Quantum Entanglement and Local Operators Tadashi Takayanagi Yukawa - - PowerPoint PPT Presentation

Strings 2014 @ Princeton, June 23-27th, 2014

Quantum Entanglement and Local Operators

Tadashi Takayanagi

Yukawa Institute for Theoretical Physics (YITP), Kyoto University

Based on arXiv:1401.0539 [PRL 112(2014)111602]

arXiv:1403.0702, arXiv:1405.5946 (see also arXiv: arXiv:1302.5703 [JHEP05(2013)080])

with Pawel Caputa (YITP), Song He (YITP), Masahiro Nozaki (YITP) Tokiro Numasawa (YITP) and Kento Watanabe (YITP).

In QFTs, the entanglement entropy (EE) provides us a universal physical quantity (~order parameter). For example, we can characterize the degrees of freedom

• f CFTs (~central charges) from the EE for ground states.

(i) 2d CFT (ii) 3d CFT (iii) 4d CFT

① Introduction

[Holzhey-Larsen-Wilczek 94, Calabrese-Cardy 04,..] [Ryu-TT 06, Solodukhin 08, Sinha-Myers 10, Casini-Huerta-Myers 11,…] [F-th: Jafferis-Klebanov-Pufu-Safdi 11, Entropic proof: Casini-Huerta 12]

It is also helpful to look at (n-th) Renyi entanglement entropy (REE) which generalizes the EE : If we know all of , we find all eigenvalues of . (so called entanglement spectrum) A B .

B A tot

H H H  

In gravity, we might expect that quantum entanglement gives a quantum bit of spacetime (~ a plank size unit) . (i) BH entropy

[Bekenstein 73, Hawking 75,…]

(ii) Holographic EE (HEE) (iii) Entanglement/Gravity

[Ryu-TT 06, Hubeny-Rangamani-TT 07,…] [Swingle 09, Raamsdonk 09, Myers 12, … ]

B

A



A

Planck length

A

Entangler

= AdS

MERA

[Vidal 05]

The entanglement entropy is also a useful quantity to characterize excited states. Well-studied examples are quantum quenches:

[Calabrese-Cardy 05, 07, …., Liu’s talk]

(a) Global quantum quenches (b) Local quantum quenches Here we want to focus on more elementary excited states: (c) Local operator insertions at a time ⇒ (The main aim of this talk)

  .

/ t c S A  

  .

/ log

2

 t c S

d A

 

m(t) t t*

? 

A

S

~ Loss of information when we assume that the region B is invisible. ~ ``degrees of freedom’’ of the operator O.

) (n A

S

    .

) (

) ( ) ( ) ( n A n A n A

S x O S S   

. ) ( O(x) x O 

Consider excited states defined by local operators: We study

Two limits

(1) In this case, we find a property analogous to the first law of thermodynamics:

[Bhattacharya-Nozaki-Ugajin-TT 12, Blanco-Casini-Hung-Myers 13, Wong-Klich-Pando Zayas-Vaman 13 …, Raamsdonk’s talk]

(2) This leads to a very `entropic’ quantity ! ⇒ The main purpose of this talk.

[Nozaki-Numasawa-TT 14, He-Numasawa-Watanabe-TT 14, Caputa-Nozaki-TT 14]

 

A n A

E O S   

) (

A B

Contents

① Introduction ② Replica Calculations of EE for locally excited states ③ Case 1: Free scalar CFTs in any dimensions ④ Case 2: Rational 2d CFTs ⑤ Case 3: Large N CFTs and AdS/CFT ⑥ Conclusions

(2-1) Replica method for ground states A basic method to find EE in QFTs is the replica method. In the path-integral formalism, the ground state wave function can be expressed as follows:

  

x

t

  ,

t 

  

integrate Path

  t

② Replica Calculations of EE for locally excited states

Then we can express as follows:

  

B A

Tr 



ab A]

[

A b

t 

x

  

t

  

n A

 Tr

 

a a b

b

ly. successive boundaries each Glue

. ) ( ) (

1 n n

Z Z   

sheets n

cut

n

surface Riemann sheeted

• 

n

(2-2) Replica Method for Excited States We want to calculate for

• perator.

for the regulator UV the is where ), ( ), , ( ) , ( ) ( ) ( ) , (    

 

it ε τ it, ε τ x O x O e e x O x O e e x t

l e l e iHt H H iHt A

       

  

. set We R ) , , , , ( . R

• n

CFT dim. 1 a consider we Here

1 1 2 1 1 

 

i d d d

re i x x x x d     

 



In this way, the Renyi EE can be expressed in terms of correlation functions (2n-point function etc.) on Σn : . ) , ( ) , ( log ) , ( ) , ( ) , ( ) , ( log 1 1

1

1 1 ) (

           

  e e l l e e l l n e e n l l n A

r O r O n r O r O r O r O n S

n

      

Σn

n-sheets

We focus on the free massless scalar field theory on Σn and calculate 2n-pt functions using the Green function:

 



  



 

 

x d S

d 1

 

. | | 1 where , / ) ( cos 2 ) / 1 ( 4 1 )] , , ( ); , , [(

2 2 2 2 / 1 / 1 / 1 / 1 2 3

s r y x rs a a n a a a a a a rs n y s x r G

n n n n d n

           

   

        

. as chosen is

• perator

The : : O O

k k

 

③ Case 1: Free scalar CFTs in any dimensions

[Numasawa-Nozaki-TT 14]

Time evolution in free massless scalar theory

) 1 (i.e. : : for

) 2 (

   k O SA 

              . and 10 with chose We

2 1 d

x x l l x 

2 dim. 4 dim. 6 dim.

l

t

Note： is `topologically invariant’ under deformations of A.

f n A

S

) (



. 2 log . .

2 2 2 ) 2 ( dim) 4 (

           l t t S g E

A

Interested quantities !

:) : (

 i

e O 

f n A

S

) (



dim. 2 1 d in for

) (

   

k f n A

O S 

Renyi Entropy EE

EPR state !

[For a proof: Nozaki 14]

Heuristic Explanation First , notice that in free CFTs, there are definite (quasi) particles moving at the speed of light.

. 2 vac ) ( ) ( vac

2 / R L k j j k k j k R k j j L j k k

j k j C C     

 

   

  

.

moving

• right

moving

• left

R L

     

L=A R=B

 

]. [ log 2 2 log , ) ( 2 log 1 1

) ( j k k j j k k f A k j n j k nk f n A

C C k S C n S

 

   

       

Agree with replica Calculations !

④ Case 2: Rational 2d CFTs

(4-1) Free Scalar CFT in 2d Consider following two operators in the free scalar CFT: (i) (ii)

: :

1  i

e O  .

) (

  

f n A

S

R i L i

R L

e e O

1  

 

R L R L R i L i R i L i

R L R L

e e e e O          

  2    

: : : :

2   i i

e e O



 

. 2 log

) (

  

f n A

S

⇒Direct product state ⇒ EPR state

[He-Numasawa-Watanabe-TT 14]

(4-2) General Results for 2d Rational CFTs First, focus on n=2 REE and assume O = a primary op. We can employ the following conformal map: It is straightforward to rewrite the n=2 REE in terms of 4-pt functions on .

.

 i

re w z  

1 2

   C

1 



). , ( | | ) , ( ) , ( ) , ( ) , (

4 24 13 4 4 3 3 2 2 1 1

2

z z G z z w w O w w O w w O w w O

O

O 



  

           . ,

24 13 34 12 j i ij

z z z z z z z z

. , . ) ( , ) ( , ) ( , ) (

2 2 4 1 2 3 2 2 1 1

w e w w e w l it i w l it i w l it i w l it i w

i i  

                   

We can show that the limit leads to (i) Early time: (ii) Late time: Note: It is straightforward to confirm at early time (i).

