Danny Terno Entropy and entanglement on the horizon joint work with Etera Livine gr-qc/ gr-qc/ Phys. Rev. A (2005)
Outline LQG in 5 minutes Horizon: global SU(2) invariance of states Qubit black hole: entropy Entanglement 101 Qubit black hole: entanglement &ct Fundamental scales & universality Area renormalization
Loop quantum gravity Quantization of GR in 3+1 dimensions Action Time function & partial gauge fixing: SO(3) so(3)- valued connection on extrinsic curvature of 3D manifold M Canonical variables 1/3
Gauge freedom: su(2) Constraints: Gauss/gauge, vector/diffeo, scalar/Hamiltonian Configuration space: graphs & holonomies More math Hilbert space: functions of holonomies, basic operators Poisson brackets Quantization Phase space: holonomies & fluxes Renormalization 1 Decomposition : example Peter- Weyl 2/3
spin networks Decomposition Gauge & diffeomorphism constraints: Intertwiners: Coupling theory Area operator Volume operator Cutting edges Counting vertices Problems… 3/3
Black holes
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 Comment 3: mixed microscopic states, J>0, etc., for later
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
density matrix Standard counting story area constraint 2n spins number of states entropy Fancy counting story entropy
Combinatorics Schur’s duality is the irrep of the permutation group Example: =# reps in the universal rep?? =#standard tableaux
Entanglement a brief history Ancient times: “The sole use of entanglement was to subtly humiliate the opponents of QM” Modern age: Resource of QIT Teleportation, quantum dense coding, quantum computation…. Postmodern age: 1986 (2001)- Entanglement in physics 1/3
Entanglement a closer encounter Pure states Mixed states hierarchy Direct product Separable Entangled 2/3
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
Entanglement calculation Clever notation
2 vs 2n-2 States Unentangled fraction Entanglement degeneracy indices Entanglement
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
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/ relation between the intertwiners. No area change
Entropy Explanation: a random walk with a mirror Practical calculation: RWM(0)=RW(0)-RW(1) Universality and the random walks
Calculations & asymptotics Asymptotics Entanglement: n vs n
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
The most probable spin: maximal degeneracy different options The average spin: Area rescaling:
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
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)
Entanglement calculation Alternative decomposition: linear combinations Its reduced density matrices: mixtures Entropy: concavity Clever notation (2): Clever notation (3): Coup de grâce: