operators iis givenbysimplematrixoperations:H0 QlHQm, (36)H1 PlHQm, (37)H2 QlHPm, (38)H3 PlHPm, (39)wherePkI Qk, (40)(Qk)ij1k(41)arekkprojectionmatricessatisfyingP2k=Pk, Q2k=Qk, PkQk=QkPk=0, Pk+ Qk=I. (Toavoidconfu-sion,weareusingboldfaceforn nmatricesandplainfontforsmallermatrices involving onlythe eigenvalues.)Forthen = 2 2exampleofequation(32),wehaveP2=_12121212_, Q2=12_12121212_, (42)andageneraldiagonalHisdecomposedintofourtermsH= H0 +H1 +H2 +H3asfollows:H=_tttt_+_a aa a_+_b bb b_+_v vv v_. (43)Asexpected, onlythelastmatrixisnon-separable, andtherow/columnsumsvanishforthetwopreviousmatri-ces,correspondingtovanishingpartialtraces.NotethatweareherechoosingthenbasisstatesofthefullHilbertspacetobeproductsofbasisstatesfromthetwofactorspaces. Thisiswithoutlossofgenerality,since any other basis states can be transformed into suchproductstatesbyaunitarytransformation.Finally, note that the theorem above applies only to ex-actnite-dimensionalHamiltonians,nottoapproximatediscretizations of innite-dimensional ones suchas arefrequentlyemployedinphysics. Ifnisnotfactorizable,theH-factorizationproblemcanberigorouslymapped18ontoaphysicallyindistinguishable one withaslightlylarger factorizable nbysettingthe correspondingnewrowsandcolumnsofthedensitymatrixequaltozero,sothatthenewdegreesoffreedomareallfrozenoutwe will discuss this idea in more detail in in Section IVF.L. UltimateindependenceandtheQuantumZenoparadoxFIG. 9: If theHamiltonianof asystemcommuteswiththeinteractionHamiltonian([H1, H3] =0), thendecoherencedrivesthesystemtowardatime-independentstatewherenothing ever changes. The gure illustrates this for the BlochSphere of a single qubit starting in a pure state and ending upin a fully mixed state = I/2. More general initial states endupsomewherealongthez-axis. HereH1 z, generatingasimpleprecessionaroundthez-axis.In Section III G, we began exploring the idea that if wedivide the world into maximally independent parts (withminimal interaction Hamiltonians), then the observed ob-ject hierarchyfromFigure1wouldemerge. TheHDTtells us that this decomposition (factorization) into max-imally independent parts can be performed in the energyeigenbasisofthetotalHamiltonian. Thismeansthatallsubsystem Hamiltonians and all interaction Hamiltonianscommutewithoneanother, correspondingtoanessen-tiallyclassical worldwherenoneof thequantumeectsassociated with non-commutativity manifest themselves!Incontrast, manysystemsthatwecustomarilyrefertoas objects inour classical worlddonot commutewiththeirinteractionHamiltonians: forexample, theHamil-toniangoverningthedynamicsofabaseballinvolvesitsmomentum,whichdoesnotcommutewiththeposition-dependentpotentialenergyduetoexternalforces.AsemphasizedbyZurek[33], statescommutingwiththe interactionHamiltonianforma pointer basis ofclassicallyobservablestates, playinganimportant rolein understanding the emergence of a classical world.Thefactthattheindependenceprincipleautomaticallyleads to commutativity with interaction Hamiltoniansmight therefore be takenas anencouragingindicationthat we are on the right track. However, whereasthepointerstatesinZureksexamplesevolveovertimeduetothesystemsownHamiltonianH1, thoseinourindependence-maximizing decomposition do not, becausethey commute also with H1. Indeed, the situationis even worse, as illustrated in Figure 9: any time-dependent systemwill evolve into a time-independentone,asenvironment-induceddecoherence[3437,39,40]drivesittowardsaneigenstateoftheinteractionHamil-tonian,i.e.,anenergyeigenstate.12ThefamousQuantumZenoeect, wherebyasystemcan cease to evolve in the limit where it is arbitrar-ilystronglycoupledtoitsenvironment[41], thushasastrongerandmoreperniciouscousin,whichwewilltermtheQuantumZenoParadoxortheIndependencePara-dox.QuantumZenoParadox:If wedecomposeouruniverseintomaximallyindependentobjects,thenall changegrindstoahalt.Insummary, wehavetriedtounderstandtheemer-gence of our observedsemiclassical world, withits hi-erarchyof moving objects, bydecomposing the worldinto maximally independent parts, but our attempts havefaileddismally,producingmerelyatimelessworldremi-niscentofheat death. In SectionII G, we saw thatusingtheintegrationprinciplealoneledtoasimilarlyembar-rassingfailure, withnomorethanaquarterof abitofintegratedinformationpossible. Atleastonemoreprin-cipleisthereforeneeded.IV. DYNAMICSANDAUTONOMYLetusnowexploretheimplicationsof thedynamicsprinciplefromTableII, accordingtowhichaconscioussystemhas thecapacitytonot onlystoreinformation,butalsotoprocessit. Aswejustsawabove,thereisaninterestingtensionbetweenthisprincipleandtheinde-pendenceprinciple,whoseQuantumZenoParadoxgivesthe exact opposite: no dynamics and no information pro-cessingatall.12Forasystemwithaniteenvironment,theentropywilleventu-allydecreaseagain,causingtheresumptionoftime-dependence,but this Poincare recurrence time grows exponentially with envi-ronmentsizeandisnormallylargeenoughthatdecoherencecanbeapproximatedaspermanent.19We will term the synthesis of these two competing prin-ciples theautonomyprinciple: aconscious systemhassubstantial dynamicsandindependence. Whenexplor-ingautonomoussystemsbelow, wecannolongerstudythe state and the Hamiltonian H separately, since theirinterplay is crucial. Indeed, we well see that there are in-terestingclassesofstatesthatprovidesubstantial dy-namics and near-perfect independence even when the in-teractionHamiltonianH3isnotsmall. Inotherwords,forcertainpreferredclassesof states, theindependenceprinciple no longer pushes us to simply minimize H3andfacetheQuantumZenoParadox.A. ProbabilityvelocityandenergycoherenceToobtainaquantitativemeasureof dynamics, letusrst dene the probability velocity v p, where the prob-abilityvectorpisgivenbypi ii. Inotherwords,vk= kk= i[H, ]kk. (44)Sincevisbasis-dependent, weareinterestedinndingthebasiswherev2

kv2k=

k( kk)2(45)is maximized, i.e., the basis where the sums of squares ofthediagonal elementsof ismaximal. Itiseasytoseethatthisbasisistheeigenbasisof :v2=

k( kk)2=

jk( jk)2

j=k( jk)2=[[ [[2

j=k( jk)2(46)is clearly maximized in the eigenbasis where all o-diagonal elements in the last termvanish, since theHilbert-Schmidt norm [[ [[ is the same ineverybasis;[[ [[2=tr 2, whichissimplythesumof thesquaresoftheeigenvaluesof .LetusdenetheenergycoherenceH12[[ [[ =12[[i[H, ][[ =_tr [H, ]22=_tr [H22HH]. (47)Forapurestate= [[, thisdenitionimpliesthatH H,whereHistheenergyuncertaintyH=_[H2[ [H[21/2, (48)sowecanthinkof Has thecoherent part of theen-ergy uncertainty, i.e., as the part that is due to quantumratherthanclassicaluncertainty.Since [[ [[= [[[H, ][[=2H, weseethatthemaxi-mumpossibleprobabilityvelocityvissimplyvmax=2 H, (49)sowecanequivalentlyuseeitherof vorHasconve-nient measures of quantumdynamics.13Whimsicallyspeaking, thedynamics principlethus implies that en-ergyeigenstatesareasunconsciousasthingscome, andthatifyouknowyourownenergyexactly,youredead.Although it is not obvious from their denitions, thesequantities vmaxandHareindependent of time(eventhoughgenerallyevolves). This is easilyseenintheenergyeigenbasis,wherei mn= [H, ]mn= mn(EmEn), (51)wheretheenergiesEnaretheeigenvaluesofH. Inthisbasis,(t) = eiHt(0)eiHtsimpliesto(t)mn= (0)mnei(EmEn)t, (52)Thismeansthatintheenergyeigenbasis,theprobabili-tiespn nnareinvariantovertime. Thesequantitiesconstitutetheenergyspectraldensityforthestate:pn= En[[En. (53)Intheenergyeigenbasis,equation(48)reducestoH2= H2=