 

l t  

). , ( )) ( ), ( ( ) , (

2 2

    O O z z

l t 

). , 1 ( )) ( ), ( 1 ( ) , (

2 2

     O O z z

Chiral Fusion Transformation z→1-z

) (

 

n A

S

O(x)

l 

x

Subsystem A

In terms of conformal block, we find at late time: where is so called the fusion matrix, defined by

, ) 1 ( ] [ ) | ( ) | ( ) | ( ) | ( ) , (

2 2 , ) , 1 ( ) , (

O O z

z O F z I F z I F z p F z p F C z z G

I I O O z z O p O p OO O     

         ] [

, O

F

q p

). | ( ] [ ) 1 | (

,

z q F O F z p F

O q q p O

  



p



O O O O

p

O O O O

q

O O O O

p

Then the n=2 REE is simply expressed at late time: In rational 2d CFTs, we can rewrite this in term of the quantum dimension as follows: Actually, more generally we can prove for any n.

]. [ log

, ) 2 (

O F S

I I f A

   , ] [ 1

, , ,

O F S S d

I I I I O I O

 

[Moore-Seiberg 89]

. log

) 2 ( O f A

d S   log

) ( O f n A

d S  

O

d

Example: Ising model

3 conformal blocks:

]. [ ], [ ], [   I

. 1 ]. [ ] [ ] [ ] [ ] [ ] [       



d d I ε ε , I I I

I

. 2 ] [ 2 ] [ 2 ]) [ ] ([ ] [ ]. [ ] [ ] [ ] [

particles 2 1 1 2

         

  

     d I I ε I

N

N N N N

      

. 2 log ] [ , ] [ ] [ : find we Thus     

(n) A (n) A (n) A

ΔS ΔS I ΔS

⑤ Case 3: Large N CFTs and AdS/CFT

(5-1) Free U(N) Yang-Mills at large N

 

. 2 2 log 1 1 : result exact the find we , 2 when example, For

) 1 ( 2 2 1 ) ( n n n f n A

N n S J

  

     

]. ) ( Tr[ ) ( choose We

J

x x O  

scalar) matrix Hermitian ( N N   

can be neglected only if n>1

). ( log 2 1 if ). ( 2 log 1 1 1 if find we general, In

2 ) 1 ( 2 ) (  

            N O N J S n N O n Jn S n

f A f n A

Enhance at n=1 ~deconfinement ?

[Caputa-Nozaki-TT 14]

Actually, the behavior is easy to explain.

• cf. Log [N] behavior for a heavy quark

[Lewkowycz-Maldacena 13]

N J S

f A

log

) 1 (

 

 

            

 

R a a R a a R L a a L a a L J R

J J J J

) ( ) ( ) ( ) ( ) ( Tr

1 2 1 1 2 1

L

EE ~ J・Log[N2]

(5-2) Holographic Results from AdS/CFT

CFTs dim in

) 2 (

d S n

A 



⇒ Holographic 2n-point functions in (d+1) dim. topological AdS BH

. log ) 1 ( 4

) (

           t n d n S

O n A

This calculation is based on naïve large N limit. Thus the n=1 limit and the late time limit t=∞ are not trustable. For n=1 (EE), we can employ the HEE formula to find directly. [Nozaki-Numasawa-TT 13]

) 1 ( A

S  . log 6 ), CFT (AdS CFT 2d For

) 1 ( 2 3

         t c S /

A

⑥ Conclusions

In the large limit of A, the (Renyi) EEs for a locally excited state describe the `degrees of freedom’ of a given local operator.

• Monotonic time evolution describes entangled pair propagation.
• The final values can be explained by entanglement
• f finite number of states such as EPR states.
• They are topological invariant against deformations of A.
• In 2d rational CFTs, is given by the log of quantum

dimension.

[cf. Topological EE: Kitaev-Preskill, Levin-Wen 05]

• In large N CFTs, 1/N subleading terms get important at n=1.

The von-Neumann EE sees N2 degrees of freedom, while REE not.

• In strongly coupled large N CFTs, we find a logarithmic time evolution.

(Does it approach to finite value or not ? –future problem.)

f n A

S

) (



f n A

S

) (



) (n A

S 

One lesson: The Renyi EE and von-Neumann EE behave differently ! In QFTs, the Renyi EE (REE) is easier to compute. In Gravity, the von-Neumann EE (EE) is simpler. ⇒ Why ??

~High temp. ~Low temp.