Holography, Quantum Entanglement and Higher Spin Gravity@YITP ,Feb 6 ,2017

Relative Entropy and Conformal Interfaces

Tokiro NumasawaYukawa Institute for Theoretical Physics,

Kyoto University

Based on arXiv1702.*****Collaboration with

S.Ryu(Chicago),T.Ugajin(KITP) and X.Wen(UIUC)

→Osaka Univ (From April)→McGill Univ(From Septemer)

Relative Entropy

, :density matrices� ⇢

Relative Entropy

・Distinguishability or “Distance” between quantum states

・Related to Holography

・ S(⇢||�) = 0 $ ⇢ = �

[Dong-Harlow-Wall, 15][Jafferis-Lewcowycz-Maldacena-Suh, 15]

・used to the entropic proof of g-theorem[Casini-Landea-Torroba, 16]

S(⇢||�) ⌘ Tr⇢ log ⇢� Tr⇢ log �

[Blanco-Casini-Hung-Myers, 13]

[cf: Ugajin’s talk]

Renyi Version of Relative Entropy

We choose

S(n)(⇢||�) = 1

1� n

"log

Tr(⇢�n�1)

Tr⇢(Tr�)n�1� log

Tr(⇢n)

(Tr⇢)n

#

S(n)(⇢||�) = 1

1� n

⇥log Tr(⇢�n�1

)� log Tr(⇢n)⇤

For normalized and ,⇢ �

Conformal Interface

L(1)n � L̃(1)

�n = L(2)n � L̃(2)

�n

Conformal invariance

Interface

CFT1 CFT2⇒ continuity cond.

at the interface

Folding trick[Affleck-Oshikawa, 96]

[Bachas- de Boer-Dijkgraaf-Ooguri, 02]

Interface boundary

CFT1 CFT2

in CFT1 CFT2⌦

⌦CFT1

CFT2

2) If and

1) If and

Condition for “Interface state” in CFT1 CFT2:

⇒ |Bi = |B1i ⌦ |B2i

=

⇒ Interface is topological

=

Special cases

⌦

(perfectly reflective)

(perfectly transmissive)

L(1)n + L(2)

n � L̃(1)�n � L̃(2)

�n |Bi = 0

L(1)n � L̃(1)

�n |Bi = 0 L(2)n � L̃(2)

�n |Bi = 0

L(1)n � L̃(2)

�n |Bi = 0 L(2)n � L̃(1)

�n |Bi = 0

Interface on the center

Interface

Branch Cut

[Gutperle-Miller, 16]

[Brehm-Brunner-Jaud and Schidt-Colinet, 16]

Boudnary

SA =

c

3

log

l

✏+ log gB

l0

[Calabrese-Cardy, 04]

Z( ) Z( )Z( )

Z( )=

vacuum ptexpectation value

of iterface

Interface

Branch Cut

l0

w =z + l/2

z � l/2

exp(2⇡w/n)

= hIi

I : interface operator

after folding

= h0|Bi = gB

※ If the interface is not centered,

or, by unfolding

Relative Entropy of Interfaces

Interface a Interface b

→We need to connect two interfaces

Interface b

Tr(⇢a⇢n�1b )

…=

[TN-Ryu-Ugajin-Wen, Work in Progress]

=Change of interface at some points

iab

introduce an “Interface Changing Operator”

a b�iwith dim.

After the folding,

iab

a b �iwith dim.

⇒ : Boundary Changing Operator iab

To avoid a divergence at , we introduce a cutoff n ! 1

a

Branch Cut

a

a

b

iab

iba

to ⇢aS(n)

(⇢a||⇢b) =1

1� n

"log

Tr(⇢a⇢n�1b )

Tr⇢a(Tr⇢b)n�1� log

Tr(⇢na)

(Tr⇢a)n

#

p

|0i h0| ! e�p2H i

ab |0i h0| ibae

� p2H

Z( ) =vacuum pt

exp. value of iterface

Z( ) · Z( )

Z( )· Z( )

Z( )exp. value

of ICO

= Zground

n

· hIa

i · h i

ab

i

ba

i

Evaluation of )Tr(⇢a⇢n�1b )

Interface b Interface b

…

a

a

b

iab

iba

+ folding

two point funcof Boundary ops

p

z =

w + l/2

w � l/2

! 1n

Correlation function of BCO

aa b

iab(x)

iba(y)

=A

iab

|x� y|�i

✓1

✓2

h iab(x)

iba(y)iUHP

h iab(x)

iba(y)iD

=Ai

ab

| sin(✓1 � ✓2)|�i

tan�(p)

2=

p

l

1 sheet ( )

�(p)

n sheet ( )

Tr⇢a

Tr(⇢a⇢n�1b )

✓+(n)

✓�(n)

✓+(n) =2⇡

n� �(p)

n

✓�(n) =�(p)

n

Using these ingredients,

log Tr⇢a

⇢n�1b

= logZground

n

+ log hIa

i+ log h i

ab

i

ba

i

= logZground

n

+ log ga

+ log

⇣4l

n(p2 + l2)

⌘2�i Ai

ab

| sin( 2⇡n

� �(p)n

)|2�i

log

Tr(⇢a⇢n�1b )

Tr⇢a(Tr⇢b)n�1

= log

Zground

n

(Zground

1 )

n

+ (1� n) log gb

� 2�

i

n� 1

log

n| sin( 2⇡n

� �(p)n

)|sin 2�(p)

S(n)(⇢||�) = 1

1� n

"log

Tr(⇢�n�1)

Tr⇢(Tr�)n�1� log

Tr(⇢n)

(Tr⇢)n

#

( )

log

Tr⇢na(Tr⇢a)n

= log

Zground

n

(Zground

1 )

n

+ log

hIa

ihI

a

in+ log

h i

ab

i

ba

· · · i

ab

i

ba

ih i

ab

i

ba

in

cancel at p ! 0

= log

Zground

n

(Zground

1 )

n

+ (1� n) log ga

does not depend on ICO

S(n)(⇢a||⇢b) = log gb � log ga +

2�i

n� 1

log

n| sin( 2⇡n � �(p)n )|

sin 2�(p)

Relative Renyi Entropy is

Relative Entropy is

S(⇢a||⇢b) = log gb � log ga +⇡l

p�i

S(⇢a||⇢b) = log gb � log ga +⇡l

p�i

S(⇢b||⇢a) = log ga � log gb +⇡l

p�i

Both of them are positive

⇒ If ,then ga 6= gb �i 6= 0

We cannot connect two interface without introducing

ICO with (even if interfaces are topological) �i 6= 0

Another placementInterface

Branch Cut

l0We can also consider an interface on the entangling surface

・Entanglement Entropy

(1)Topological case

SA =

c

3

log

l

✏� 2

X

i

|Sai|2 logSai

S0i

[Gutperle-Miller, 16][Brehm-Brunner-Jaud and Schidt-Colinet, 16]

(2)Conformal interface in Free boson [Sakai-Satoh, 08]

s : control permeability

�(s) : monotonic, , �(0) = 0

SA =

�(|s|)3

log

l

✏� log k1k2

�(1) = 1

・Relative Entropy [TN-Ryu-Ugajin-Wen, Work in Progress]

(1)Topological case

(2)Conformal Interface in free bosonFor same transmission coeff

Interface a Interface b

R1 R3

k1 k3k2

R2

S(⇢a||⇢b) =X

i

|Sai|2 log|Sai|2

|Sbi|2

R1

k1

general case, the computation is difficult except for n=2

S(n)(⇢a||⇢b) = log

k2k3

� 1

1� nlog

k1gcd(k1, k2)

k2R2 = k3R3

Conclusion /Future Works

・We computed relative entropy between interfaces located on

the center.

・Positivity of relative entropy leads that the dimension of Interface changing operator should not be 0

・If we computed relative entropy when we put interface on the entangling surface.

・Holographic dual ?

・Can we compute for non-centered case?