The holographic dual of an EPR pair (has a wormhole!) Kristan - - PowerPoint PPT Presentation

The holographic dual of an EPR pair (has a wormhole!)

Kristan Jensen - University of Victoria

Gauge/Gravity Duality 2013 - MPI Munich

based on: arXiv:1307.1132 (see also arXiv:1306.0533, 1307.6850) arXiv:1308.XXXX and due to collaboration with A. Karch and A. O’Bannon

1 Thursday, August 1, 13

A year of entanglement!

2 Thursday, August 1, 13

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Firewall controversy

[AMPS] and many others

Derivation of RT in AdS3

[Faulkner], [Hartman]

ER=EPR conjecture

[Maldacena, Susskind]

Quantum corrections

[Barrella, et al] [Faulkner, et al]

Near-derivation of RT

[Lewkowycz, Maldacena]

EE and excited states

[J. Bhattacharya, et al], others

A year of entanglement!

Thursday, August 1, 13

4

Firewall controversy

[AMPS] and many others

Derivation of RT in AdS3

[Faulkner], [Hartman]

ER=EPR conjecture

[Maldacena, Susskind]

Near-derivation of RT

[Lewkowycz, Maldacena]

And many others

A year of entanglement!

Quantum corrections

[Barrella, et al] [Faulkner, et al]

EE and excited states

[J. Bhattacharya, et al], others

Thursday, August 1, 13

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ER=EPR conjecture

[Maldacena, Susskind]

I’m going to talk about the Maldacena-Susskind proposal

A year of entanglement!

Thursday, August 1, 13

The plan for today

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• 1. Review of EPR correlations
• 2. Brief summary of Maldacena-Susskind
• 3. Entangled pairs via holography

Thursday, August 1, 13

The EPR “paradox” and Einstein locality

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As you all know, Einstein was disturbed by entanglement

• Basic issue: faster-than-light correlations between entangled spins

Sharpened his discomfort into a “paradox” with Podolsky and Rosen (EPR) Einstein wanted to demand local realism: Suppose A and B are spacelike separated physical systems. In a complete physical description, any action on A cannot change the description of B.

Thursday, August 1, 13

The Bell inequalities - I

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Consider a maximally entangled state (following [Preskill]’s lecture notes): You can easily show: Define projection operators to spin up/down along h |(ˆ n · ~ (1))( ˆ m · ~ (2))| i = ˆ n · ˆ m = cos ✓ ˆ n P(ˆ n, ±)(i) = 1 2(1 ± ˆ n · ~ (i)) This gives probabilities, via hP(ˆ n, ±)(1)P( ˆ m, ±)(2)i = 1 4(1 cos θ)

Equal time, spatially separated spins

Thursday, August 1, 13

The Bell inequalities - II

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Find probabilities: ⇢

1 2(1 − cos θ),

same spins

1 2(1 + cos θ),

• pposite spins

Take three axes all separated by ˆ ni cos θ = 1 2 ˆ n1 ˆ n2 ˆ n3 Corresponding Bell inequality: Psame(1, 2) + Psame(2, 3) + Psame(3, 1) = 1 4 + 1 4 + 1 4 = 3 4 ⇤ 1 Signals breakdown of local realism!

Fun side note: Bell’s inequalities are usually experimentally tested with photons (CH74, CHSH)

Thursday, August 1, 13

A simpler diagnostic

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However, in a general setting it’s difficult to pose an analogue of Bell’s inequalities, partially because local hidden variable theories are nasty. Much simpler way to characterize the existence of entanglement:

Thursday, August 1, 13

A simpler diagnostic

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However, in a general setting it’s difficult to pose an analogue of Bell’s inequalities, partially because local hidden variable theories are nasty. Much simpler way to characterize the existence of entanglement: Connected n-point function with spacelike separated insertions, e.g. h~ (1)(t1)~ (2)(t2)iconn = h~ (1)(t1)~ (2)(t2)idisconn h~ (1)(t1)ih~ (2)(t2)i Local realism tells you that these vanish

• Nonzero then implies entanglement

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Basics of the ER=EPR proposal

12 Thursday, August 1, 13

Basic idea

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We just understood EPR correlations as a phenomenon which looks non-local, but in reality is not. (Can’t transmit FTL information.) There is a similar challenge to locality in GR in the form of Einstein-Rosen (ER) bridge or non-traversable wormhole

Path 1 Path 2

ER bridge L1 L2 PA PB

[MS] make ambitious claim

ER = EPR

Thursday, August 1, 13

Eternal AdS black holes

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Recall Kruskal extension of AdS BH: Dual interpretation:

• 1. Thermofield double (CTP) of single CFT
• 2. Maximally entangled doubled CFT

|Ψi = X e−βEn|n, ni

Thursday, August 1, 13

Eternal AdS black holes

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Recall Kruskal extension of AdS BH: Dual interpretation:

• 1. Thermofield double (CTP) of single CFT
• 2. Maximally entangled doubled CFT

|Ψi = X e−βEn|n, ni Boundaries are causally disconnected;

• nly way to communicate is through the

interior regions (spacelike trajectories) Boundary CFTs exhibit EPR correlations!

Thursday, August 1, 13

The role of the ER bridge

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In light of earlier discussion about spacelike correlations.. NOTE: the existence ER bridge is precisely what allows them

• Causal correlators between boundaries = 0
• unordered ones not 0 from spacelike

trajectories across bridge connecting both boundaries entanglement, unordered correlations O(N 2)

Thursday, August 1, 13

Non-trivial example: black hole pair production

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[Garfinkle, Strominger] [Garfinkle, Giddings, Strominger] long ago found instanton solutions

describing near-extremal BH circulating in a B-field

• upon analytic continuation, solution describes Schwinger pair

production of BHs in E-field, which then accelerate away from each other (BHs are entangled, but out of causal contact) Lorentzian wormhole connecting the two horizons!

