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