1. Danny Terno Entropy and entanglement on the horizon joint work with Etera Livine gr-qc/0508085 gr-qc/0505068 Phys. Rev. A 72 022307 (2005)

2. Gauge invariance: SU(2) invariance at each vertex becomes SU(2) invariance for the horizon states Object: static black hole States: spin network that crosses the horizon in LQG Black hole Comment 1: no dynamics Comment 2: closed 2-surface Definition of a “black hole” : complete coarse-graining of the spin network inside Microscopic states: intertwiners

3. Features & assumptions Area spectrum The probing scale The flow: scaling and invariance of physical quantities We work at fixed j Comment: reasons to be discussed For starters: a qubit black hole

4. Summary Qubit black hole Spin-j black hole Entanglement between halves of the horizon Logarithmic correction = quantum mutual information Area rescaling

5. density matrix Standard counting story area constraint 2n spins number of states entropy Fancy counting story entropy

6. Combinatorics Schur’s duality is the irrep of the permutation group Example: =#standard tableaux 1 3 2 4 1 2 3 4

7. Entanglement a brief history Ancient times: 1935-1993 “The sole use of entanglement was to subtly humiliate the opponents of QM” Modern age: 1993- Resource of QIT Teleportation, quantum dense coding, quantum computation…. Postmodern age: 1986 (2001)- Entanglement in physics 1/3

8. Entanglement a closer encounter Pure states 0.2 0.4 0.6 0.8 1 1 Mixed states hierarchy Direct product Separable Entangled 2/3

9. Entanglement of formation Minimal weighted average entanglement of constituents Entanglement measures “Good” measures of entanglement: satisfy three axioms Coincide on pure states with Do not increase under LOCC Zero on unentangled states Almost never known 3/3

10. Entanglement calculation Clever notation

11. 2 vs 2n-2 States Unentangled fraction Entanglement degeneracy indices Entanglement

12. n vs n Entropy of the whole vs. sum of its parts Reduced density matrices BH is not made from independent qubits, but… Logarithmic correction equals quantum mutual information

13. Why qubits (fixed j)? Answer 1: Dreyer, Markopoulou, Smolin Comment: spin-1 Answer 2: if the spectrum is Answer 3: irreducibility Decomposition into spin-1/2. 1-1 relation between the intertwiners. No area change

14. Entropy 1 2 34 Explanation: a random walk with a mirror -4 -3 -2 Practical calculation: RWM(0)=RW(0)-RW(1) Universality and the random walks

15. Calculations & asymptotics Asymptotics Entanglement: n vs n

16. Area renormalization Generic surface, 2n qubits Complete coarse-graining The most probable spin: maximal degeneracy Horizon, 2n qubits split into p patches of 2k qubits

17. The most probable spin: maximal degeneracy different options The average spin: Area rescaling:

18. Open questions

19. Dynamics: evolution of entanglement dynamical evolution of evaporation "H=0" section & the number of states Semi-classicality: requiring states to represent semi-classical BH rotating BH Open questions

20. Evaporation A model for Bekenstein-Mukhanov spectroscopy (1995) Minimal frequency <= fundamental j Probability for the jump is proportional to the unentangled fraction number of blocks unentangled fraction (of 2-spin blocks)

21. Entanglement calculation Alternative decomposition: linear combinations Its reduced density matrices: mixtures Entropy: concavity Clever notation (2): Clever notation (3): Coup de grâce: