Quantum Entanglement of locally perturbed thermal states

Transcription

1 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: and work in progress with P. Caputa, A. Štikonas, T. Takayanagi & K. Watanabe Simón (Edinburgh) Entanglement excited states Swansea / 44

2 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 / 44

3 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 / 44

4 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 / 44

5 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 / 44

6 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 / 44

7 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 / 44

8 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 / 44

9 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 / 44 i

10 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 / 44

11 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 / 44

12 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 / 44

13 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 / 44

14 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 / 44

15 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 / 44

16 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 / 44

17 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 z 2 G 1 (z, z) Simón (Edinburgh) Entanglement excited states Swansea / 44

18 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 / 44

19 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 / 44

20 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 / 44

21 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 / 44

22 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 / 44

23 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 / 44

24 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 / 44

25 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 / 44

26 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 / 44

27 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 / 44

28 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 / 44

29 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 / 44

30 Cross-ratios The cross-ratios are z = z = sinh ( πx12 sinh ( πx πiɛ 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 / 44

31 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 / 44

32 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 / 44

33 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 / 44

34 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 / 44

35 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 / 44

36 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 / 44

37 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 / 44

38 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 ) H 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 / 44

39 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 / 44

40 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 / 44

41 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 / 44

42 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 / 44

43 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 / 44

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 / 44