Quantum Gravity and Quantum Entanglement (lecture 1)

Dmitri V. Fursaev

Joint Institute for Nuclear ResearchDubna, RUSSIA

Talk is based on hep-th/0602134 hep-th/0606184 Dubna, July 25, 2007

Helmholtz International Summer School onModern Mathematical PhysicsDubna July 22 – 30, 2007

a recent review

L. Amico, R. Fazio, A. Osterloch, V. Vedral,

“Entanglement in Many-Body Systems”,

quant-ph/0703044

What do the following problems have in common?

• finding entanglement entropy in a spin chain near a

critical point

• finding a minimal surface in a curved space

(the Plateau problem)

plan of the 1st lecture

● quantum entanglement (QE) and entropy (EE): general properties

● EE in QFT’s: functional integral methods

● geometrical structure of entanglement entropy

● entanglement in spin chains: 2D critical phenomena CFT’s

● (fundamental) entanglement entropy in quantum gravity

● the Plateau problem

Lecture 1

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 ofentanglement

density matrix of particle «2» under integration over the states of «1»

«2» is in a mixed state when information about «1» is not availableS – measures the loss of information about “1” (or “2”)

a general definition

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

consequence: the entropy is a function of the characteristics of the separating

surface

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)

subadditivity of the entropy

1 2 1 2

1 2 1 2

| | , lnS S S S S S Tr

S S S

1 2

strong subadditivity

1 2 1 2 1 2S S S S

equalities are applied to the von Neumann entropyand are based on the concavity property

effective action approach to EE in a QFT

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

- “inverse temperature”

1 1 1 2

1 2

( ) lim lim 1 ln ( , )

( , )

ln ( , )

2

nnS T Tr Z T

n

Z T Tr

Z T

n

- “partition function”

theory at a finite temperature T

1 2

1 2

1 2

1 2

/

{ ' },{ ' }[ ]

1 2 1 2

{ },{ }

1 2

{ ' },{ }[ ]

1 1 1 2

{ },{ }

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

[ ]

1{ } { ' } [ ]

H T

I

I

e

D eN

I

Tr

d D eN

classical Euclidean action for a given model

1{ ' }

1{ }

2{ }

2{ }0

1/T

1

1

2

2these intervals are identified

Example: 2D case

the geometrical structure

3n

31 1Tr

case

conical singularity is located at the separating point

( 2 , )

( )

Z T

Z T

- standardpartition function

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!

summary of calculation:

1

1 2 1

1)

2) ( , )

2

3) ( ) lim 1 ln ( , )

Z T

n

S T Z T

find a family of manifolds corresponding to a given system

compute

- “geometrical” inverse temperature

- partition function,

have conical singularities on a co-dimension 2 hypersurface(separating surface)

Spectral geometry: example of calculation

2

2

2

20 1 2/ 2

1

2 2 212 2

2 2 22

1( ) ...

(4 )

2( )

3 2

1( )

2

1 1ln

32

1 1ln ( )

48

tLD

tL tm

L

Tr e A At A tt

A vol B

dtTr e et

m m A

S m m vol B

many-body systems in higher dimensions

a

spin lattice continuum limit

2

AS

a A – area of a flat separation surface which divides

the system into two parts (pure quantum states!)

entropy per unit area in a QFT is determined by a UV cutoff!

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)

(method of derivation: spectral geometry)

(Fursaev, hep-th/0602134)

for ground statea is a cutoff

C – topological term (first pointed out in D=3 by Preskill and Kitaev)

Ising spin chains

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

RG-evolution of the entropy

entropy does not increase under RG-flow (as a result of integration of high energy modes)

IR IR

UV

1 is UV fixed point

Explanation

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

What is the entanglement entropy in a fundamental theory?

CONJECTURE(Fursaev, hep-th/0602134)

3

4FUNDN

cs

G

FUNDs - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space

( 4)d

arguments:

● entropy density is determined by a UV-cutoff

● entanglement entropy can be derived from

the effective gravity action

● the conjecture is valid for area density of the entropy of black holes

BLACK HOLE THERMODYNAMICS

3

4BH H

N

AS c

G

HA

Bekenstein-Hawking entropy

- area of the horizon

BHS - measure of the loss of information about states underthe horizon

some references: ● black hole entropy as the entropy of entanglement (Srednicki 93, Sorkin et

al 86, Frolov & Novikov 93)

● iduced gravity (Sakharov 68) as a condition (Jacobson 94, Frolov, Fursaev, Zelnikov 96)

● application to de Sitter horizon (Hawking, Maldacena, Strominger 00)

● entropy of certain type black holes in string theory as the entanglement entropy in 2- and 3- qubit systems (Duff 06, Kallosh & Linde 06)

● yields the value for the fundamental entropy in flat space in terms of gravity coupling

● horizon entropy is a particular case

our conjecture :

Open questions:

● Does the definition of a “separating surface” make sense in a quantum

gravity theory (in the presence of “quantum geometry”)?

● Entanglement of gravitational degrees of freedom?

● Can the problem of UV divergences in EE be solved by the standard

renormalization prescription? What are the physical constants which

should be renormalized?

the geometry was “frozen” till now:

assumption

... ...fundamental low energydof dof

Ising model:

“fundamental” dof are the spin variables on the lattice

low-energies = near-critical regime

low-energy theory = QFT (CFT) of fermions

B

1

B

2

at low energies integration over fundamental degrees of freedom is equivalent to the integration over all low energy fields, including fluctuations of the space-time metric

This means that:

(if the boundary of the separating surface is fixed)

the geometry of the separating surface is determined by a quantum problem

B

B

Bfluctuations of are induced by fluctuations of the space-time geometry

entanglement entropy in the semiclassical approximation

[ , ]

4 3

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

1 1[ ] ,

16 8

( ) ln ( ) [ , ],

n n

I gmatter

M M

Z T Dg D e Z T I g I g I g

I g R gd x K hd yG G

F T Z T I g

a standard procedure

( , , )

4

1 1 1 2

( , )

2(2 ) ( ),

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

8

lim lim 1 ln ( , ) ,

( )

4

( )

n

I g

B

regular

M

regular

nn g m

m

g

Z T e

R gd x R A B

A BI g I g

G

S Tr Z T S Sn

S

A BS

G

A B

fix n and “average” over all possible positionsof the separating surface on

- entanglement entropy of quantum matter (if one goes beyond the semiclassical approximation)

- pure gravitational part of entanglement entropy

- some average area

what are the conditions on the separating surface?

conditions for the separating surface

( )( 2 )( , , ) ( , , ) 4

( )( )( 2 ) 44

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 minimalco-dimension 2 hypersurface in

, ,

2 2

0

,

1, 0,

0,

0.

iji j

ij

n

p

X X X

n p

n p np

k n

k p

- induced metric on the surface

- normal vectors to the surface

- traces of extrinsic curvatures

Equations

NB: we worked with Euclidean version of the theory (finite

temperature), stationary space-times was implied;

In the Lorentzian version of the theory space-times: the

surface is extremal;

Hint: In non-stationary space-times the fundamental

entanglement should be associated with extremal surfaces

A similar conclusion in AdS/CFT context is in (Hubeny,

Rangami, Takayanagi, hep-th/0705.0016)

Quantum corrections

2

4

4 4

g q

g

q

div finq q q

divq

divq

bare ren

S S S

AS

GS

S S S

AS

A AS

G G

the UV divergences in the entropy areremoved by the standard renormalization of thegravitational couplings;

the result is finite and is expressed entirely in terms of low-energy variables and effective constants like G

2, , ;

; ;

0

0

iji j

B

t

A d y X X X

X

t

B

a Killing vector field

- a constant time hypersurface (a Riemannian manifold)

is a co-dimension 1 minimal surface on a constant-time hypersurface

Stationary spacetimes: simplification

the statement is true for the Lorentzian theory as well !

variational formulae for EE

S M

M

S

- change of the entropy per unit length (for a cosmic string)

- string tension

-change of the entropy under the shift of a point particle

-mass of the particle

- shift distance

other approaches

• Jacobson :

- entanglement is associated with a local causal structure of a space-time; we consider more general case;

- space-like surface is arbitrary, it is considered as a local Rindler horizon for a family of accelerated observers;

we: the surface is minimal (extremal), black hole horizon is a particular case;

- evolution of the surface is along light rays starting at the surface; we study the evolution leaving the surface minimal (extremal).

the Plateau Problem (Joseph Plateau, 1801-1883)

It is a problem of finding a least area surface (minimal surface)for a given boundary

soap films:1 2

1

1 2

( )k h p p

k

h

p p

- the mean curvature

- surface tension

-pressure difference across the film

- equilibrium equation

the Plateau Problem there are no unique solutions in general

the Plateau Problem simple surfaces

The structure of part of a DNA double helix

catenoid is a three-dimensional shape made by rotating a catenary curve (discovered by L.Euler in 1744) helicoid is a ruled surface, meaning that it is a trace of a line

the Plateau Problem

Costa’s surface (1982)

other embedded surfaces

the Plateau Problem

A minimal Klein bottle with one end

Non-orientable surfaces

A projective plane with three planar ends. From far away the surface looks like the three coordinate plane

the Plateau Problem

Non-trivial topology: surfaces with hadles

a surface was found by Chen and Gackstatter

a singly periodic Scherk surface approaches two orthogonal planes

the Plateau Problem a minimal surface may be unstable against small perturbations

plan of the 2d lecture

● entanglement entropy in AdS/CFT: “holographic formula”

● derivation of the “holographic formula” for EE

● some examples: EE in 2D CFT’s

● conclusions