quantum teleportation and traversable...

Quantum Teleportation and TraversableWormholes

Dongsu Bak

University of Seoul

KIAS, November 2018

Dongsu Bak Quantum Teleportation and Traversable Wormholes

Quantum teleportation

I Alice located in L sends a quantum state to Bob in a remoteplace R.

I The main holographic picture is that the qubit is sent throughbulk traversable wormhole.


Holographic Decoding

I It has an important implication in holographic decoding problems inAdS/CFT.

I We are interested in the decoding of the behind horizon degrees andin the wormhole spacetime the horizon can disappear.

I One can in principle understandwhat happens to behind horizondegrees and the fate of futurespacelike singularities.


I At this stage the total state becomes

|T 〉|Ψ〉LR =1

2

∑θ1,θ2,i,k

|θ1θ2〉M(U−1{θ1θ2})ikck |i〉R

where U{θ1θ2} = σθ11 σ

θ23 .

I Now Alice on the left side makes a measurement in the M basisending up with a particular state |θ1θ2〉M and sends its result{θ1, θ2} to Bob on the right side via a classical channel. At thisstage, the state becomes

∑i,k

|θ1θ2〉M(U−1{θ1θ2})ikck |i〉R

I Once Bob gets the message, he performs a unitary transform of hisstate by the action Vθ = U{θ1θ2} and then the resulting statebecomes

|θ1θ2〉M∑k

ck |k〉R


I Of course one can consider more general setup where one has anL-R entanglement involving many qubits and teleports more thanone qubit.

I In particular when one uses our Einstein-Rosen bridge as the L-Rentanglement based on the so called ER=EPR relation [Maldacena,

Susskind 2013], there will be in general a thermailzation of |T 〉 state afterits inclusion to the left side.

RL

I Measurement can be made by picking up an arbitrary qubit of Lsystem. Alice again sends the result of measurement to Bob. Bobthen recovers the |T 〉 by the action of an appropriate unitarytransformation.


Measurement Mθ and Recovery Vθ

I The measurement in general makes the L-R entanglement reducedand the L-T system entangled instead.

I The second essential feature is the L-R coupling by the measurementMθ on the left side and the recovery action Vθ on the right side.

I This coupling basically makes the wormhole traversable as we shallsee in the following.


Gravity description [Jackiw-Teitelboim]

I The AdS2 dilaton gravity is described by

I = Itop −1

16πG

∫M

d2x√g φ

(R +

2

`2

)+ IM(g , χ)

where

Itop = − φ0

16πG

∫M

d2x√gR

IM =1

2

∫M

d2x√g(∇χ · ∇χ+ m2χ2

)I The equations of motion read

R +2

`2= 0, ∇2χ−m2χ = 0

∇a∇bφ− gab∇2φ+ gabφ = −8πGTab

with

Tab = ∇aχ∇bχ−1

2gab(∇χ · ∇χ+ m2χ2

)Dongsu Bak Quantum Teleportation and Traversable Wormholes

I The geometry is solved by the AdS2 space

ds2 =`2

cos2 µ

(−dτ 2 + dµ2

)where the coordinate µ is ranged from−π/2 to +π/2.

I The most general vacuum solution for the dilaton field is given by

φ = φ̄ L(b + b−1) cos(τ − τB)− (b − b−1) sinµ

2 cosµ

I Make the coordinate transformation

r

L=

(b + b−1) cos(τ − τB )− (b − b−1) sinµ

2 cosµ

tanhtL

`2=

2 sin(τ − τB )

(b + b−1) sinµ− (b − b−1) cos(τ − τB )


AdS2 Black holes

I One is led to the corresponding AdS black hole metric and thedilaton field:

ds2 = − r2 − L2

`2dt2 +

`2

r2 − L2dr2

φ = φ̄ r

We shall set b = 1 and τB = 0.

tL tR

I The singularity is determined by Φ2 = φ0 + φ = 0 where Φ2 playsthe role of radius squared in the transverse space when viewed fromthe point of view of the dimensional reduction from the higherdimensions as in

ds24 = ds2

AdS2+ Φ2(dθ2 + sin2θdϕ2)


AdS2 Black holes

I The Gibbons-Hawking temperature of this black hole can beidentified as

T =1

2π

L

`2

I The energy and entropy are then

E =1

2CT 2

S = S0 + CT

which is expected with the AdS2 conformal symmetries.

