entanglement in quantum gravity and space-time topology
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Entanglement in Quantum Gravity and Space-Time Topology
Dmitri V. Fursaev
Joint Institute for Nuclear Research Dubna, RUSSIA
the talk is based on hep-th/0602134, hep-th/0606184,
arXiv:0711.1221 [hep-th]
Quarks-08Sergiev Posad24.05.08
quantum entanglement: statesof subsystems cannot described independently
1 2
entanglement has to do with quantum gravity:
● possible source of the entropy of a black hole (states inside and outside the horizon);
● d=4 supersymmetric BH’s are equivalent to 2, 3,… qubit systems
● entanglement entropy allows a holographic interpretation for CFT’s with AdS duals
Holographic Formula for the Entropy
B
5AdS
B
4d space-time manifold (asymptotic boundary of AdS)
(bulk space)
separating surface
minimal (least area) surface in the bulk
Ryu and Takayanagi,hep-th/0603001, 0605073
entropy of entanglement is measured in terms of thearea of
( 1)dG is the gravity coupling in AdS
( 1)4 d
AS
G
Suggestion (DF, 06,07): EE in quantum gravitybetween degrees of freedom separated by a surface B is
entanglement entropy in quantum gravity
conditions:
● static space-times
● slices have trivial topology
● the boundary of the slice is simply connected
B is a least area minimal hypersurface in a constant-time slice
entropy of fundamentald.of f. is UV finite
1 2
( )( )
4
A BS B
G
aim of the talk
extension to problems with non-trivial topology:
slices which admit closed least area surfaces;
plan
● motivations for entanglement entropy (EE)
● problems with non-trivial topology
● tests of the suggestions
entanglement entropy
A a
1
2
1 2 2 1
1 1 1 1 2 2 2 2
/
1 2/
( , | , )
( | ) ( , | , ),
( | ) ( , | , ),
, ,
ln , ln
a
A
H T
H T
A a B b
A B A a B a
a b A a A b
Tr Tr
S Tr S Tr
eS S
Tr e
• for realistic condensed matter systems the entanglement entropy is a non-trivial function of both macroscopical and microscopical parameters;
• its calculation is technically involved, it does not allow an analytical treatment in general
• DF: 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
Motivations:effective action approach to EE in a QFT
-effective action is defined on manifolds with cone-like singularities
- “inverse temperature”
1 2 2 2
2 1
( ) lim lim 1 ln ( , )
( , )
ln ( , )
2
nnS T Tr Z T
n
Z T Tr
Z T
n
- “partition function”
2{ ' }1{ }
1{ } 2{ }0
1/T
1
1
2
2these intervals are identified
Example: finite temperature theory on an interval
32 2Tr
conical singularity is located at the separating point
the geometrical structure for
“gravitational” entanglement entropy(semiclassical approximation)
4 31 1 2[ , ] ( )
16 8 8
0, 2
( )( ) ( 1) [ , ]
4
( )
n nM MI g R gd x K hd y A B
G G G
R n in considered example
A BS B I g
G
A B area of B
the “gravitational”entropy appears from the classical gravity action(which is a low-energy approximation of the effective action in quantum gravity)
B
1
B
2
the geometry of the separating surface is determined by a quantum problem
Bfluctuations of are induced by fluctuations of the space-time geometry
conditions for the “separating” surface
( )( 2 )( , , ) ( , , ) 8
( )( ) ( 2 )( 2 ) 88
2
( , ) ,
, 2
( ) 0, ( ) 0
regularA B
I g I g G
B B
A BA BGG
B
Z T e e e
e e
A B A B
the separating surface is a minimalleast area co-dimension 2 hypersurface
slices with non-trivial topology
• slices with handles
(regions where states are integrated out are dashed)
1 nR Sslices which locally are
the work is done withA.I. Zelnikov
2 1 2 1
• slices with wormhole topology
closed least area surfaces
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
1 nR S
slices with a single handle
suggestion: EE in quantum gravity on a slice with a handle is
11 2 0
021 2 0
( ), ( ) ( ) ( )
4
( )( ), ( ) ( ) ( )
4 4
A BS A B A B A B
G
A BA BS A B A B A B
G G
1 2 1 2, ,B B D Dare homologous to , respectively
we follow the principle of the least total area
EE in quantum gravity is:
11 2 0
021 2 0
( ), ( ) ( ) ( )
4
( )( ), ( ) ( ) ( )
4 4
A BS A B A B A B
G
A BA BS A B A B A B
G G
1 2,B B are least area minimal hypersurfaces homologous, respectively, to
1 2,D D
slices with wormhole topology
observation:
if the EE is
• black holes: EE reproduces the Bekenstein-Hawking entropy
• wormholes may be characterized by an intrinsic entropy
1D 0( )
4
A BS
G
Araki-Lieb inequality
1 2| |S S S
1 2
strong subadditivity property
1 2 1 2 1 2S S S S
equalities are applied to the von Neumann entropyand are based on the concavity property
inequalities for the von Neumann entropy
strong subadditivity: 1 2 1 2 1 2S S S S
a b
c d
f a b
c d
f1 2
1 2
1 2
1 2 1 2
, , (4 1)
( ) ( )
ad bc
ad bc af fd bf fc
af bf fd fc ab dc
S A S A G
S S A A A A A A
A A A A A A S S
generalization in the presence of closed least areasurfaces is straightforward
entire system is in a mixed state because the states on the other part of the throat are unobervable
1 2S S S
2 0
1 2 0 1 1 2 1 0 2 1 0
1 2 0 1 2 0 2 2 1 2 0
2 0 1 2 0 1 1 2 2
2 1 2 1 0 0 2 1
1 2 1 2 0 0 1 2
( ) , 0,1,2,
1) ,
2) ,
3) ,
,
,
k kA B A k assume that A A
A A A then S A S A A and S S S
A A A then S A A S A and S S S
A A A A A then S A S A and
S S A A A S if S S
S S A A A S if S S
Araki-Lieb inequality, case ofslices with a wormhole topology
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);
• “the least area principle” can be used to generalize the entropy definition for slices with non-trivial topology;
• the principle can be tested by the entropy inequalities;
• BH entropy is a particular case of EE in quantum gravity;
• wormholes can be characterized by an intrinsic entropy determined by the least area surface at the throat
thank you for attention
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