Quantum Entanglement and Gravity

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Gravity in three dimensions, ESI Workshop,
Vienna, 24.04.09
Quantum Entanglement and Gravity

Dmitri V. Fursaev Joint Institute for Nuclear
Research and Dubna University
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plan of the talk

• Part I (a
  review)
• ? general properties and examples (spin chains,
  2D CFT, ...)
• ? computation partition function approach
• ? entanglement in CFTs with AdS gravity duals (a
  holographic formula
• for the entropy)
• Part II (entanglement
  entropy in quantum gravity)
• ? suggestions and motivations
• ? tests
• ? consequences

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Quantum Entanglement
Quantum state of particle 1 cannot be
described independently from particle 2 (even
for spatial separation at long distances)
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measure of entanglement

• entropy of entanglement

density matrix of particle 2 under integration
over the states of 1
2 is in a mixed state when information
about 1 is not available S measures
the loss of information about 1 (or 2)
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definition of entanglement entropy
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symmetry of EE in a pure state
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Entanglement in many-body systems
spin lattice
continuum limit
Entanglement entropy is an important physical
quantity which helps to understand better
collective effects in stringly correlated systems
(both in QFT and in condensed matter)
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spin chains (Ising model as an example)
off-critical regime at large N
critical regime

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• Near the critical point the Ising model is
  equivalent to a 2D
• quantum field theory with mass m proportional to
  
• At the critical point it is equivalent to a 2D
  CFT with 2 massless
• fermions each having the central charge 1/2

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Behavior near the critical pointand
RG-interpretation
UV
is UV fixed point
IR
IR
The entropy decreases under the evolution to IR
region because the contribution of short wave
length modes is ignored (increasing the mass
is equivalent to decreasing the energy
cutoff)
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more analytical results in 2D
ground state entanglement on an interval
Calabrese, Cardy hep-th/0405152
is the length of
a is a UV cutoff
massive case
massless case
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analytical results (continued)
is the length of
ground state entanglement for a system on a
circle
system at a finite temperature
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Entropy in higher dimensions
in a simple case the entropy is a fuction of the
area A
- in a relativistic QFT (Srednicki 93, Bombelli
et al, 86)
- in some fermionic condensed matter systems
(Gioev Klich 06)
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geometrical structure of the entropy
edge (L number of edges)
separating surface (of area A)
sharp corner (C number of corners)
for ground state
a is a cutoff
(DF, hep-th/0602134)
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partition function and effective action
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replica method
- partition function (a path integral)

• effective action is defined on manifolds
• with cone-like singularities

- inverse temperature
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theory at a finite temperature T
classical Euclidean action for a given model
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Example 2D case
these intervals are identified
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the geometrical structure for
conical singularity is located at the separating
point
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effective action on a manifold with conical
singularities is the gravity action
(even if the manifold is locally flat)
curvature at the singularity is non-trivial
derivation of entanglement entropy in a flat
space has to do with gravity effects!
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entanglement in CFTs and a holographic
formula
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Holographic Formula
Ryu and Takayanagi, hep-th/0603001, 0605073
(bulk space)
minimal (least area) surface in the bulk
4d space-time manifold (asymptotic boundary of
AdS)
separating surface
is measured in terms of the area of
entropy of entanglement
is the gravity coupling in AdS
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Holographic formula enables one to compute
entanglement entropy in strongly correlated
systems with the help of classical methods (the
Palteau problem)
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2D CFT on a circle
is the length of
ground state entanglement for a system on a
circle
c is a central charge
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gravity
- AdS radius
minimal surface a geodesic line
A is the length of the geodesic
- UV cutoff

