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