Quantum Entanglement of locally perturbed thermal states

Quantum Entanglement of locally perturbed thermal states Joan Simón University of Edinburgh and Maxwell Institute of Mathematical Sciences Holography, strings and higher spins Swansea, March 20, 2015 Based on arxiv:1410.2287 and work in progress with P. Caputa, A. Štikonas, T. Takayanagi & K. Watanabe Simón (Edinburgh) Entanglement excited states Swansea 2015 1 / 44

Motivation Consider a critical physical system in 1+1 dimensions in some thermal state ρ Perturb the state by a local primary operator O w (x 0, 0) ρ O w (x 0, 0) Evolve the system unitarily ( ) e iht O w (x 0, 0) ρ O w (x 0, 0) Question : Is there any sense in which subsystems behave thermally after some time scale t ω? I A:B (t ω) = 0 e iht Simón (Edinburgh) Entanglement excited states Swansea 2015 2 / 44

Holographic Motivation 1 Eternal BH thermo field double (Maldacena) Recently re-interpreted in terms of EPR=ER (Maldacena-Susskind) Local Perturbation of this scenario : time evolution of the throat? Improvement in the holographic dictionary 2 BH physics suggest speed at which thermality is regained is faster than in diffusive systems (scrambling) (Susskind-Sekino) No-cloning argument & causality bounds t ω log S Small perturbations get blue shifted near horizon (Shenker-Stanford) t log m p Question : Any CFT evidence for any of these bulk effects? Simón (Edinburgh) Entanglement excited states Swansea 2015 3 / 44

Entanglement vs Correlations Question : Is there any relation between quantum entanglement and correlation lengths? Consider as measure of entanglement, mutual information I (C : D) = S(ρ C ) + S(ρ D ) S(ρ CD ) Using Pinksler s inequality, one can show (Wolf, Verstraete, Hastings, Cirac) I (C : D) ( O C O D O C O D ) 2 2 O C 2 O D 2 Simón (Edinburgh) Entanglement excited states Swansea 2015 4 / 44

Connected correlators as geodesics in AdS/CFT The connected 2-pt correlation function of a heavy operator behaves like (Balasubramanian & Ross) O C (x c )O D (x D ) O C (x c ) O D (x D ) e ml bulk(x C, x D ) L bulk (x C, x D ) is the bulk geodesic distance between the boundary points x C and x D O = ml 1 (not scaling with N or c) Holographic dual correlation only depends mildly on the dual operator (through O ) Simón (Edinburgh) Entanglement excited states Swansea 2015 5 / 44

Entanglement entropy in AdS/CFT Entanglement entropy in an strongly coupled CFT vs bulk geometry. Ryu & Takayanagi S(ρ B ) Area( B) Area(Σ bulk ) where Σ bulk is a bulk minimal surface anchored to B Non-local diffeomorphism invariant observables Deep relation between the set of minimal surfaces and Einstein s equations Simón (Edinburgh) Entanglement excited states Swansea 2015 6 / 44

Entanglement vs spacetime connectedness Consider a full quantum system described by A B Study the limit of vanishing entanglement holographically Spacetime connectedness (van Raamsdonk) Sending entanglement to zero, requires : 1 Proper bulk distance to infinity 2 Area of the common boundary to zero pinching Simón (Edinburgh) Entanglement excited states Swansea 2015 7 / 44

Consequences Quantum mechanics Consider a Hilbert space H 1 H 2 with no interactions 1 Product states have vanishing connected correlators 2 Entangled states have non-vanishing correlators!! AdS/CFT Consider 2 decoupled CFTs 1 Product states having holographic duals correspond to disconnected asymptotically AdS spacetimes Example : vac vac 2 disconnected AdS spacetimes 2 Entangled states correlations connected geometry!! Example : eternal AdS black hole Simón (Edinburgh) Entanglement excited states Swansea 2015 8 / 44

Eternal AdS BH revisited 1 Classical maximal extension of the eternal AdS BH 2 Connectedness through BH event horizon For certain observables and low energies, an observer in H R measures a thermal state : ρ BH = 1 e E i E i E i, E i H R Z() i Can we interpret ρ BH as a reduced density matrix? (Maldacena) 1 ρ BH = tr HL Ψ Ψ with Ψ = e E i /2 E i E i H L H R Z() Quantum entanglement is responsible for the existence of correlations Simón (Edinburgh) Entanglement excited states Swansea 2015 9 / 44 i

EPR = ER (Maldacena & Susskind) Eternal black hole re-interpreted 1 Non-vanishing correlators between H L and H R are due to quantum entanglement (EPR) 2 These correlations are holographically captured by the bulk geodesic distance between opposite boundaries length of the ER bridge 3 Entanglement entropy = black hole entropy maximal cross-section of the ER bridge Simón (Edinburgh) Entanglement excited states Swansea 2015 10 / 44

Outline 2d CFT set-up Free scalar 2d CFT at T = 0 (warm-up) Large c 2d CFTs at finite T & thermo field double Holographic remarks Final remarks Simón (Edinburgh) Entanglement excited states Swansea 2015 11 / 44

Set-up (single CFT) Consider an excited state in a 2d CFT Ψ O (t) = N e iht e ɛh O(0, l) 0 O is inserted at t = 0 and x = l and dynamically evolved afterwards ɛ is a small parameter smearing the UV behaviour of the local operator Density matrix : ρ(t) = N e iht e ɛh O(0, l) 0 0 O (0, l) e iht e ɛh = N O(ω 2, ω 2 ) 0 0 O (ω 1, ω 1 ) where ω 1 = l + i(ɛ it), ω 2 = l i(ɛ + it) ( ω 1 = l i(ɛ it)) Simón (Edinburgh) Entanglement excited states Swansea 2015 12 / 44

Set-up (notation) Our calculations will be done in euclidean signature : ω = x + iτ, ω = x iτ We use the euclidean continuation : τ = it The normalization factor N is fixed by Tr(ρ(t)) = 1 The cut-off ɛ can be viewed as a separation in the insertion time appearing in ρ(t) Simón (Edinburgh) Entanglement excited states Swansea 2015 13 / 44

Replica trick - I Following Cardy & Calabrese S (n) A = 1 1 n log ( Trρn A Tr ρ (0) = 1 1 n log A ) n [ O(ω1, ω 1 )O (ω 2, ω 2 )...O (ω 2n, ω 2n ) Σn ( O(ω 1, ω 1 )O (ω 2, ω 2 ) Σ1 ) n Notice no twisted operators but CFT defined on a Riemann surface ] ω 2k+1 = e 2πik ω 1 ω 2k+2 = e 2πik ω 2 Simón (Edinburgh) Entanglement excited states Swansea 2015 14 / 44

Replica trick - II Following Cardy & Calabrese Trρ n A ψ σ(ω 1, ω 1 ) σ(ω 2, ω 2 ) ψ ψ stands for whatever CFT state you want to consider (vacuum or excited state) Non-trivial topology replaced by twist operators Calculation done in n-copies of the original CFT Twist operators emerge because of the existence of some internal symmetry when swapping these copies Simón (Edinburgh) Entanglement excited states Swansea 2015 15 / 44

Free scalar 2d CFT & finite region A Compute the Renyi entropy variation for n = 2 Strategy : Map Σ 2 into Σ 1 using the conformal transformation (uniformization) ω ω L = z2 The 4-pt function determining the Renyi entropy will equal O (ω 1, ω 1 )O(ω 2, ω 2 )O (ω 3, ω 3 )O(ω 4, ω 4 ) Σ2 4 = dω i O dz 2 O (z 1, z 1 )O(z 2, z 2 )O (z 3, z 3 )O(z 4, z 4 ) Σ1 i i=1 4 = dω i O dz 2 z 13 z 24 4 O G(z, z) i i=1 where the cross-ratio z = z 12z 34 z 13 z 24, with z ij = z i z j Simón (Edinburgh) Entanglement excited states Swansea 2015 16 / 44

