Herman Verlinde. Simons Symposium. February, Quantum Entanglement: From Quantum Matter to String Theory PCTS. Princeton University

Transcription

1 How to Fall into a Black Hole Princeton University Herman Verlinde Simons Symposium Quantum Entanglement: From Quantum Matter to String Theory Feruary, 2013 PCTS

2 The AdS/CFT correspondence has convinced us that lack hole physics follows the rules of QM.? Can a finite temperature CFTdescrie an oject falling through the horizon of an AdS lack hole?

3 The AdS/CFT correspondence has convinced us that lack hole physics follows the rules of QM.? Can a finite temperature CFTdescrie an oject falling through the horizon of an AdS lack hole?

4 The AdS/CFT correspondence has convinced us that lack hole physics follows the rules of QM. Ikepod Horizon Black Hole RS (LNIB) HHB10? Approx Retail: $14, Price: $9, SEND E Details Guarantee All new watches purchased from Collector guaranteed to e original genuine articles. Our sourced from authorized memers of the Can a finite temperature CFTdescrie an oject falling through the horizon of an AdS lack hole?

5 horizon a Hawking pair r e Figure 1: The firewall argument in a nut shell: quanta r and e can escape the lack hole thanks to entanglement with Hawking partners and a. Monogamy prevents the partners to get entangled with other interior quanta. So after the Page time, the lack hole is in a maximally mixed state, and a and can not escape via the normal Hawking process. 0U = n Wn n a n.

6 a FIREWALL r a e Figure 2: When the Hawking partner leaves the lack hole, it can not e entangled with any mode a right ehind the horizon: the horizon turns into a firewall. To avoid this conclusion, one needs a mechanism for allowing entanglement to re-estalish inside the lack hole.

7 What are the characteristics of a lack hole?

8 i HM N =dimh M = e S BH, S =4πM 2. BH A lack hole is always in a mixed state! Ψ = a i i H code i Φ i ρ = BH p i i H code i i

9 Hilert space splits up into three sectors: H = H H H early A B R H = H, A BH with H = H late B R. H = H + H + H BH R int, Hamiltonian evolution in Interaction Picture

10 Hilert space splits up into three sectors: H = H H H early A B R H = H, A BH with H = H late B R. H = H + H + H BH R int, Hamiltonian evolution in Interaction Picture

11 Initial mixed state ρ AB (0) = i p i i i 0 0, evolves into ρ AB (τ) = U ρ(0)u = n,m ρ nm n m ρ B = n Wn n n ; Wn = e βe n Z S B =logz + βē B = Hawking atmosphere

12 Time evolution in terms of Kraus Operators C U (τ) i 0 n = n Unitary: C nc n = M. C n i n. cf. loss of coherence of a quantum computer BAD GOOD C n C m = 1 Z δ nm M E n i C n C k m = Wn δ nm δ i k. C s are ergodic `random matrices with known statistical properties!

13 We can compute the Page curve of our model! ρ(e) 2 = we +2 d B(E) N c NZ ρ(e) db (E) NZ 2 M E S(E) =w E d B (E) NZ y log d B (E) 2+y log 1+y + 2y + y 2 y = N N c e βe 2d B (E), Two limiting cases: young BH : S AB =logn c S A =logn c +logz + βē old BH : S AB =logn, S A =logn βē

14 Key Tool: Recovery Super Operator R R n = 1 Wn i i C n, i R j 0 = a n R n j n a RU i 0 0 = a i 0U if i Hcode 0 if i / Hcode 0U = n Wn n a n. Produces the Unruh Vacuum!

15 Reconstruction of Interior Operators: φ(y) = a 0 R φ(y)r 0 a. A acts on lack hole Hilert space satisfies QFT operator algera

16 i U 0 U U + a0 i R + A 0 U φ 0U B R 0 a i U horizon 0 tr ρ φ a (y)φ (x) = 0 U φa (y)φ (x) 0U,

17 Error of the Recovery Operation: ΠU i 0 = U i 0 + error. error = EU i 0, with E = n N code M E N n. n Reconstruction reaks down for maximally mixed states! Firewall Paradox AMPS

18 Error of the Recovery Operation: ΠU i 0 = U i 0 + error. error = EU i 0, with E = n N code M E N n. n Reconstruction reaks works for down all states for maximally Psi) that fit inside mixed a code states! space! Firewall Paradox AMPS?

19 Solving the prolem: O BH = c O (c) BH Ωc Ωc Does this preserve linearity and locality?

20 Solving the prolem: O BH = c O (c) BH Ωc Ωc Does this preserve linearity and locality?

21 Some conclusions: * * * Using QECC, we ve given a general construction of local QFT operators relative to which the horizon is smooth. Generic BH states descrie a semi-classical space-time with an interior geometry and a smooth event horizon. Maximally mixed BH state = sum of semi-classical states Einstein locality + general logic => old BHs have no firewall Some outstanding challenges: Formulate of the rules of oserver complementarity Trace the flow of quantum information and entanglement Quantify violations of locality and limitations of effective QFT.

22 a FIREWALL horizon a Hawking pair r e r a e

23 horizon Code its r Measurement: entangles quits with environment, produces classical information. a e

24 a horizon Code its assists QT protocol r Measurement: enales the quantum teleportation of Hawking quanta ack into the lack hole interior a e

25 Space-Time Complementarity: Space-time complementarity: Avarialecut-off scale (x) on a Cauchy surface Σ provides a permissile semi-classical description of the second quantized Hilert space, only when the quantum fluctuations of the local ackground geometry induced y the corresponding stress-energy fluctuations do not exceed the cut-off scale itself. All critical cut-off scales that saturate this requirement provide complete, complementary descriptions of the Hilert space. Kiem, EV & HV, 1995

26 Event Horizon! out final " # t " in 2 E ~ cm #t e " in /4M " out