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

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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

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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

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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

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spin networks Decomposition Gauge & diffeomorphism constraints: Intertwiners: Coupling theory Area operator Volume operator Cutting edges Counting vertices Problems… 3/3

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Black holes

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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

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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

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

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Combinatorics Schur’s duality is the irrep of the permutation group Example: =# reps in the universal rep?? =#standard tableaux

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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

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Entanglement a closer encounter Pure states Mixed states hierarchy Direct product Separable Entangled 2/3

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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

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Entanglement calculation Clever notation

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

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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

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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

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Entropy Explanation: a random walk with a mirror Practical calculation: RWM(0)=RW(0)-RW(1) Universality and the random walks

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Calculations & asymptotics Asymptotics Entanglement: n vs n

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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

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

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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

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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)

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

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