Quantum Entanglement and Local Excitations Pawe Caputa HMSCS, - PowerPoint PPT Presentation

Quantum Entanglement and Local Excitations Pawe ł Caputa HMSCS, GGI, 12/03/2015

Based on : “Entanglement of local operators in large-N conformal field theories” • with Masahiro Nozaki, Tadashi Takayanagi PTEP 2014 (2014) 9, 093B06 “Quantum Entanglement of Localised Excited States at Finite Temperature” • with Joan Simon, Andrius Stikonas, Tadashi Takayanagi JHEP 1501 (2015) 102 “To appear…” • with Joan Simon, Andrius Stikonas, Tadashi Takayanagi, Kento Watanabe

Entanglement Renyi Entropies B A ρ = | ψ i h ψ | ρ A = Tr B ρ Renyi Entropies 1 S ( n ) 1 − n ln Tr ( ρ n = A ) A von-Neumann S (1) = − Tr ( ρ A ln ρ A ) A

� � � � � � � � � � � � � � � � � � � � � � � � � � � � Entanglement Entropy in AdS/CFT � � � � � � � � � � � � � � � � � � � � [Ryu,Takayanagi’06] � � � � CFT � � (We omit the time direction. ) d 1 S A = Area( γ d A ) � � � � � � � � � 4 G d +2 � � � � � A N � � � � [Hubeny,Rangamani,Takayanagi’07] AdS � z B d 2 � � � � Covariant � � � � � � � � � � z Disconnected regions (Mutual Information) � � � � � � I A : B = S A + S B − S A ∪ B

Question: CFT in 1+1d [see Cardy,Calabrese…] 3 log | A | S A ∼ c A ε o h A ρ ( t ) = e − iHt O ( x ) | 0 i h 0 | O † ( x ) e iHt S A ( t ) ?

Motivation (CFT): Characterise operators from the perspective of quantum entanglement Motivation (AdS/CFT): This Talk: Modest step towards this…

Plan • Entanglement and locally exited states • Large c limit and AdS/CFT • Finite temperature • Mutual information • Scrambling time

Entanglement and locally exited states ρ ( t, x ) = N e − iHt e − ✏ H O ( x ) | 0 i h 0 | O ( x ) e − ✏ H e iHt Tr ( ρ n A ) √ ! Tr( Ω n w 1 ) O † ( w 2 , ¯ ∑ h O ( w 1 , ¯ ∏ 1 A ) 1 w 2 ) ...O ( w 2 n , ¯ w 2 n ) i Σ n ∆ S ( n ) = 1 ° n log = 1 ° n log w 2 ) i Σ 1 ) n A Tr( Ω (0) ( h O ( w 1 , ¯ w 1 ) O † ( w 2 , ¯ A ) n [Sierra et al.,’12] [Nozaki,Numasawa,Takayanagi,’13]

Rational CFT [He,Numasawa,Takayanagi,Watanabe’14] (n=2) h O ( w 1 , ¯ w 1 ) O ( w 2 , ¯ w 2 ) O ( w 3 , ¯ w 3 ) O ( w 4 , ¯ w 4 ) i Σ 2 = | z | 2 ∆ O | 1 � z | 2 ∆ 0 G O ( z, ¯ z ) w 2 ) i Σ 1 ) 2 ( h O ( w 1 , ¯ w 1 ) O ( w 2 , ¯ � � � � � � � � � � � � � � � � � � � � � At late time ( z, ¯ z ) → (1 , 0) � ed tities ! In rational CFTs z ) ' F 00 [ O ] · (1 � z ) − 2 ∆ O ¯ z − 2 ∆ O , G ( z, ¯ � � � � � � � � � � � ： � S A � 2 � � `topologically invariant’ ∆ S (2) A = � log F 00 [ O ] = log d O , 1.0 0.8 0.6 d ∆ = S 0 ∆ quantum dimension 0.4 S 00 0.2 “EPR pair propagating through the system” 3.0 t 0.0 0.5 1.0 1.5 2.0 2.5

Large c [PC,M.Nozaki,T.Takayanagi14] Conformal block expansion OO † ) 2 F O ( b | z ) ¯ X ( C b G ( z, ¯ z ) = F O ( b | ¯ z ) b at large central charge c [Fateev,Ribault’11] F O ( b | z ) ' z ∆ b − 2 ∆ O · 2 F 1 ( ∆ b , ∆ b , 2 ∆ b , z ) H 2 L D S A at late time ' 4 ∆ O · log 2 t ∆ S (2) A ✏ similar to a local quench t t l 10

Energy density ⟨ T tt ⟩ = ⟨ O † ( w 2 , ¯ w 2 ) T tt ( x, x ) O ( w 1 , ¯ w 1 ) ⟩ � 1 1 � = ∆ O � 2 (( x + l − t ) 2 + � 2 ) 2 + (( x + l + t ) 2 + � 2 ) 2 ⟨ O † ( w 2 , ¯ w 2 ) O ( w 1 , ¯ w 1 ) ⟩ (39 N w 1 = i ( � − it ) − l, w 2 = − i ( � + it ) − l, w 1 = − i ( � − it ) − l, ¯ w 2 = i ( � + it ) − l. ¯ E ∼ ∆ O ✏

Falling particle of mass m in AdS [Nozaki,Numasawa,Takayanagi’13] [PC,Nozaki,Takayanagi’14] CFT (chain) L In our setup: ↵ ≡ ✏  sin ⇡ a �  t �  sin ⇡ a � t ( L − t ) ∆ S (1) ∼ c → ∆ S (1) ∼ c + c 6 log 6 log 6 log a ✏ L a ✏ r 1 − µ a = R 2

Twist operators [Bernamonti et al.’14] ρ ( t ) = Ne − iHt O ( x 4 , ¯ x 1 ) e iHt x 4 ) | 0 i h 0 | O ( x 1 , ¯ A = h O ( x 1 , ¯ x 1 ) σ ( x 2 , ¯ x 2 )˜ σ ( x 3 , ¯ x 3 ) O ( x 4 , ¯ x 4 ) i CF T n /Z n Tr ρ n x 4 ) i n h O ( x 1 , ¯ x 1 ) O ( x 4 , ¯ x 2 = l 1 − t, x 3 = l 2 − t x 1 = i ✏ , x 4 = − i ✏ ¯ ¯ x 2 = l 1 + t, x 3 = l 2 + t x 1 = − i ✏ , x 4 = i ✏ ¯ ¯ Tr ρ n A = | x 23 | − 4 ∆ n | 1 − z | 4 ∆ n G n ( z, ¯ z )

Large c limit of conformal blocks [Zamolodchikov….] z ) ∼ e f ( z, ¯ z ) G ( z, ¯ c → ∞ Two-heavy and two light operators [Fitzpatrick et al.’14] h/c → 0 ∆ O /c − fixed ! − 2 h 1 − α 1 − α r 2 (1 � z α )¯ 2 (1 � ¯ z α ) 1 − 24 ∆ O z z G ( z, ¯ z ) ' α = α 2 c Using this we can compute  sin ⇡↵ � t ( L − t ) ∆ S (1) ∼ c 6 log t < L ✏ L ↵

Back-reaction from a point particle in AdS [Horowitz,Itzhaki’99] R 2 dr 2 � � r 2 + R 2 − M ds 2 = − d τ 2 + R 2 + r 2 − M/r d − 2 + r 2 d Ω 2 d − 1 . r d − 2 m = ( d − 1) π d/ 2 − 1 M G N R 2 . · 8 Γ ( d/ 2) In order to find a back-reaction from a particle in AdS we “just” have to find the map to the r=0 solution in global AdS and insert to the above metric

Details: dz 2 − dt 2 + � d − 1 � � i =1 dx 2 ds 2 = R 2 i z 2 � z ( t ) 2 1 − ˙ � ( t − t 0 ) 2 + α 2 , � S = − mR dt . z ( t ) = z ( t )

Map: R 2 + r 2 cos τ = R 2 e β + e − β ( z 2 + x 2 − t 2 ) � , 2 z R 2 + r 2 sin τ = Rt � ↵ = ✏ = Re β z , r Ω i = Rx i ( i = 1 , 2 , · · · , d − 1) , z r Ω d = − R 2 e β + e − β ( z 2 + x 2 − t 2 ) . 2 z Back reacted metric after inserting: r = 1 � R 4 e 2 β + e − 2 β ( z 2 + x 2 i − t 2 ) 2 − 2 R 2 ( z 2 − x 2 − t 2 ) , 2 z d d τ 2 = d (cos τ ) 2 + d (sin τ ) 2 , � d Ω 2 ( d Ω i ) 2 . d − 1 = i =1 we can check that we get the appropriate energy density

Entanglement Entropy (d=2) √ √ ✓ ◆ ✓ ◆ 2 3 R 2 − µ R 2 − µ 2 cos | ∆ ˜ ⌧ ∞ | − 2 cos | ∆ � ∞ | R R S A = c 6 r (1) ∞ · r (2) 7 � � 4 log + log 6 7 R 2 − µ 6 ∞ 5 where 2 Rt tan τ ( i ) ∞ = R 2 e β + e − β (( l ( i ) ) 2 − t 2 ) , 2 Rl ( i ) tan θ ( i ) ∞ = − e − β (( l ( i ) ) 2 − t 2 ) − R 2 e β , � ∞ = 1 R 2 ( l ( i ) ) 2 + 1 e − β (( l ( i ) ) 2 − t 2 ) − R 2 e β � 2 . r ( i ) � 4 z ∞  sin ⇡ a �  t �  sin ⇡ a � t ( L − t ) ∆ S (1) ∼ c → ∆ S (1) ∼ c + c 6 log 6 log 6 log a ✏ L a ✏