Free scalar 2d CFT Altogether O (ω 1, ω 1 )O(ω 2, ω 2 )O (ω 3, ω 3 )O(ω 4, ω 4 ) Σ2 ( O (ω 1, ω 1 )O(ω 2, ω 2 ) Σ1 ) 2 = z 4 O 1 z 4 O G(z, z) We will consider two different excitations with O = 1 8 When O 1 = e iφ/2, then G 1 (z, z) = 1 z 1 z When O 2 = 1 2 ( e iφ/2 + e iφ/2), G 2 (z, z) = z + 1 + 1 z 2 G 1 (z, z) Simón (Edinburgh) Entanglement excited states Swansea 2015 17 / 44

Specific details Our points (z i, z i ) equal In the limit of small ɛ we obtain l t iɛ z 1 = z 3 = l + L t iɛ, l t + iɛ z 2 = z 4 = l + L t + iɛ. (z, z) (0, 0) when 0 < t < l or t > L + l z L 2 ɛ 2 4(l t) 2 (L + l t) 2, z L 2 ɛ 2 4(l + t) 2 (L + l + t) 2. (z, z) (1, 0) when l < t < L + l z 1 L 2 ɛ 2 4(l t) 2 (L + l t) 2, z L 2 ɛ 2 4(l + t) 2 (L + l + t) 2. Simón (Edinburgh) Entanglement excited states Swansea 2015 18 / 44

Results & interpretation S (2) A (O 1) = 0 all times { S (2) 0 0 < t < l, or t > l + L A (O 2) = log 2 l < t < l + L S (2) A (O 1) = 0 because it can be viewed as a direct product state e iφ L/2 0 L e iφ R/2 0 R Since O 2 creates a maximally entangled state at x = l propagating in opposite directions 1 ( ) e iφl/2 0 L e iφr/2 0 R + e iφl/2 0 L e iφr/2 0 R 2 Causality makes both pairs to be in the complement of A for 0 < t < l and t > l + L for l < t < l + L one member of the pair lies in A. Simón (Edinburgh) Entanglement excited states Swansea 2015 19 / 44

Excitations at finite temperature Same set-up as before, but now 1 we perturb a thermal state : ρ(t) N O(ω 2, ω 2 ) e H O (ω 1, ω 1 ) with ω 1 = x 0 + t + t ω + iɛ ω 1 = x 0 t t ω iɛ ω 2 = x 0 + t + t ω iɛ ω 2 = x 0 t t ω + iɛ. 2 A pair of operators will be inserted on a cylinder, separated 2iɛ Simón (Edinburgh) Entanglement excited states Swansea 2015 20 / 44

Our calculation & notion of scrambling Consider a thermofield double set-up. Perturbed the system at t ω by a primary localised operator O Evolve unitarily Measure the amount of entanglement at t = 0 using the mutual information I (A : B; t ω ) = S A + S B S A B We can ask what the time scale t ω has to be so that the perturbation can not be distinguished from the original thermal state (scrambling time) I (A : B; t ω ) = S A + S B S A B = 0 Simón (Edinburgh) Entanglement excited states Swansea 2015 21 / 44

What our condition boils down to Hartman & Maldacena showed that in the absence of perturbation : at early times, mutual information decreases linearly at late times, i.e. t > L 2, S A B = S A + S B saturates and the mutual information vanishes. Thus, if we assume tω > L 2, our condition reduces to I (A : B; t ω) = 0 This is what was analysed by Shenker & Stanford and what we will end up discussing today. Simón (Edinburgh) Entanglement excited states Swansea 2015 22 / 44

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 Ψ = e 2 En n L n R Z() 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 ρ R () = tr HL ( Ψ Ψ ) = 1 Z() n n H R e En n R n R, Simón (Edinburgh) Entanglement excited states Swansea 2015 23 / 44

Thermofield double : observables Single sided correlators are thermal Ψ O R (x 1, t 1 )... O R (x n, t n ) Ψ = tr HR (ρ R ()O R (x 1, t 1 )... O R (x n, t n )). Two sided correlators : by analytic continuation Ψ O L (x 1, t)... O R (x n, t n) Ψ = ( tr HR ρr ()O R (x 1, t i/2)... O R (x n, t n) ). Will use this observation when computing Renyi entropies Simón (Edinburgh) Entanglement excited states Swansea 2015 24 / 44

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 + two-sided correlators involve a path integral over a cylinder with the same periodicity τ τ +, where all operators O R are inserted at τ = i/2, whereas O L are inserted at τ = 0 Set-up : Consider thermofield double state two finite intervals: A = [L 1, L 2 ] in the left CFT L and B = [L 1, L 2 ] 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 Simón (Edinburgh) Entanglement excited states Swansea 2015 25 / 44

Bulk interpretation 1 Single BH in thermal equilibrium : evolution by a boost (H R H L ) H tf = I L H R H L I R. Time propagates upwards in H R and downwards in H L. Thermofield double is (boost) invariant 2 Approximate description of the state at t = 0 of two AdS black holes (H R + H L ) H = I L H R + H L I R H R + H L. Time propagates upwards in both boundaries Simón (Edinburgh) Entanglement excited states Swansea 2015 26 / 44

Calculation of S A where 1 S A = lim n 1 n 1 log (Tr ρn A (t)) Tr ρ n A (t) = ψ(x 1, x 1 )σ(x 2, x 2 ) σ(x 3, x 3 )ψ (x 4, x 4 ) Cn ( ψ(x, x 1 )ψ (x 4, x 4 ) C1 ) n with the insertion points x 1 = iɛ, x 2 = L 1 t ω t, x 3 = L 2 t ω t, x 4 = +iɛ x 1 = +iɛ, x 2 = L 1 + t ω + t, x 3 = L 2 + t ω + t, x 4 = iɛ with conformal dimensions H ψ = nh ψ, H σ = nh σ = n c 24 ( n 1 ) n Simón (Edinburgh) Entanglement excited states Swansea 2015 27 / 44

Conformal maps 1 From the cylinder to the plane ω(x) = e 2πx/ 2 Standard map : ω 1 0, ω 2 z, ω 3 1 and ω 4 where the cross-ratio satisfies z(ω) = (ω 1 ω)ω 34 ω 13 (ω ω 4 ) z = ω 12ω 34 ω 13 ω 24 Simón (Edinburgh) Entanglement excited states Swansea 2015 28 / 44

Result where S (n) A = c(n + 1) 6 ( log sinh π(l 2 L 1 ) πɛ UV 1 n 1 log ( 1 z 4Hσ G(z, z) 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 ( S A = c 1 ) z 6 log 2 (1 α ψ) z 1 2 (1 ᾱψ) (1 zψ α )(1 zᾱψ) α ψ ᾱ ψ (1 z)(1 z) where α ψ = 1 h ψ c. ) ) Simón (Edinburgh) Entanglement excited states Swansea 2015 29 / 44

Cross-ratios The cross-ratios are z = z = sinh ( πx12 sinh ( πx13 1 + 2πiɛ sinh ( π x12 sinh ( π x13 1 2πiɛ ) ) ( ) sinh πx34 ( ) sinh πx24 sinh π(l 2 L 1 ) sinh π(l 2 t t ω) sinh π(l 1 t t ω) ) ) sinh ( π x34 sinh ( π x24 ) ) sinh π(l 2 L 1 ) sinh π(l 2+t+t ω) sinh π(l 1+t+t ω) + O(ɛ 2 ) + O(ɛ 2 ) Simón (Edinburgh) Entanglement excited states Swansea 2015 30 / 44

Final result Analysing the imaginary parts, we reach the conclusions : (z, z) (1, 1) for t + t ω < L 1 and t + t ω > L 2 (z, z) (e 2πi, 1) for L 1 < t + t ω < L 2 The importance of this monodromy has been emphasized by several groups including Asplund, Bernamonti, Galli & Hartman and Roberts & Stanford S A = 0, t + t ω < L 1 and t + t ω > L 2 S A = c 6 log πɛ where L = L 2 L 1 sin πα ψ α ψ L 1 < t + t ω < L 2 sinh ( ) π(l t t ω) sinh ( πl sinh ) ( π(t +t ω) ) Simón (Edinburgh) Entanglement excited states Swansea 2015 31 / 44

