Quantum Entanglement and Gravity

Dmitri V. Fursaev

Joint Institute for Nuclear Research and

Dubna University

“Gravity in three dimensions”,ESI Workshop, Vienna, 24.04.09

plan of the talk Part I (a review)

● general properties and examples (spin chains, 2D CFT, ...)

● computation: “partition function” approach

● entanglement in CFT’s with AdS gravity duals (a holographic formula for the entropy)

Part II (entanglement entropy in quantum gravity)

● suggestions and motivations

● tests

● consequences

Quantum Entanglement

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

1 2 1 2

1| (| | | | )

2

measure of entanglement

2 2 2

2 1

( ln )

(| |)

S Tr

Tr

- 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”)

definition of 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

“symmetry” of EE in a pure state

1 1

2 2

2 1

1 2

( | )

( | ) ,

, ( 0)

AaaA

Aa Baa

TAa Ab

A

C A a

A B C C CC

a b C C C C

if d Ce e e d d

S S

Entanglement in many-body systems

a

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)

spin chains (Ising model as an example)

11

( )N

X X ZK K K

K

H

2

1( , ) log

6 2

NS N

2

1( , ) log | 1|

6S N

1 | 1| 1 off-critical regime at large N

critical regime 1

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

| 1|

ln6

ln6

cS ma

c LS

a

Behavior near the critical pointand RG-interpretation

1

IRIR

UV

is UV fixed point

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)| 1|

more analytical results in 2D

1ln

6

cS

ma

1ln6

LcS

a

1L

1ln sin 26

Lc LS g

a L

L

11/ m La is a UV cutoff

Calabrese, Cardyhep-th/0405152

ground state entanglementon an interval

massive case:

massless case:

is the length of

analytical results (continued)

1ln sin3

Lc LS

a L

1ln sinh3

LcS

a

1/T

ground state entanglement for asystem on a circle

system at a finite temperature

1L is the length of

Entropy in higher dimensions

1 2 ( )S S f A

in a simple case the entropyis a fuction of the area A

ln

S A

S A A

- in a relativistic QFT (Srednicki 93, Bombelli et al, 86)

- in some fermionic condensed matter systems (Gioev & Klich 06)

geometrical structure of the entropy

2ln

A LS C a

a a

edge (L = number of edges)

separating surface (of area A)

sharp corner (C = number of corners)

(DF, hep-th/0602134)

for ground statea is a cutoff

“partition function” and effective action

replica method

-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” (a path integral)

theory at a finite temperature T

2 1

2 1

2 1

2 1

/

{ ' },{ ' }[ ]

2 1 2 1

{ },{ }

2 1

{ ' },{ }[ ]

2 2 2 1

{ },{ }

1{ },{ } { ' },{ ' } [ ]

[ ]

1{ } { ' } [ ]

H T

I

I

e

D eN

I

Tr

d D eN

classical Euclidean action for a given model

2{ ' }1{ }

1{ } 2{ }0

1/T

1

1

2

2these intervals are identified

Example: 2D case

32 2Tr

conical singularity is located at the separating point

the geometrical structure for

effective action on a manifold with conical singularities is the gravity action (even if the manifold is locally flat)

(2)2(2 ) ( )R B

curvature at the singularity is non-trivial:

derivation of entanglement entropy in a flat space has to do with gravity effects!

entanglement in CFT’s and a “holographic formula”

Holographic Formula

B

1dAdS

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 the area of

( 1)dG is the gravity coupling in AdS

( 1)

( )

4 d

A BS

G

B B

B

Holographic formula enables one to compute entanglement entropyin strongly correlated systems with the help of classical

methods (the Palteau problem)

2D CFT on a circle

1ln sin3

Lc LS

a L

ground state entanglement for asystem on a circle

1L is the length of

c – is a central charge

gravity

0

0

0

2 2 2 2 2 2 2

2 211

2 2 10

1

3

3

cosh sinh

2

cosh 1 2sinh sin

ln sin4 3

3

2

CFT

ds l d dt d

l

Lds ds

LLA

l L

Le

a

LA cS e

G L

lc

G

- AdS radius

A is the length of the geodesic

- UV cutoff

-holographic formula

- central charge

minimal surface =a geodesic line

a finite temperature theory: a black hole in the bulk space

1 21 2

3 3

( ) ( )