npnE2n_

npnEn_2, (54)whichistime-invariantbecausethespectral densitypnis. Forgeneralstates,equation(47)simpliestoH2=

mn[mn[2En(EnEm). (55)This is time-independent because equation(52) showsthat mnchanges merely by a phase factor, leaving [mn[invariant. In other words,when a quantum state evolvesunitarilyintheHilbert-Schmidtvectorspace, boththepositionvectorandthevelocityvector retaintheirlengths: both [[[[and [[ [[remaininvariantovertime.B. DynamicsversuscomplexityOur results above show that if all we are interested in ismaximizingthemaximalprobabilityvelocityvmax,thenweshouldndthetwomostwidelyseparatedeigenval-uesofH, EminandEmax, andchooseapurestatethatinvolvesacoherentsuperpositionofthetwo:[ = c1[Emin +c2[Emax, (56)13Thedelitybetweenthestate(t) andtheinitial state0isdenedasF(t) 0|(t), (50)anditiseasytoshowthatF(0)=0andF(0)= (H)2, sotheenergyuncertaintyisagoodmeasureofdynamicsinthatitalsodetermines thedelityevolutiontolowest order, for purestates. For a detailedreviewof relatedmeasures of dynam-ics/informationprocessingcapacity,see[16].20FIG. 10: Time-evolution of Bloch vector tr 1for a single qubit subsystem. We saw how minimizing H3leads to a static statewithnodynamics,suchastheleftexample. MaximizingH,ontheotherhand,producesextremelysimpledynamicssuchastherightexample. ReducingHbyamodestfactoroforderunitycanallowcomplexandchaoticdynamics(center); shownhereisa2-qubitsystemwherethesecondqubitistracedout.where [c1[ = [c2[ =1/2. This gives H=(Emax Emin)/2,thelargestpossiblevalue,butproducesanex-tremelysimpleandboringsolution(t). Sincethespec-tral densitypn=0except for thesetwoenergies, thedynamics is eectivelythat of a2-statesystem(asin-glequbit)nomatterhowlargethedimensionalityofHis, correspondingtoasimpleperiodicsolutionwithfre-quency=Emax Emin(acircular trajectoryintheBloch sphere as in the right panel of Figure 10). This vi-olates the dynamics principle as dened in Table II, sinceno substantial information processing capacity exists: thesystem is simply performing the trivial computation thatipsasinglebitrepeatedly.To perform interesting computations, the systemclearlyneeds toexploit asignicant part of its energyspectrum. As canbe seenfromequation(52), if theeigenvaluedierencesareirrational multiplesofonean-other, thenthetimeevolutionwill neverrepeat, andwill eventuallyevolvethroughall partsofHilbertspaceallowedbytheinvariants [Em[[En[. ThereductionofHrequiredtotransitionfromsimpleperiodicmotiontosuchcomplexaperiodicmotionisquitemodest. Forexample,iftheeigenvaluesareroughlyequispaced,thenchanging the spectral density pn from having all weight atthetwoendpointstohavingapproximatelyequalweightforalleigenvalueswillonlyreducetheenergycoherenceHbyaboutafactor3, sincethestandarddeviationof auniformdistributionis3times smaller thanitshalf-width.C. Highlyautonomoussystems: slidingalongthediagonalWhatcombinationsofH,andfactorizationproducehighlyautonomous systems? Abroadandinterestingclasscorrespondstomacroscopicobjectsaroundusthatmoveclassicallytoanexcellentapproximation.Thestatesthataremostrobusttowardenvironment-induceddecoherencearethosethatapproximatelycom-mutewiththeinteractionHamiltonian[36]. Asasimplebut important example, let us consider aninteractionHamiltonianofthefactorizableformH3= AB, (57)andworkinasystembasiswheretheinteractiontermAis diagonal. If 1is approximatelydiagonal inthisbasis, thenH3haslittleeectonthedynamics, whichbecomesdominatedbytheinternal subsystemHamilto-nianH1. TheQuantumZenoParadoxweencounteredinSectionIII LinvolvedasituationwhereH1wasalsodiagonal inthis samebasis, sothat weendedupwithnodynamics. Aswewillillustratewithexamplesbelow,classicallymovingobjectsinasenseconstitutetheop-posite limit: the commutator 1= i[H1, 1] is essentiallyaslargeaspossibleinsteadofassmall aspossible, con-tinuallyevadingdecoherencebyconcentratingaroundasinglepointthatcontinuallyslidesalongthediagonal,asillustratedinFigure11. Decohererencerapidlysup-presseso-diagonalelementsfarfromthisdiagonal,butleaves the diagonal elements completelyunaected, so21Dynamics from H1slides along diagonalij0High-decoherence subspaceHigh-decoherence subspaceDynamics from H3 suppresses off-diagonal elements ijDynamics from H3 suppresses off-diagonal elements ijLow-decoherence subspaceijFIG. 11: Schematicrepresentationof thetime-evolutionofthedensitymatrixij for ahighlyautonomous subsystem.ij 0 except for a single region around the diagonal(red/greydot), andthisregionslidesalongthediagonal un-dertheinuenceofthesubsystemHamiltonianH1. Anyij-elements far from the diagonal rapidly approach zero becauseof environment-decoherence caused by the interaction Hamil-tonianH3.there exists a low-decoherence band around the diagonal.Suppose, forinstance, thatoursubsystemisthecenter-of-masspositionxofamacroscopicobjectexperiencingaposition-dependentpotentialV (x)causedbycouplingto the environment, so that Figure 11 represents the den-sity matrix 1(x, x

