Download Wormholes and quantum cloning

Wormholesandquantumcloning
Thesurprisingsymplicity ofnear-AdS2
JuanMaldacena
BasedonworkwithDouglasStanfordand
Zhenbin Yang.
Thisisalsobasedonarecentpaperby
Gao,Jafferis andWall
PuntadelEste,April2017
Firstastory….
The Cult of the Thermofield Double
Iran
India
Iran
India
Twoentangledfires
The Rite of Passage
Iran
Catthrownin
Iran
Waitawhiletillthecatistotallyburnt.
Pictureofasmallportionofthefire.
Iran
India
HP
HP
PicturesentovertheinternetfromIrantoIndia
India
Picturefedintothemechanism
controllingthefire.
Waitawhile.
India
Catcomesout!
End
Thistalkwillbeonwhetherthisispossibleaccordingtothescienceweknow.
Theanswerisyes.
1) Yes,itispossibleaccordingtotheprinciplesofquantummechanics. Hayden
andPreskill argument&quantumteleportation.
2)Yes,itisalsopossibleaccordingtotheprinciplesofgeneralrelativity.
UsingdynamicsinaKruskal-Schwarzschild-AdS wormhole.Firesà blackholes.
Thetwoarerelatedbythecentraldogma(assumption):
Thethermofield doubleisthesameastheKruskal (Schwarszschild)wormhole.
egions. It is important not to confuse the future interior with the left exterior.
he left exterior is referred colloquially as the “interior” of the right black hole,
it is important not to do that. Note that no signal from the future interior
either of the two exteriors.
Kruskal-Schwarzschild-AdS blackhole
L
Left
ER
Exterior
Geometric connection
from entanglement
Future
Interior
Right
ER
Exterior
Past
Interior
R
Entangled state in
two non-interacting
quantum systems.
enrose diagram of the eternal black hole in AdS. 1 and 2, or Left and Right,
wo boundaries and the two CFT’s that the system is dual to.
En /2
| i=
X
e
|Ēn iL ⇥ |En iR
em is described by two identical
uncoupled CFTs defined on disconnected
n
heres. We’ll call them the Left and Right sectors. The energy levels of the
e discrete. The corresponding eigenstates are denoted |n⇤L , |n⇤R . To simplify
Israel
JM
EPR
• Wewillstartfromthewormholeside.
• Itwillbenecessarytoconsidergravitational
backreaction.
• Thiscanbestudiedingeneral,butitis
particularlysimpleforapairofnearextremal
blackholes.TheirnearhorizonisanearlyAdS2 wormhole.
NearlyAdS2
Z
Keeptheleadingeffectsthatperturbawayfrom
AdS2
2
p
d x g (R + 2) +
0
Z
2
p
Teitelboim Jackiw
Almheiri Polchinski
d x gR
Groundstateentropy
Comesfromtheareaoftheadditionaldimensions,ifwearegettingthisfrom4d
gravityforanearextremal blackhole.
Z
p
g (R + 2) +
Bulk
Equationofmotionfor
b
Z
K + Smatter [g, ]
Bdy
à metricisAdS2.Rigidgeometry!
Onlydynamicalinformationà locationoftheboundary.
b
Z
K
Bdy
Localactionontheboundary
Onesolution
ADMmassà size
Restofsolutions
RelatedbyAdS2
isometries
Euclideanblackhole
Kruskal SchwarzschildAdS2
wormhole
Dynamics
Bulkfieldspropagate
onarigidAdS2space.
Boundariesalso
moveinarigid
AdS2 space,following
localdynamicallaws.
UVparticleorUVbraneasin
aRandall-Sundrum model
Dynamics
Emissionofabulkexcitation
Newpositionofthehorizon
Theboundarytrajectorygets
a“kick”determinedbylocal
energymomentumconservation.
Dynamics
Absoption ofabulkexcitation
Newpositionofthehorizon
Theboundarytrajectorygets
a“kick”determinedbylocal
energymomentumconservation.
Interactionbetweenthetwoboundaries
GaoJafferis Wall
Insertthisinthepath
integral
eig
L (tL ) R (tR )
approximate
eigh
L (tL ) R (tR )i
Forcebetweenthetwo
boundaries.