4 4

A B A BS S

G G

Entropies are different (as they should be) because there are topologically inequivalent minimal surfaces

1 1

1ln sinh

3

cS TL

Ta

a simple example for higher dimensions

B

22 2 25 42

3

2

3 2

2 25 5

32

5

( )

4

, ( ( ))

lds dz ds

z

lA A

a

A l NS A A

G a G a

lN SU N

G

2

2

1

– is IR cutoffa

Motivation of the holographic formula

DF, hep-th/0606184

Low-energy approximation

( )1

[ , ]

1

1

( , ) ( , )

( , ) [ ][ ] , 2

ln ( , ) [ , , ] [ , ] [ , ],

1[ , ] . . ,

16

( , ) ln ( , ) [ , ],

nd

CFT AdS

I gAdS

AdS matter

d

Md

AdS

Z T Z T

Z T Dg D e n

Z T I g I g I g

I g R gd x b tG

F T Z T I g

Partition function for the bulk gravity (for the “replicated” boundary CFT)

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

( ) ( )1

n nd dM M

( )1

1 1

1

1 1 1 2

1

2(2 ) ( ),

( )( , , ) ( , , ) (2 ) ,

8

lim lim 1 ln ( , ),

( )

4

nd

d regular d

M

regular

d

nn AdS

d

R gd x R gd x A B

A BI g I g

G

S Tr Z Tn

A BS

G

- holographic entanglement entropy

Semiclassical approximation

conditions for the singular surface in the bulk

1

11

( )( 2 )( , , ) ( , , ) 8

( )( ) ( 2 )( 2 ) 88

2

( , ) ,

, 2

( ) 0, ( ) 0

regular

d

dd

A BI g I g G

AdSB 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

Part IIentanglement entropy

in quantum gravity

entanglement has to do with quantum gravity:

● entanglement entropy allows a holographic interpretation for CFT’s with AdS duals

● 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

● S(B) is a macroscopical quantity (like thermodynamical entropy);

● S(B) can be computed without knowledge of a microscopicalcontent of the theory (for an ordinary quantum system it can’t)

● the definition of the entropy is possible for surfaces B of a certain type

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

Suggestion (DF, 06,07): EE in quantum gravitybetween degrees of freedom separated by a surface B is

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

1

2

( )( )

4

A BS B

G

the system is determind by a set of boundary conditions;subsets, “1” and “2” , in the bulkare specified by the division of theboundary

a Killing symmetry + orthogonality of the Killing field to constant-time slices:

a hypersurface minimal in a constant time slice is minimal inthe entire space-time

a “proof” of the entropy formula is the same as the motivation

of the “holographic formula”

Higher-dimensional (AdS) bulk -> physical space-timeAdS boundary -> boundary of the physical space

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

1 nR S

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

we follow the principle of the least total area

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

1D 0( )

4

A BS

G

Entropy of a wormhole: analogous conclusion (S. Hayward, P. Martin-Moruno and P. Gonzalez-Diaz) is based on variational formulae

tests

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

variational formulae

• 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

simple variational formulae

3710 1 , 1

(1)

S M z

M mass of a particle

z shift

S if M g z cm

S O if z is a Compton wavelength

S l z

string tension

l lenght of the segment

variational formula for a wormhole2

2 2 2 2 2 2 2 2

3

2

22

22 ( )

1

: ( ) 0

21

( )2

( )

4

4

3

4

rrH H

H H

H

H

HH

drds e dt r d e dx dx r d

E rr

r r g r

E S w V

E E r r

A rS

G

rV

w e T

Ee r r w

r

- 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

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]

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?....)

B

Extension of the formula for entanglement entropy to non-static space times?

minimal surfaces on constant time sections