Danny Terno

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Entropy and entanglementon the horizon

• Danny Terno

joint work with Etera
Livine
gr-qc/0508085 gr-qc/0505068 Phys. Rev. A 72
022307 (2005)
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Black hole
in LQG
Object static black hole
Comment 1 no dynamics Comment 2 closed 2-surface
States spin network that crosses the horizon
Definition of a black hole complete
coarse-graining of the spin network inside
Gauge invariance SU(2) invariance at each
vertex becomes SU(2) invariance for the horizon
states
Microscopic states intertwiners
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Features assumptions
Area spectrum
The probing scale
We work at fixed j
Comment reasons to be discussed
The flow scaling and invariance
of physical quantities
For starters a qubit black hole
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Summary
Qubit black hole
Spin-j black hole
Entanglement between halves of the horizon
Logarithmic correction quantum mutual
information
Area rescaling
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Standard counting story
2n spins
area
constraint
number of states
entropy
Fancy counting story
density matrix
entropy
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Combinatorics
Schurs duality
is the irrep of the permutation
group
Example
standard tableaux
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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
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Entanglement
a closer encounter
Pure states
Mixed states hierarchy
Direct product
Separable
Entangled
2/3
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Entanglement
Entanglement of formation
measures
Minimal weighted average entanglement of
constituents
Good measures of entanglement satisfy three
axioms
Almost never known
Coincide on pure states with
Zero on unentangled states
Do not increase under LOCC
3/3
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Entanglement
calculation
Clever notation
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Entanglement
2 vs 2n-2
States
degeneracy indices
Unentangled fraction
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
Decomposition into spin-1/2. 1-1 relation
between the intertwiners. No area change
Answer 3 irreducibility
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Universality
and the random walks
Entropy
Explanation a random walk with a mirror
Practical calculation RWM(0)RW(0)-RW(1)
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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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different options
The most probable spin maximal degeneracy
The average spin
Area rescaling
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Open questions
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Open questions
Dynamics evolution of entanglement
dynamical evolution of evaporation
"H0" section the number of states
Semi-classicality requiring states to
represent
semi-classical BH
rotating BH
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Evaporation
A model for Bekenstein-Mukhanov spectroscopy
(1995)
Minimal frequency lt fundamental j
Probability for the jump is proportional to the
unentangled fraction
unentangled fraction (of 2-spin blocks)
number of blocks
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Entanglement

• Alternative decomposition linear combinations
• Its reduced density matrices mixtures
• Entropy concavity

calculation
Clever notation (2)
Clever notation (3)
Coup de grâce