I This entropy can be written as a Beckenstein formula

S =φ0 + φ̄ L

4G

I This two-sided AdS black hole is dual to the so-called thermofielddouble of CFT1L × CFT1R

|Ψ(0)〉 =1√Z

∑n,n′〈n|U|n′〉 |n′〉 ⊗ |n〉 =

1√Z

∑n

e− β

2En |n〉 ⊗ |n〉


Deformations

I Let us turn on a scalar field χ with mass m. This matter field isdual to a scalar primary operator O∆(t), where its dimension isrelated to the mass squared by

∆ =1

2

(1 +

√1 + 4m2

)for nontachyonic case of m2 ≥ 0.

I When 0 > m2 > −1/4, both possibilities of operator dimensions areallowed

∆ = ∆± =1

2

(1±

√1 + 4m2

)I For the double trace deformation of the traversable wormhole, ∆−

will be the relevant one.


Deformations

I We solve the scalar field equation for χ with boundary conditions.

I The asymptotic behavior of the scalar field in the right/left wedgesin the Penrose diagram is given by

χ(t, r)|R/L =αR/L

r∆+ · · ·+

βR/L

r1−∆+ · · ·

where where α plays a source term and β is then the expectationvalue in the conventional case.

I Once the matter part is solved, we solve the dilaton with thecorresponding energy momentum tensor. Thus the problem isalmost linear.


Thermalization of excited states

I In this case we turn on the coefficient β without turning on thesource term. The Hamiltonian of the system remains undeformedwhile the thermofield double state will be excited.

I This is describing a thermalization of an initial excitation above thethermal vacuum. The left and the right system remains independentand one cannot send a signal from one side to the other side as aneffective elongation of the Penrose diagram horizontally.


Janus deformation [Bak Kim Yi 18]

I We consider m2 = 0 with ∆ = 1 and turn on the source term whichmakes left and right Hamiltonians differ from each other. ThePenrose diagram is similar to the above.

I One may consider a time dependent Janus deformation where theleft side Hamiltonian is deformed at a certain time. Then thisdeformation falls into the horizon and singularity and in the righthand side there is no signal of the deformation.

I Later we shall use this to send a teleportee state to the other side.


Double trace deformation and traversable wormholes

I For the double trace deformation, one has the mixed boundarycondition

βL(t) = h(−t)αR(−t) , βR(t) = h(t)αL(−t)

I According to the AdS/CFT correspondence, this mixed bccorresponds to the double trace deformation in the boundary theoryof the form [Witten:2001]

δH(t) = −h(t)OR(t)OL(−t)

I The change in the two point function G = 〈χ(t)χ(t ′)〉 can beevaluated by

G = i

∫ t

t0

ds 〈[δH(s), χ(t)], χ(t ′)〉+ i

∫ t′

t0

ds 〈χ(t)[δH(s), χ(t ′)]〉


One-loop stress tensor and ANEC

I The 1-loop stress tensor can be computed through the bulk 2-pointfunction G (x , x ′) as

Tab = limx′→x

[∂a∂′bG (x , x ′)− 1

2gabg

ρσ∂ρ∂′σG (x , x ′)− 1

2gabm

2G (x , x ′)]

which is linear in the coupling h.

I Thus depending on the signature of h, the averaged null energycondition can be violated∫ ∞

−∞TUUdU < 0

which makes the wormhole traversable. [Jafferis Gao Wall 2016]


Traversable wormholes

I The change in the dilaton field can be identified explicitly and thelocation of singularity is delayed by ∆τ in τ coordinate leading tothe traversable wormhole solution. We shall call this ∆τ as awormhole opening parameter. [Bak Kim Yi 18]

tL tR

∆τ

t0

t1

I One can show that the horizon size and the corresponding entropy isreduced if the wormhole becomes traversable. With this traversablewormhole, one can send a signal from the left to the right.


I We add the teleportee state T to the left side, which will appear tothe right side after some time which is describing the full quantumteleportation.

I The measurement in general makes the L-R entanglement reducedand the L-T system entangled instead as we discussed previously.

I The essential feature is the L-R coupling by the measurement MθL

and the recovery action V θR . This coupling basically makes the

wormhole traversable which is modelled by the L-R interactionhOLOR .


Observability of bulk!

I Let us now turn to the case where Bob also throws a matter intothe horizon from the right. If the extra matter is not too big, thewormhole remains traversable and the teleportee can be sent fromthe left to the right sides.

I It is clear that the teleportee will meet the matter from the rightside while transported. It can record and report this encounter toBob on the right side. Hence one may conclude that the bulkwormhole is experimentally observable. [Susskind 2017]


Eternally traversable wormholes

I The eternal traversable wormhole geometry [Maldacena Qi 18], is quiteinteresting.

I We simply adjust h(t) in an appropriate manner which makes thewormhole eternally traversable and one may verify explicitly thesingularity disappears completely. [Bak Kim Yi]

I Hence there are no trapped region and all the information behindthe horizon comes out to L or R. This is an example of black holeevaporation.


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