Scrambling of locally perturbed thermal states Joan Sim´ on University of Edinburgh and Maxwell Institute of Mathematical Sciences Topics in 3D Gravity ICTP, Trieste, March 22nd 2016 Based on arXiv:1503.08161 with P. Caputa, A. ˇ Stikonas, T. Takayanagi & K. Watanabe Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 1 / 39
Outline Motivating & quantifying scrambling 1 2d CFT discussion 2 ◮ Set-up ◮ Large c 2d CFTs at finite T & thermo field double Brief holographic discussion 3 Quantum chaos & butterfly effect vs scrambling 4 Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 2 / 39
Motivation Question If we input some localised information in a quantum system, such as a perturbation, does it remain localised or does it spread over the entire system ? ∃ delocalisation ∼ scrambling Measures of scrambling Any arbitrary subsystem up to half of the state’s dof is nearly 1 maximally entangled : Page scrambling If ∃ scrambling ⇒ information about input can not be deduced by 2 local output measurements ⇒ mutual information may quantify this Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 3 / 39
Quantifying scrambling Any unitary operator 2 n − 1 � u ij | i �� j | U ( t ) = i , j =0 can be mapped into a 2n-qubit state treating input and output legs equally 2 n − 1 1 � | U ( t ) � = u ij | i � in ⊗ | i � out 2 n / 2 i , j =0 This is an example of the channel-state duality in QI. Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 4 / 39
Quantifying scrambling Given some local disturbance in A , ∃ scrambling ⇒ not measurable in C ⇓ I ( A : C ) = S A + S C − S A ∪ C small Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 5 / 39
Quantifying scrambling Same argument and conclusion for local region D ⇒ I ( A : D ) small Amount of information that is non-locally hidden in CD by computing I ( A : CD ) − I ( A : C ) − I ( A : D ) In QI, this is captured by the tripartite information I 3 ( A : C : D ) = I ( A : C ) + I ( A : D ) − I ( A : CD ) being very negative (Hosur, Qi, Roberts & Yoshida) Today, we will focus on I ( A : C ) ≈ 0 Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 6 / 39
BHs : scrambling vs quantum cloning BH physics suggest speed at which thermality is regained is faster than in diffusive systems (scrambling) (Susskind-Sekino) scrambling time 1 from causality bound preventing quantum cloning τ scrambling ∼ β log S Faster than 2 diffusion τ diff ∼ S 2 / d ≫ log S Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 7 / 39
Perturbing eternal BH (Shenker & Stanford) Perturbation turned on at time t 1 on the left boundary Backreaction can be non-trivial, no matter how light the perturbation is, depending on the t 1 scale Shock-wave description M+E M+E t 1 M M α M+E t 1 Small perturbations get blue shifted near horizon (Shenker-Stanford) t ⋆ ∼ β log m p β Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 8 / 39
CFT calculation Consider a 2d CFT at finite temperature ρ β Perturb the thermal state by a local primary operator O w ( x 0 , 0) ρ β O † w ( x 0 , 0) Evolve the system unitarily e − iHt � � O w ( x 0 , 0) ρ β O † e iHt w ( x 0 , 0) Question Time scale t ⋆ at which I ( A : B )( t ⋆ ) = 0 Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 9 / 39
CFT : overall strategy Non-compact 2d CFT ⇒ no Poincar´ e recurrences Logic : Describe density operator and its regularisation 1 (on top of the UV cut-off) Use replica trick to compute entanglement entropy ⇒ correlators in 2 2d CFT Use large c limit to compute these correlators analytically 3 Solve for the scrambling time scale t ⋆ 4 Remark : this main set-up was also considered in Roberts & Stanford Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 10 / 39
CFT set-up Consider a perturbation, generated at time t = 0 at x = − ℓ , created by a primary operator O acting on the vacuum of the 2d CFT : √ N e − iHt e − ǫ H O (0 , − ℓ ) | 0 � | Ψ O ( t ) � = O is inserted at t = 0 and x = − ℓ and dynamically evolved afterwards ǫ is a small parameter smearing the UV behaviour of the local operator (separation in euclidean time) Density matrix : ρ ( t ) = N e − iHt e − ǫ H O (0 , − ℓ ) | 0 �� 0 |O † (0 , − ℓ ) e iHt e − ǫ H ω 2 ) | 0 �� 0 |O † ( ω 1 , ¯ = N O ( ω 2 , ¯ ω 1 ) where ω 1 = − ℓ + i ( ǫ − it ), ω 2 = − ℓ − i ( ǫ + it ) (¯ ω 1 = − ℓ − i ( ǫ − it )) Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 11 / 39
