1. Black Hole Evaporation, Unitarity, and Final State Projection Daniel Gottesman Perimeter Institute

2. Black Hole Evaporation Black holes emit Hawking radiation at a temperature T = 1/(8  M), where M = black hole mass in Planck units Black holes have an entropy S = A/4 associated with this temperature Outgoing Hawking radiation is entangled with some infalling Hawking radiation carrying negative energy, reducing black hole mass

3. Information Loss Problem When a black hole evaporates, what happens to information about the matter that formed it or fell into it? Does quantum mechanics need to be modified to describe black hole evaporation? Is black hole evaporation unitary? “When you have eliminated the impossible, whatever remains, however improbable, must be the truth” Sherlock Holmes, The Sign of Four

4. Solution 1: Information is Lost Advantages: Agrees with semiclassical calculation Disadvantages: Trouble with energy conservation? Lose either time reversal invariance or predictability Implies quantum gravity includes non-unitary processes. Why don’t we see them in atomic physics? When a black hole evaporates, the information is really gone.

5. Variant Solution: Baby Universe A baby universe is born at a black hole singularity and the information goes there. Advantages: Overall unitarity is preserved Disadvantages: Doesn’t explain why we don’t see non-unitary effects in atomic physics

6. Solution 2: Black Hole Remnants Black hole evaporation leaves a remnant particle, with mass comparable to the Planck mass, containing all information. Advantages: No need to modify either quantum mechanics or semiclassical calculation Disadvantages: Very peculiar particles, with fixed mass, but unlimited entropy Why don’t we see effects of virtual remnant production in particle physics?

7. Solution 3: Information Escapes The information escapes with the Hawking radiation, subtly encoded in correlations between particles. Advantages: Preserves unitarity Explains entropy as microstates of black hole horizon Disadvantages: Escaping information seems to require either quantum cloning or faster-than-light travel

8. Penrose Diagram - Flat Space Light moves along 45º lines time Massive objects move slower than light, at less than 45º from vertical Future timelike infinity Past timelike infinity Spacelike infinity Future null infinity Past null infinity

9. Penrose Diagram - Black Hole Future timelike infinity Spacelike infinity Past timelike infinity r=0 Singularity Event horizon: not even light can escape An object which stays outside the black hole An object which falls into the black hole

10. Why quantum cloning? For a large black hole, the horizon seems (locally) like nothing special: infalling object should not be destroyed. (Copy 1) But the escaping Hawking radiation also has a copy of the information. (Copy 2) There exist spacelike slices that include both copies: quantum mechanics is violated on them.

11. Quantum Teleportation time Bell measurement: produces 2 classical bits (a,b) 1 quantum bit Alice  BobXaZbXaZb  

12. Black Hole Final State Black hole singularity projects onto some maximally entangled final state of matter + plus infalling Hawking radiation. (Horowitz & Maldacena, hep-th/0310281, JHEP 2004) Acts like quantum teleportation, but with some specific measurement outcome Outgoing state rotated by some complicated unitary from original matter, so looks thermal, but is actually unitary Strangeness only required at singularity, which is strange anyway

13. Black Hole Final State (I  U T )(  ) (Final state projection)  U   Infalling matter Infalling Hawking radiation Outgoing Hawking radiation No communication needed

14. Problem with Interactions (DG, Preskill, hep-th/0311269, JHEP 2004) (I  U T )(  ) (Final state projection)  U   Suppose the infalling matter interacts with the infalling Hawking radiation before hitting the singularity: Similar to Bennett & Schumacher, “Simulated time travel”

15. Problem with Interactions   00  We can absorb the interactions into the final state. The resulting evolution need not be unitary! For instance:  (Final state projection) Note: Input state of  1  not allowed.

16. Faster-Than-Light Communication (Yurtsever & Hockney, hep-th/0402060)  00   (Final state projection)  Hawking radiation Inside black hole Bob 00 Alice If Alice drops her qubit into the black hole, Bob always sees  0 . Otherwise, Bob sees a mixture of  0  and  1 .

17. Summary Is black hole information unitary or not? There is no consensus. Black hole final state proposal pushes new physics to the black hole singularity while allowing information to escape. However, under perturbation, non-unitarity reappears and can even leak outside the black hole.