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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?