Finite Temperature I A : B = S A + S B − S A ∪ B

Eternal BH-TFD duality [Maldacena’01] t − H L H R Eternal BH t + 1 2 E n | n i L | n i R e � β X | Ψ β i = TFD p Z ( β ) n

[Maldacena Hartman] Evolution of EE in TFD [Morrison,Roberts] B t < L/ 2 S A ∪ B ' t A t > L/ 2 S A ∪ B ' 2 S th I A : B = S A + S B − S A ∪ B I A : B H L H R H L - H R H L + H R t L / 2

[P.C,Simon,Stikonas,Takayanagi’14] Operator Insertion to TFD ? � � � � Eternal BH O L | ψ β > TFD

| ψ 0 i = e � iH L t w O ( x ) e iH L t w | ψ i [Shenker,Stanford] [Roberts,Stanford] I A : B ( t w ) = 0? [+ Susskind] t w ∼ β log c ∼ β log S

Point particle in BTZ [PC,Simon,Stikonas,Takayanagi’14] ds 2 = R 2 dz 2 ✓ ◆ dt 2 + � 1 � Mz 2 � (1 � Mz 2 ) + dx 2 � z 2 s Z d ⌧ z ( ⌧ ) 2 ˙ 1 � Mz ( ⌧ ) 2 � S p = � mR z ( ⌧ ) 1 � Mz ( ⌧ ) 2 T v ◆ 2 ! ✓ u ✓ 2 ⇡✏ ✓ 2 ⇡⌧ ◆◆ z ( ⌧ ) = � u 1 � tanh 2 t 1 � 1 � . X 2 ⇡ � �

Check: Entanglement Entropy gravity sinh π ( t + t w ) sinh π ( L − t − t w ) " # ∆ S A ' c � sin a β β 6 log sinh π L ⇡✏ a β CFT large c x 3 ) † ( x 4 , ¯ A ( t ) = h ( x 1 , ¯ x 1 ) � ( x 2 , ¯ x 2 )˜ � ( x 3 , ¯ x 4 ) i C n 2 π Tr ⇢ n w ( x ) = e β x x 4 ) i C 1 ) n x 1 ) † ( x 4 , ¯ ( h ( x 1 , ¯ O ≡ ψ ⇣ ⌘ ⇣ ⌘ 2 3 π ( L − t − t w ) π ( t + t w ) sinh sinh sin ⇡↵ ψ ∆ S A = c 4 � β β 6 log 5 ⇣ ⌘ ⇡✏ ↵ ψ π L sinh β

Point particle in Kruskal coordinates = R 2 � 4 dT 2 + 4 dX 2 + (1 � T 2 + X 2 ) 2 d � 2 ds 2 = R 2 � 4 dudv + ( � 1 + uv ) 2 d � 2 (1 + T 2 � X 2 ) 2 (1 + uv ) 2 1 � Mz 2 = (1 � M ✏ 2 ) cosh � 2 ⇣ p ⌘ t � = ˜ ⌧ , ✓ = 0 , M (˜ ⌧ + t ω ) our solution in v(u) or T(X) is valid everywhere T T v ( u ) = − a 1 u − 1 u v , u + a 2 we can compute the back reaction using a map with two parameters X X X h - X h + √ λ 1 = Mt w p 1 − M ✏ 2 tanh � 2 =

Large t w T u v X X m X h - X s - X h + X s + T m

Mutual Information CFT [PC,Simon,Stikonas,Takayanagi,Watanabe] S A ∪ B B A O ≡ ψ x 6 ) † ( x 4 , ¯ A ( t ) = h ( x 1 , ¯ x 1 ) � ( x 2 , ¯ x 2 )˜ � ( x 3 , ¯ x 3 ) � ( x 5 , ¯ x 5 )˜ � ( x 6 , ¯ x 4 ) i Tr ⇢ n x 4 ) i C 1 ) n x 1 ) † ( x 4 , ¯ ( h ( x 1 , ¯

Mutual Information results [PC,Simon,Stikonas,Takayanagi,Watanabe] I A : B ( t − , t + , t w , L, a ) = I A : B ( t − , t + , t w , L, α ) I A : B ( t ∗ w ) = 0? S ! = f ( L, � ) + � t ? 2 ⇡ log ⇡ E ✓ ◆ q 1 � 24 ∆ O sin ⇡ c ' 3 � ∆ O � = ⇡ E O 4 ⇡✏ q c ✏ S 1 � 24 ∆ O c