) in the position basis. If the potentialV (x) has a at (V

= 0) bottom of width L, then 1(x, x

)will be completely unaected by decoherence for the band[x

x[ < L. For a generic smooth potential V , the deco-herencesuppressionof o-diagonal elementsgrowsonlyquadratically with the distance [x

x[ from the diagonal[4,35],againmakingdecoherencemuchslowerthantheinternaldynamicsinanarrowdiagonalband.Asaspecicexampleofthishighlyautonomoustype,letusconsiderasubsystemwithauniformlyspaceden-ergyspectrum. Specically, consider ann-dimensionalHilbertspaceandaHamiltonianwithspectrumEk=_k n 12_= k +E0, (58)k=0, 1, ..., n 1. Wewill oftenset =1forsimplic-ity. For example, n=2gives the spectrum 12,12like the Pauli matrices divided by two, n =5 gives2, 1, 0, 1, 2andn givesthesimpleHarmonicoscillator (since the zero-point energy is physically irrele-vant, we have chosen it so that tr H =

Ek= 0, whereasthecustomarychoicefortheharmonicoscillatorissuchthatthegroundstateenergyisE0= /2).Ifwewantto,wecandenethefamiliarpositionandmomentumoperatorsxandp,andinterpretthissystemasaHarmonicoscillator. However, theprobabilityve-locityvis not maximizedineither thepositionor themomentumbasis, except twiceper oscillationwhentheoscillatorhasonlykineticenergy, vismaximizedinthex-basis, andwhenithasonlypotential energy, vismaximizedinthep-basis, andwhenithasonlypoten-tialenergy. IfweconsidertheWignerfunctionW(x, p),whichsimplyrotatesuniformlywithfrequency, itbe-comesclearthattheobservablewhichisalwayschang-ingwiththemaximalprobabilityvelocityisinsteadthephase, theFourier-dual of theenergy. Letusthereforedenethephaseoperator FHF, (59)whereFistheunitaryFouriermatrix.PleaserememberthatnoneofthesystemsHthatweconsider have any a priori physical interpretation; rather,theultimategoalofthephysics-from-scratchprogramisto derive any interpretation from the mathematics alone.Generally,anythusemergentinterpretationofasubsys-temwill dependonitsinteractionswithothersystems.Since we have not yet introduced any interactions for oursubsystem, we are free to interpret it in whichever way isconvenient. Inthisspirit, anequivalentandsometimesmoreconvenientwaytointerpretourHamiltonianfromequation (58) is as a massless one-dimensional scalar par-ticle,forwhichthemomentumequalstheenergy,sothemomentumoperatorisp=H. If weinterpretthepar-ticleasexistinginadiscretespacewithnpointsandatoroidal topology (which we can think of as n equispacedpoints on a ring), then the position operator is related tothe momentum operator by a discrete Fourier transform:x = FpF, Fjk 1Neijk2n. (60)Comparingequations (59) and (60), weseethat x=. Since Fis unitary, the operators H, p, xandall havethesamespectrum: theevenlyspacedgridofequation(58).As illustratedinFigure 12, the time-evolutiongen-eratedby Hhas a simple geometric interpretationinthespacespannedbythepositioneigenstates [xk, k=1, ...n: thespaceisunitarilyrotatingwithfrequency,soafter atime t =2/n, astate [(0) = [xk hasbeenrotatedsuchthat it equals the next eigenvector:[(t)= [xk+1, wheretheadditionismodulon. ThismeansthatthesystemhasperiodT 2/, andthat[ rotates througheachof thenbasis vectors duringeachperiod.Let us now quantify the autonomy of this system, start-ingwiththedynamics. SinceapositioneigenstateisaDiracdeltafunctioninpositionspace,itisaplanewavein momentum space and in energy space, since H = p.22xzyzxyz2x1x5x8x7x4x6x3xFIG.12: Forasystemwithanequispacedenergyspectrum(suchasatruncatedharmonicoscillatororamasslessparticleinadiscrete1-dimensionalperiodicspace),thetime-evolutionhasasimplegeometricinterpretationinthespacespannedbytheeigenvectors xkof the phase operator FHF,the Fourier dual of the Hamiltonian,corresponding to unitarily rotating the entirespacewithfrequency,where istheenergylevelspacing. Afteratime2/n,eachbasisvectorhasbeenrotatedintothesubsequentone, asschematicallyillustratedabove. (TheorbitinHilbertspaceisonlyplanarforn 3, sothegureshouldnotbetakentooliterally.) Theblackstardenotesthe = 1apodizedstatedescribedinthetext,whichismorerobusttowarddecoherence.Thismeansthatthespectral densityispn=1/nforaposition eigenstate. Substituting equation (58) into equa-tion(54)givesanenergycoherenceH= _n2112. (61)Forcomparison,[[H[[ =_n1