• holographic formula

- central charge
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a finite temperature theory a black hole in the
bulk space
Entropies are different (as they should be)
because there are topologically inequivalent
minimal surfaces
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a simple example for higher dimensions
2
is IR cutoff
1
2
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Motivation of the holographic formula
DF, hep-th/0606184
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Low-energy approximation
Partition function for the bulk gravity (for the
replicated boundary CFT)
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Boundary conditions
The boundary manifold has conical singularities
at the separating surface. Hence, the bulk path
integral should involve manifolds with conical
singularities, position of the singular surfaces
in the bulk is specified by boundary conditions
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Semiclassical approximation
- holographic entanglement entropy
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conditions for the singular surface in the bulk
the separating surface is a minimal least area
co-dimension 2 hypersurface
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Part IIentanglement entropy in quantum gravity
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entanglement has to do with quantum gravity
? entanglement entropy allows a holographic
interpretation for CFTs with AdS duals ?
possible source of the entropy of a black hole
(states inside and outside the horizon) ? d4
supersymmetric BHs are equivalent to 2, 3,
qubit systems
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quantum gravity theory Can one define an
entanglement entropy, S(B), of fundamental
degrees of freedom spatially separated by a
surface B? How can the fluctuations of the
geometry be taken into account?
the hypothesis
? S(B) is a macroscopical quantity (like
thermodynamical entropy) ? S(B) can be
computed without knowledge of a
microscopical content of the theory (for an
ordinary quantum system it cant) ? the
definition of the entropy is possible for
surfaces B of a certain type
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Suggestion (DF, 06,07) EE in quantum
gravity between degrees of freedom separated by
a surface B is
1
2
B is a least area minimal hypersurface in a
constant-time slice
the system is determind by a set of boundary
conditions subsets, 1 and 2 , in the
bulk are specified by the division of the boundary

• conditions
• ? static space-times
• ? slices have trivial topology
• ? the boundary of the slice is simply connected

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a Killing symmetry orthogonality of the Killing
field to constant-time slices a hypersurface
minimal in a constant time slice is minimal
in the entire space-time
a proof of the entropy formula is the same as
the motivation of the holographic formula
Higher-dimensional (AdS) bulk -gt physical
space-time AdS boundary -gt boundary of
the physical space
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Slices with wormhole topology (black holes,
wormholes)
on topological grounds, on a space-time slice
which locally is there are
closed least area surfaces example for
stationary black holes the cross-section of the
black hole horizon with a constant-time
hypersurface is a minimal surface there are
contributions from closed least area surfaces to
the entanglement
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slices with wormhole topology
we follow the principle of the least total area
EE in quantum gravity is
are least area minimal hypersurfaces
homologous, respectively, to
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consequences

• if the EE is
• for black holes one reproduces the
  Bekenstein-Hawking formula
• wormholes may be characterized by an intrinsic
  entropy associated to
• the area of he mouth

Entropy of a wormhole analogous conclusion (S.
Hayward, P. Martin-Moruno and P. Gonzalez-Diaz)
is based on variational formulae
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tests
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Araki-Lieb inequality
inequalities for the von Neumann entropy
strong subadditivity property
equalities are applied to the von Neumann
entropy and are based on the concavity property
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strong subadditivity
c
d
c
d
1
2
f
f
b
a
a
b
generalization in the presence of closed least
area surfaces is straightforward
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Araki-Lieb inequality, case of slices with a
wormhole topology
entire system is in a mixed state because the
states on the other part of the throat are
unobervable
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variational formulae
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• for realistic condensed matter systems the
  entanglement entropy is a non-trivial function of
  both macroscopical and microscopical parameters
• entanglement entropy in a quantum gravity theory
  can be measured solely in terms of macroscopical
  (low-energy) parameters without the knowledge of
  a microscopical content of the theory

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simple variational formulae
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variational formula for a wormhole
- position of the w.h. mouth (a marginal sphere)

• a Misner-Sharp energy (in static case)

stress-energy tensor of the matter on the mouth
- a surface gravity
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For extension to non-static spherically symmetric
wormholes and ideas of wormhole
thermodynamics see S. Hayward 0903.5438
gr-qc P. Martin-Moruno and P. Gonzalez-Diaz
0904.0099 gr-qc
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conclusions and future questions

• there is a deep connection between quantum
  entanglement and gravity which goes beyond the
  black hole physics
• entanglement entropy in quantum gravity may be a
  pure macroscopical quantity, information about
  microscopical structure of the fundamental theory
  is not needed (analogy with thermodynamical
  entropy)
• entanglement entropy is given by the
  Bekenstein-Hawking formula in terms of the area
  of a co-dimensiin 2 hypersurface black hole
  entropy is a particular case
• entropy formula passes tests based on
  inequalities
• wormholes may possess an intrinsic entropy
  variational formulae for a wormhole might imply
  thermodynamical interpretation
• (microscopical derivation?, Cardy
  formula?....)

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Extension of the formula for entanglement entropy
to non-static space times?
minimal surfaces on constant time sections