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
Near-derivation of RT [Lewkowycz, Maldacena] Derivation of RT in AdS3 Firewall controversy [Faulkner], [Hartman] [AMPS] and many others A year of entanglement! ER=EPR conjecture Quantum corrections [Maldacena, Susskind] [Barrella, et al] [Faulkner, et al] EE and excited states [J. Bhattacharya, et al], others 3 Thursday, August 1, 13
Near-derivation of RT [Lewkowycz, Maldacena] Derivation of RT in AdS3 Firewall controversy [Faulkner], [Hartman] [AMPS] and many others A year of entanglement! ER=EPR conjecture Quantum corrections [Maldacena, Susskind] [Barrella, et al] [Faulkner, et al] EE and excited states [J. Bhattacharya, et al], others And many others 4 Thursday, August 1, 13
A year of entanglement! ER=EPR conjecture I’m going to talk about the [Maldacena, Susskind] Maldacena-Susskind proposal 5 Thursday, August 1, 13
The plan for today 1. Review of EPR correlations 2. Brief summary of Maldacena-Susskind 3. Entangled pairs via holography 6 Thursday, August 1, 13
The EPR “paradox” and Einstein locality 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. 7 Thursday, August 1, 13
The Bell inequalities - I Consider a maximally entangled state (following [Preskill] ’s lecture notes): Equal time, spatially separated spins � (1) )( ˆ � (2) ) | i = � ˆ You can easily show: h | (ˆ n · ~ m · ~ n · ˆ m = � cos ✓ Define projection operators to spin up/down along ˆ n n, ± ) ( i ) = 1 � ( i ) ) P (ˆ 2(1 ± ˆ n · ~ m, ± ) (2) i = 1 This gives probabilities, via h P (ˆ n, ± ) (1) P ( ˆ 4(1 � cos θ ) 8 Thursday, August 1, 13
The Bell inequalities - II ˆ n 1 1 ⇢ 2 (1 − cos θ ) , same spins Find probabilities: 1 2 (1 + cos θ ) , opposite spins ˆ n 2 ˆ n 3 cos θ = 1 Take three axes all separated by ˆ n i 2 Corresponding Bell inequality: P same (1 , 2) + P same (2 , 3) + P same (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) 9 Thursday, August 1, 13
A simpler diagnostic 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: 10 Thursday, August 1, 13
A simpler diagnostic 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. � (1) ( t 1 ) ~ � (2) ( t 2 ) i conn = h ~ � (1) ( t 1 ) ~ � (2) ( t 2 ) i disconn � h ~ � (1) ( t 1 ) ih ~ � (2) ( t 2 ) i h ~ Local realism tells you that these vanish - Nonzero then implies entanglement 11 Thursday, August 1, 13
Basics of the ER=EPR proposal 12 Thursday, August 1, 13
Basic idea 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 P A [MS] make ambitious claim P B ER = EPR Path 2 ER bridge L 1 � L 2 13 Thursday, August 1, 13
Eternal AdS black holes Recall Kruskal extension of AdS BH: Dual interpretation: 1. Thermofield double (CTP) of single CFT 2. Maximally entangled doubled CFT X e − β E n | n, n i | Ψ i = 14 Thursday, August 1, 13
Eternal AdS black holes Recall Kruskal extension of AdS BH: Dual interpretation: 1. Thermofield double (CTP) of single CFT 2. Maximally entangled doubled CFT X e − β E n | n, n i | Ψ i = Boundaries are causally disconnected; only way to communicate is through the interior regions (spacelike trajectories) Boundary CFTs exhibit EPR correlations! 15 Thursday, August 1, 13
The role of the ER bridge 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 ) 16 Thursday, August 1, 13
Non-trivial example: black hole pair production [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! 17 Thursday, August 1, 13
Potential problem: EPR correlations exist in non-gravitational theories. Then what? 18 Thursday, August 1, 13
Basic idea 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? 19 Thursday, August 1, 13
EPR pairs in holography 20 Thursday, August 1, 13
Quarks from strings Let’s make an EPR pair out of a quark and anti-quark. Recall that to do so we add F1 string to AdS 5 × S 5 C1: string goes to boundary -> test quarks C2: string ends on flavor brane [Karch, Katz] -> dynamical quarks z = 0 z = z m z 21 Thursday, August 1, 13
Color singlet pairs (although it’s fun to note that Two possibilities: one never really gets colored states; in global AdS, you 1. Connected string => color singlet always find the other endpoint) 2. Disconnected strings => colored state NOTE: test quarks z = 0 are really quasiparticles , z = z m z with “ “ gluons √ O ( λ ) Conclude this from T=0 entropy of single quark = straight string √ λ [Karch, O’Bannon, S T =0 = Thompson] 2 22 Thursday, August 1, 13
Color singlet pairs (although it’s fun to note that Two possibilities: one never really gets colored states; in global AdS, you 1. Connected string => color singlet always find the other endpoint) 2. Disconnected strings => colored state NOTE: test quarks z = 0 are really quasiparticles , z = z m z with “ “ gluons √ O ( λ ) Conclude this from T=0 entropy of single quark = straight string Color singlet gives entangled QPs! √ λ [Karch, O’Bannon, S T =0 = Thompson] 2 23 Thursday, August 1, 13
Some subtleties Color singlet looks entangled (analogous to SU(N) spin singlet).. BUT describing the entanglement is tricky: 24 Thursday, August 1, 13
Some subtleties Color singlet looks entangled (analogous to SU(N) spin singlet).. BUT describing the entanglement is tricky: 1. We can’t trace over internal dofs of a colored QP! 2. No gauge-invariant way to measure color orientations of remaining QP Consequently, no reduced density matrix describing entanglement of the internal dofs in the pair. q − ¯ q 25 Thursday, August 1, 13
Characterizing the pair entanglement We are forced to employ other observables. Our approach: - study a variety of configurations with q − ¯ q - compute the leading correction to various position-space EEs a priori no crisp connection to pair entanglement.. but our results suggest the EE characterizes it anyway 26 Thursday, August 1, 13
Summary of results √ λ S EE = S N =4 + We find two main classes of solution: 3 - connected string means the pair is always entangled in causal contact, e.g. not in causal contact, e.g. q − ¯ q − ¯ q q quark anti-quark 𝒜 = 𝒄 𝒜 = 𝒄 World volume horizons 27 Thursday, August 1, 13
Summary of results √ λ S EE = S N =4 + We find two main classes of solution: 3 - connected string means the pair is always entangled in causal contact, e.g. not in causal contact, e.g. q − ¯ q − ¯ q q quark anti-quark 𝒜 = 𝒄 𝒜 = 𝒄 World volume horizons No wormhole 27 28 Thursday, August 1, 13
No causal contact => wormhole t ê b 4 Expanding string 3 dual to uniformly 2 accelerating pair 1 x x 2 = t 2 + b 2 − z 2 - 4 - 2 2 4 b - 1 - 2 29 Thursday, August 1, 13
How we calculate the EE We use a result that does not yet exist. 30 Thursday, August 1, 13
How we calculate the EE 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 31 Thursday, August 1, 13
Entanglement and correlations Go back to earlier statement: entanglement signaled by connected spacelike correlations. What happens here? 32 Thursday, August 1, 13
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