Calculation of S B Very similar, but with different insertion points : Tr ρ n A (t) = ψ(x 1, x 1 )σ(x 5, x 5 ) σ(x 6, x 6 )ψ (x 4, x 4 ) Cn ( ψ(x, x 1 )ψ (x 4, x 4 ) C1 ) n with the insertion points x 1 = iɛ, x 5 = L 2 + i 2 t, x 6 = L 1 + i 2 t, x 4 = +iɛ x 1 = +iɛ, x 5 = L 2 i 2 + t, x 6 = L 1 i 2 + t, x 4 = iɛ We always obtain the expected thermal answer at all times S B = c ( 3 log sinh πl ) πɛ UV Simón (Edinburgh) Entanglement excited states Swansea 2015 32 / 44

Calculation of S A B Very similar, but with different insertion points : Tr ρ n A B (t) = ψ(x 1, x 1 )σ(x 2, x 2 ) σ(x 2 x 3 )σ(x 5, x 5 ) σ(x 6, x 6 )ψ (x 4, x 4 ) Cn ( ψ(x, x 1 )ψ (x 4, x 4 ) C1 ) n with the insertion points x 1 = iɛ, x 2 = L 1 t ω t, x 3 = L 2 t ω t, x 4 = +iɛ, x 1 = +iɛ, x 2 = L 1 + t ω + t, x 3 = L 2 + t ω + t, x 4 = iɛ, x 5 = L 2 + i 2 t +, x 6 = L 1 + i 2 t +, x 5 = L 2 i 2 + t +, x 6 = L 1 i 2 + t +. Simón (Edinburgh) Entanglement excited states Swansea 2015 33 / 44

Strategy Using conformal maps ( ) Tr ρ n A B = πl 8H σ sinh 1 z 4Hσ z 56 4Hσ πɛ UV ψ σ(z, z) σ(1, 1)σ(z 5, z 5 ) σ(z 6, z 6 ) ψ where all cross-ratios z, z i are analytically known. ψ σ(z, z) σ(1, 1)σ(z 5, z 5 ) σ(z 6, z 6 ) ψ expected 6-pt function Simón (Edinburgh) Entanglement excited states Swansea 2015 34 / 44

S-channel (I) Let us introduce a resolution of the identity ψ σ(z, z) σ(1, 1)σ(z 5, z 5 ) σ(z 6, z 6 ) ψ = α ψ σ(z, z) σ(1, 1) α α σ(z 5, z 5 ) σ(z 6, z 6 ) ψ Thus, (z, z) (1, 1) for t + t ω > L 2 use OPE!! σ(z, z) σ(1, 1) I + corrections in (z 1) r O r Orthogonality of 2-pt functions α = ψ dominant ψ σ(z, z) σ(1, 1)σ(z 5, z 5 ) σ(z 6, z 6 ) ψ ψ σ(z, z) σ(1, 1) ψ ψ σ(z 5, z 5 ) σ(z 6, z 6 ) ψ Simón (Edinburgh) Entanglement excited states Swansea 2015 35 / 44

S-channel (II) Using conformal maps ψ σ(z 5, z 5 ) σ(z 6, z 6 ) ψ = 1 z 5 4Hσ z 56 4Hσ ψ σ( z 5, z 5 ) σ(1, 1) ψ, we obtain ( ) Tr ρ n A B πl 8H σ sinh 1 z 4Hσ 1 z 5 4Hσ G(z, z)g( z 5, z 5 )+... πɛ UV Since z 5 = z 5, the cross-ratio determining S B, we derive S A B = S A + S B, and I A:B = 0 This resembles the bulk calculation from two geodesics joining pairs of points in the same boundary!! Simón (Edinburgh) Entanglement excited states Swansea 2015 36 / 44

T-channel (I) We could introduce the resolution of the identity as follows ψ σ(z, z) σ(1, 1)σ(z 5, z 5 ) σ(z 6, z 6 ) ψ = α ψ σ(z, z) σ(z 6, z 6 ) α α σ(z 5, z 5 ) σ(1, 1) ψ. (z 5, z 5 ) (1, 1) for small ɛ use OPE!! As before, α = ψ dominant contribution!! Simón (Edinburgh) Entanglement excited states Swansea 2015 37 / 44