(Canbeattractiveforthe
rightsignofg).
kicksthetrajectoriesinwards
Interactionmakesthewormholetraversable
Newpositionofthehorizon
Wecannowsendasignal
fromthelefttotheright.
Thewormholehasbeen
renderedtraversable.
Nocontradictionbecause
wehadanon-localinteraction
betweenthetwoboundaries.
Whatdoesthistake?.
Generatingtheboundaryinteraction
eig
L (tL ) R (tR )
MeasureL
!
L
Actontherightwith
eig
L
R (tR )
Fromthepointofviewofthe
rightwegetthesame,
whetherwemeasureornot.
Whatifwewanttosendtoomuch
information?
Theinsertionofthemessage
alsogivesasmallkicktothe
trajectory.
Movestheinsertionpoints
ofthenon-localoperator
awayfromeachother
h
L (tL ) R (tR )i
becomes
smaller
Attractiveforceweakensà
noopeningofthewormhole.
Preciseformulaforthe2ptfunction
C = he
igV
C⇠
igV
(t)e
R
Z
dp(p)2
L(
t)i ,
1
e
ip
V =g
e
L (0) R (0)
g
i
ig (1+pet )2
Amountofinformationwecansendisroughlyg
e
hV i = 1 ,
h
2
L (0)i
⇠1
Relationtotheblackholecloning
paradox
• Supposeyouhaveanoldblackhole.Orablack
holemaximallyentangledwithasecondsystem.
• Bobhasaccesstothesecondsystemandan
infinitelypowerfulquantumcomputer,butnot
theblackhole.(ThisistheHPcomputer)
• ThenAlicesendsinaM-bitmessage.Andwaits
forarelativelyshorttimeforthemessageto
effectivelyfallintotheblackhole(ascrambling
time).
• BobonlyneedsabitmorethanMbitsof
Hawkingradiationtodecodethemessage.
Susskind-Thorlacius
Hayden-Preskill
FigurefromthepaperofPreskill andHayden.
Thecloningpuzzleistheapparentduplicationofinformationontheredspacelike slice.
Spacelike slicecontaining
BothAlice’sinformationinside
andintheradiation.
Hereitseemsthattheinformationis
bothintheradiationandinthemessage.
Bothoutsidethehorizonandinsidethehorizon.
Wormholepicture:Bob’scomputercouldconverttheentangledstateintoa
blackholeintheTFDentangledstate.Inthatcasethegeometryshouldlookasfollows
Bob’sblack
Hole,whichis
Partof
Bob’scomputer
Alice’sblack
hole
Alice’smessage
Trajectoriesoftheboundaries
BeforeBobcatchestheHawkingmode
Bobgetssomeradiationandfeedsittohiscomputer.
Trajectoriesoftheboundaries
afterBobcatchestheHawkingmode
Bobgettingradiation
Bob’sblack
Hole,whichis
Partof
Bob’scomputer
Alice’sblack
hole
Alice’smessage
Feeding
hiscomputer=
blackhole.
Trajectoriesoftheboundaries
beforeBobcatchestheHawkingmode
BobnowgetsthemessageatP
P
Bob’sblack
Hole,whichis
Partof
Bob’scomputer
Alice’sblack
hole
Alice’smessage
Trajectoriesoftheboundaries
BeforeBobcatchestheHawkingmode
Themessageswitchedsides!
P
Bobgettingradiation
Newhorizon
Bob’sblack
Hole,whichis
Partof
Bob’scomputer
Alice’sblack
hole
Alice’smessage
Feeding
hiscomputer=
blackhole.
Trajectoriesoftheboundaries
BeforeBobcatchestheHawkingmode
Backward
extrapolation
Ofthestate
ofAlice’sboundary
afterBob’s
extraction,using
theunperturbed
Hamiltonian.
Beforetransfer:AlicehasthemessagebutBobdoesnot
Aftertransfer:BobhasitbutAlicedoesnot!
P
Bobgettingradiation
Bob
Alice’sblack
hole
Alice’smessage
Backward
extrapolation
Ofthestate
ofAlice’sboundary
afterBob’s
extraction,using
theunperturbed
Hamiltonian.
MoreliketheHPfigure
P
Bobgettingradiation
Alice’smessage
Bob
Alice’sblack
hole
NowBobcannotgetthemessage!,itisstillinAlice’spossesion.
NowAlicesendislater…
Bobwillnotgetithere. P
Bobgetsthe
machinerythatsent
Alice’smessage,but
withnomessage.
Alice’smessage
Bob
Alice’sblack
hole
Bobgettingradiation
NowAlicesendislater…
Bobwillnotgetithere. P
Bobcanextract
thatmachinery
Alice’smessage
Bob
Alice’sblack
hole
Bobgettingradiation
NowAlicesendislater…
Bobwillnotgetithere. P
Alice’smessage
Bobcanextract
thatmachinery
Bobgettingradiation
Bob
Alice’sblack
hole
NowAlicesendislater…
Alice’smessage
Bobgettingradiation
Bob
Bobcanevolve
hissidebackwardsin
timeandrecoverthemessage
here
Alice’sblack
hole
• Theprocessofextractingthemessageputsit
outofreachfromAlice.
• Themessageisneverduplicatedinthebulk
picture.
• Noneedtoinvokeunkown new
transplanckian physicstosolvetheno-cloning
problem.
• Allunderstandablefromstandardrulesof
gravityonthewormholegeometry.
Quantummechanicalmodel
TheSYKmodel
NMajorana fermions
H=
X
{ i,
J i1 i2 i3 i4
j}
i1
=
i2
Sachdev YeKitaev
Georges,Parcollet
ij
i3
i4
i1 ,··· ,i4
Randomcouplings,gaussian distribution.
hJi21 i2 i3 i4 i = J 2 /N 3
Toleadingorderà treatJijkl asanadditionalfield
J=dimensionful coupling.Wewillbeinterestedinthestrongcouplingregion
1 ⌧ J, ⌧ J ⌧ N
Definenewvariable
1 X
G(⌧, ⌧ ) =
h i (⌧ ) i (⌧ 0 )i
N i
0
IntegrateoutfermionsandgetanactionintermsofanewfieldG
S = N f [G(⌧, ⌧ 0 )]
Isanalogoustothefullbulkgravity+matteraction.
ThereisaparticularGthatminimizestheaction.ItisSL(2)invariant.
(analogoustothevacuumAdS geometry)
Setoflowactionfluctuationsofthissolution.Parametrizedbyafunctionofasinglevariable.
Reparametrization mode.
Gf = (f 0 (⌧ )f 0 (⌧ 0 )) G(f (⌧ ), f (⌧ 0 ))
Lowenergyaction:
Z
N
S=
{f, ⌧ }d⌧
J
SameastheactionfortheUVboundary
inthegravitydescription.
H=
X
J i1 i2 i3 i4
i1
i2
i3
i1 ,··· ,i4
N
S=
J
Z
{f, ⌧ }d⌧
SameastheactionfortheUVboundary
inthegravitydescription.
i4
• Allthatwesaidbeforeinthewormhole
contextdependedonlyonthemotionofthe
UVboundaryandthepropagationinanAdS2
bulk.
• Wegotthesameactionfortheboundary.The
othermodesofGà conformalinvariant
whichleadtocorrelatorsasinAdS2
Samepreciseformulaforthe2pt
function
C = he
igV
C = he
C⇠
Z
igV
igV
(t)e
R
j R (t)e
dp(p)2
igV
L(
j L(
1
e
t)i ,
V =g
K
1 X
V =g
K j=1
t)i ,
ip
e
g
i
ig (1+pet )2
e
Amountofinformationwecansendisroughlyg
L (0) R (0)
L (0) R (0)
Conclusions
• Traversability understoodfromboththebulk
andtheboundary.Bothingravityandinthe
dualquantumsystem.
• Applicationtothecloningparadox.
• Wediscussedhowtothinkabouttheprocess
ofinformationextractionfromablackhole.
• Informationgoesoutviathewormhole.
Questions
• Isthereanotherwaytoextractinformation
fromablackhole?
• Aretheselessonsapplicableforanyother
conceivablewaytodoit?