k=0E2k_1/2= _n(n21)12=n H. (62)Let us nowturntoquantifyingindependence andde-coherence. Theinner product betweentheunit vector[(0)andthevector [(t) eiHt[(0)intowhichitevolvesafteratimetisfn() [eiH[ =1nn1

k=0eiEk= ein12n1

k=0eik=1nein121 ein1 ei=sin nnsin , (63)where t. ThisinnerproductfnisplottedinFig-ure13,andisseentobeasharplypeakedevenfunctionsatisfyingfn(0)=1, fn(2k/n)=0fork=1, ..., n 1and exhibiting one small oscillation between each of thesezeros. Theangle cos1fn()betweenaninitialvec-tor and its time evolution thus grows rapidly from 0 to90, then oscillates close to 90until returning to 0afterafull periodT. Aninitial state [(0)= [xkthereforeevolvesasj(t) = fn(t 2[j k]/n)1.0-0.20.80.40.20.6-100 -150 150 100 50 -50Penalty functionApodizedNon-apodizedFIG. 13: Thewiggliest(heavyblack)curveshowstheinnerproductof apositioneigenstatewithwhatitevolvesintoatimet=/laterduetoourn=20-dimensional Hamilto-nianwithenergyspacings . Whenoptimizingtominimizethe square of this curve using the 1 cos penaltyfunc-tionshown, correspondingtoapodizationintheFourierdo-main,weinsteadobtainthegreen/lightgreycurve,resultinginmuchlessdecoherence.in the position basis, i.e., a wavefunction jsharplypeakedforj k + nt/2(modn). Sincethedensitymatrixevolvesasij(t) =i(t)j(t), itwill thereforebesmallexceptfori j k + nt/2(modn), corre-spondingtotherounddotonthediagonalinFigure11.In particular, the decoherence-sensitive elements jkwillbe small far from the diagonal, corresponding to the smallvaluesthatfntakesfarfromzero. Howsmall will thedecoherencebe?Letusnowdevelopthetoolsneededtoquantifythis.23D. Theexponential growthofautonomywithsystemsizeLet us return to the most general Hamiltonian Handstudyhowaninitiallyseparablestate=1 2evolves over time. Usingthe orthogonal projectors ofSectionIII I,wecandecomposeHasH = H1I +I H2 +H3, (64)where tr1H3=tr2H3=0. By substituting equa-tion (64) into the evolution equation 1= tr2 =itr2[H, ]andusingvariouspartial