Thursday, August 1, 13

Potential problem:

EPR correlations exist in non-gravitational theories. Then what?

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Basic idea

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Two step procedure:

• 1. Build (something like) an EPR pair in our favorite non-gravitational

theory [N=4 SYM via holography]

• 2. Is there a wormhole?

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EPR pairs in holography

20 Thursday, August 1, 13

Quarks from strings

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Let’s make an EPR pair out of a quark and anti-quark. Recall that to do so we add F1 string to AdS5 × S5 C1: string goes to boundary -> test quarks C2: string ends on flavor brane [Karch, Katz] -> dynamical quarks z z = 0 z = zm

Thursday, August 1, 13

Color singlet pairs

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Two possibilities:

• 1. Connected string => color singlet
• 2. Disconnected strings => colored state

z z = 0 z = zm

(although it’s fun to note that

• ne never really gets colored

states; in global AdS, you always find the other endpoint)

NOTE: test quarks are really quasiparticles, with “ “ gluons O( √ λ) Conclude this from T=0 entropy

• f single quark = straight string

ST =0 = √ λ 2

[Karch, O’Bannon, Thompson]

Thursday, August 1, 13

Color singlet pairs

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Two possibilities:

• 1. Connected string => color singlet
• 2. Disconnected strings => colored state

z z = 0 z = zm

(although it’s fun to note that

• ne never really gets colored

states; in global AdS, you always find the other endpoint)

NOTE: test quarks are really quasiparticles, with “ “ gluons O( √ λ) Conclude this from T=0 entropy

• f single quark = straight string

ST =0 = √ λ 2

[Karch, O’Bannon, Thompson]

Color singlet gives entangled QPs!

Thursday, August 1, 13

Some subtleties

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Color singlet looks entangled (analogous to SU(N) spin singlet).. BUT describing the entanglement is tricky:

Thursday, August 1, 13

Some subtleties

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• 1. We can’t trace over internal dofs of a colored QP!
• 2. No gauge-invariant way to measure color orientations
• f remaining QP

Color singlet looks entangled (analogous to SU(N) spin singlet).. BUT describing the entanglement is tricky: Consequently, no reduced density matrix describing entanglement

• f the internal dofs in the pair.

q − ¯ q

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Characterizing the pair entanglement

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We are forced to employ other observables. Our approach:

• study a variety of configurations with
• compute the leading correction to various

position-space EEs q − ¯ q a priori no crisp connection to pair entanglement.. but our results suggest the EE characterizes it anyway

Thursday, August 1, 13

𝒜 = 𝒄 𝒜 = 𝒄

World volume horizons

anti-quark quark

Summary of results

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We find two main classes of solution:

• connected string means the pair is always entangled

in causal contact, e.g. q − ¯ q not in causal contact, e.g. q − ¯ q SEE = SN =4 + √ λ 3

Thursday, August 1, 13

Summary of results

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No wormhole

𝒜 = 𝒄 𝒜 = 𝒄

World volume horizons

anti-quark quark

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We find two main classes of solution:

• connected string means the pair is always entangled

in causal contact, e.g. q − ¯ q not in causal contact, e.g. q − ¯ q SEE = SN =4 + √ λ 3

Thursday, August 1, 13

No causal contact => wormhole

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Expanding string dual to uniformly accelerating pair x2 = t2 + b2 − z2

• 4
• 2

2 4

x b

• 2
• 1

1 2 3 4

têb

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How we calculate the EE

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We use a result that does not yet exist.

Thursday, August 1, 13

How we calculate the EE

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We use a result that does not yet exist. WIP: [KJ, O’Bannon] Feel free to ask us for details:

• we essentially generalize the result of

[Casini, Huerta, Myers] for CFTs with conformal defects and boundaries

Thursday, August 1, 13

Entanglement and correlations

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Go back to earlier statement: entanglement signaled by connected spacelike correlations. What happens here?

Thursday, August 1, 13

Entanglement and correlations

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Go back to earlier statement: entanglement signaled by connected spacelike correlations. What happens here? Any connected string has spacelike trajectories along string which admit such correlations. [Another way to see pair is entangled] For strings with worldsheet horizons, regions behind horizon (i.e. the wormhole) are necessary for them! O( √ λ) entanglement, spacelike correlations

Thursday, August 1, 13

Relation to pair production

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Julian Sonner pointed out last week that the expanding string is Lorentzian version of a Euclidean instanton solution describing Schwinger pair production of q − ¯ q Analogous to BH pair production in flat space a la [Garfinkle, Strominger]

Thursday, August 1, 13

Summary

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• 1. MS conjecture ER = EPR based on QM of cold BHs
• 2. Easy to construct EPR pairs in holography via dual strings
• 3. Entanglement with causal contact doesn’t need wormhole,

but entanglement between causally disconnected dofs does

• 4. Can understand necessity of wormhole in terms of large

spacelike correlations

• 5. Lots to do!

Thursday, August 1, 13

Summary

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• 1. MS conjecture ER = EPR based on QM of cold BHs
• 2. Easy to construct EPR pairs in holography via dual strings
• 3. Entanglement with causal contact doesn’t need wormhole,

but entanglement between causally disconnected dofs does

• 4. Can understand necessity of wormhole in terms of large

spacelike correlations

• 5. Lots to do! Maybe strings offer a nice playground for future work

[Toy model for horizon formation? Firewalls on string? Systematically characterize interior region via correlators?]

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Thank you!

Thursday, August 1, 13