T-channel (II) In this case, ( ) Tr ρ n A B πl 8H σ x sinh πɛ UV 1 x G( z 2, z 2 )G(z 5, z 5 ) +... 4H σ 1 z 5 4Hσ 1 z 2 4Hσ where (x, x) are the cross-ratios computed out of the insertion points of the four twist operators x = z 23z 56 z 25 z 36 = w 23w 56 w 25 w 36 = 2 sinh 2 π(l 2 L 1 ) cosh 2π(L 2 L 1 ) + cosh 2π(t +t ω t +) = x, Simón (Edinburgh) Entanglement excited states Swansea 2015 38 / 44

T-channel (III) For t + t ω > L 2, we derive S A B 2c ( 3 log π t cosh πɛ UV sinh π(t +t w ) + c 6 log cosh π t cosh πt+ ) + c ( 3 log πɛ sinh π(t +t w L) ) sin πα ψ α ψ cosh π t cosh π(l t+) where we set L 1 = 0, L 2 = L and t = t + t ω t + To derive this result we used Fitzpatrick, Kaplan & Walters Simón (Edinburgh) Entanglement excited states Swansea 2015 39 / 44

Mutual information & Scrambling time (I) In the regime t + t ω > L 2 > L 1, I A:B 2c ( 3 log sinh πl πɛ UV c ( ) 3 log sin πα ψ πɛ αψ sinh π(t +t ω) c 6 log cosh π t ) 2c 3 log πz cosh cosh πt+ sinh π(t +t ω L) cosh π t ( ) π t cosh π(l t+) take t = t + = 0 and look for t ω satisfying I A:B (t ω) = 0 Simón (Edinburgh) Entanglement excited states Swansea 2015 40 / 44

Mutual information & Scrambling time (II) Assuming t ω/ 1, we currently obtain t ω = L 4 2π log ( πɛ sin πα ψ α ψ ) ( ) + 4π 8 sinh 4 πl log cosh πl if h ψ c, then where we used t ω = f (L, ) + 2π sin πα ψ πɛ α ψ log S/L π E ψ πe ψ S/L with S L = πc 3 and E ψ = h ψ ɛ Simón (Edinburgh) Entanglement excited states Swansea 2015 41 / 44

Holographic considerations Main idea & strategy : Static point particle at r = 0 in global AdS 3 ds 2 = ( r 2 + R 2 µ ) dτ 2 + R 2 dr 2 r 2 + R 2 µ + r 2 dϕ 2, Holographic entanglement entropy known [ ( ) ] S A = c r (1) r (2) log 6 R 2 + log 2 cos ( τ α µ ) 2 cos ( ϕ α µ ) αµ 2 Map metric to Kruskal coordinates, while boosting the particle, to describe a free falling particle in eternal BTZ Use an initial condition ensuring the particle carries the right energy, from CFT and stress tensor perspective Map endpoints & compute entanglement entropy Simón (Edinburgh) Entanglement excited states Swansea 2015 42 / 44

Holographic comments Calculations involve many explicit technical details, leading to 1 Exact matching of dominant CFT contributions with the holographic model geodesic calculations S-channel and T-channel contributions precisely match the two dominant geodesics computing S A B 2 In the limit of large t ω : free falling particle becomes almost null with energy localised at the horizon matches the schock-wave descriptions proposed/used by Shenker, Stanford, Roberts, Susskind Simón (Edinburgh) Entanglement excited states Swansea 2015 43 / 44

Final remarks Stringy corrections (Shenker & Stanford) Our results in the CFT follow from properties of 2d correlators in the large c limit they may exist in other slicings of AdS, i.e. hyperbolic slicing responsible for AdS-Rindler physics this may be related to the bulk expectation that scrambling occurs more generally than for event horizons (Susskind, Fischler et al) Statistics of OPE coefficients : thermalisation, typicality of correlators in CFTs and validity of ensembles in CFT Simón (Edinburgh) Entanglement excited states Swansea 2015 44 / 44