Entanglement entropy : replica trick Method 1 : uniformization Tr ρ n ω 1 ) O † ( ω 2 , ¯ ω 2 ) ... O † ( ω 2 n , ¯ A ∼ �O ( ω 1 , ¯ ω 2 n ) � Σ n ◮ Σ n Riemann surface ◮ ω 2 k +1 = e 2 π ik ω 1 ◮ ω 2 k +2 = e 2 π ik ω 2 Method 2 : Twist operators Tr ρ n A ∼ � ψ | σ ( ω 1 , ¯ ω 1 )˜ σ ( ω 2 , ¯ ω 2 ) | ψ � ◮ Calculation done in n-copies of the original CFT ◮ Twist operators emerge because of the existence of some internal symmetry when swapping these copies Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 12 / 39
Perturbations at finite temperature Same set-up as before, but now we perturb a thermal state at t = − t ω : 1 ω 2 ) e − β H O † ( ω 1 , ¯ ρ ( t ) ≡ NO ( ω 2 , ¯ ω 1 ) with ω 1 = x 0 − t − t ω − i ǫ ω 1 = x 0 + t + t ω + i ǫ ¯ ω 2 = x 0 + t + t ω − i ǫ ω 2 = x 0 − t − t ω + i ǫ . ¯ A pair of operators will be inserted on a cylinder, separated 2 i ǫ 2 Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 13 / 39
Thermofield double set-up Consider two non-interacting 2d CFTs, say CFT L and CFT R , with isomorphic Hilbert spaces H L , R Thermofield double state : 1 e − β 2 E n | n � L | n � R � | Ψ β � = � Z ( β ) n | n � L is an eigenstate of the hamiltonian H L acting on H L with eigenvalue E n (and similarly for | n � R ). | n � L is the CPT conjugate of the state | n � R Notation : | n � L ⊗ | n � R as | n � L | n � R . Thermal reduced density 1 e − β E n | n � R � n | R , � ρ R ( β ) = tr H L ( | Ψ β � � Ψ β | ) = Z ( β ) n ∈H R Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 14 / 39
Thermofield double : observables Single sided correlators are thermal � Ψ β | O R ( x 1 , t 1 ) . . . O R ( x n , t n ) | Ψ β � = tr H R ( ρ R ( β ) O R ( x 1 , t 1 ) . . . O R ( x n , t n )) . Two sided correlators : by analytic continuation � Ψ β | O L ( x 1 , − t ) . . . O R ( x ′ n , t ′ n ) | Ψ β � = ρ R ( β ) O R ( x 1 , t − i β/ 2) . . . O R ( x ′ n , t ′ � � tr H R n ) . Will use this observation when computing Renyi entropies Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 15 / 39
CFT considerations As discussed by Morrison & Roberts (see also Hartman & Maldacena) : single sided thermal correlation functions are computed on a single cylinder with periodicity τ ∼ τ + β two-sided correlators involve a path integral over a cylinder with the same periodicity τ ∼ τ + β , where all operators O R are inserted at τ = i β/ 2, whereas O L are inserted at τ = 0 Set-up : Consider thermofield double state two finite intervals: A = [ y , y + L ] in the left CFT L and B = [ y , y + L ] in the right CFT R perturb the TFD by the insertion of a local primary operator O L acting on CFT L at x = 0 , t − = − t ω Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 16 / 39
Calculation of S A 1 n − 1 log (Tr ρ n S A = − lim A ( t )) n → 1 where x 3 ) ψ † ( x 4 , ¯ A ( t ) = � ψ ( x 1 , ¯ x 1 ) σ ( x 2 , ¯ x 2 )˜ σ ( x 3 , ¯ x 4 ) � C n Tr ρ n x 4 ) � C 1 ) n ( � ψ ( x , ¯ x 1 ) ψ † ( x 4 , ¯ with the insertion points x 1 = − i ǫ , x 2 = y − t ω − t − , x 3 = y + L − t ω − t − , x 4 = + i ǫ x 4 = − i ǫ ¯ x 1 = + i ǫ , ¯ x 2 = y + t ω + t − , x 3 = y + L + t ω + t − , ¯ ¯ with conformal dimensions � n − 1 � H σ = c H ψ = nh ψ , 24 n Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 17 / 39
Conformal maps From the cylinder to the plane 1 ω ( x ) = e 2 π x /β Standard map : ω 1 → 0, ω 2 → z , ω 3 → 1 and ω 4 → ∞ 2 z ( ω ) = ( ω 1 − ω ) ω 34 ω 13 ( ω − ω 4 ) where the cross-ratio satisfies z = ω 12 ω 34 ω 13 ω 24 Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 18 / 39
Result = c ( n + 1) � β sinh π L � S ( n ) log A 6 πǫ UV β 1 � � | 1 − z | 4 H σ G ( z , ¯ + n − 1 log z ) where G ( z , ¯ z ) = � ψ | σ ( z , ¯ z )˜ σ (1 , 1) | ψ � Using the large c results derived by Fitzpatrick, Kaplan & Walters in the limit n → 1 � 1 1 � 2 (1 − α ψ ) ¯ 2 (1 − ¯ α ψ ) (1 − z α ψ )(1 − ¯ z ¯ α ψ ) ∆ S A = c z z 6 log α ψ ¯ α ψ (1 − z )(1 − ¯ z ) � 1 − 24 h ψ where α ψ = . c Sim´ on (Edinburgh) Scrambling, Entanglement & CFT ICTP 19 / 39
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