Holographic Entanglement Entropy

Transcription

1 Motivation Time-dependent Multi-region Summary Holographic entanglement entropy for time dependent states and disconnected regions Durham University INT08: From Strings to Things, April 3, 2008 VH, M. Rangamani, T. Takayanagi, arxiv: VH & M. Rangamani, arxiv:

2 Motivation Time-dependent Multi-region Summary 1 Motivation & background 2 Entanglement entropy with time-dependence Candidate covariant constructions Tests & applications 3 Entanglement entropy for disconnected regions Proof for disconnected regions in dimensions Conjecture for disconnected regions in dimensions 4 Summary and future directions

3 Motivation Time-dependent Multi-region Summary Entanglement Entropy Entanglement entropy of a given region A in the bdy CFT encodes the # of entangled/operative degrees of freedom. Applications to condensed matter systems, quantum information,... It is the von Neumann entropy for reduced density matrix ρ A : S A = Tr ρ A log ρ A where ρ A is trace of density matrix over the complement of A. Depends on theory, state, and region A. FT motivation Study operative DOF under time-dependence and for non-trivial (disconnected) regions

4 Motivation Time-dependent Multi-region Summary Entanglement Entropy Entanglement entropy of a given region A in the bdy CFT encodes the # of entangled/operative degrees of freedom. Applications to condensed matter systems, quantum information,... It is the von Neumann entropy for reduced density matrix ρ A : S A = Tr ρ A log ρ A where ρ A is trace of density matrix over the complement of A. Depends on theory, state, and region A. FT motivation Study operative DOF under time-dependence and for non-trivial (disconnected) regions

5 Motivation Time-dependent Multi-region Summary Entanglement Entropy Entanglement entropy of a given region A in the bdy CFT encodes the # of entangled/operative degrees of freedom. Applications to condensed matter systems, quantum information,... It is the von Neumann entropy for reduced density matrix ρ A : S A = Tr ρ A log ρ A where ρ A is trace of density matrix over the complement of A. Depends on theory, state, and region A. FT motivation Study operative DOF under time-dependence and for non-trivial (disconnected) regions

6 Motivation Time-dependent Multi-region Summary Entanglement Entropy Entanglement entropy of a given region A in the bdy CFT encodes the # of entangled/operative degrees of freedom. Applications to condensed matter systems, quantum information,... It is the von Neumann entropy for reduced density matrix ρ A : S A = Tr ρ A log ρ A where ρ A is trace of density matrix over the complement of A. Depends on theory, state, and region A. FT motivation Study operative DOF under time-dependence and for non-trivial (disconnected) regions

7 Motivation Time-dependent Multi-region Summary Probing AdS/CFT AdS/CFT correspondence provides a useful framework for addressing questions in quantum gravity e.g. emergence of spacetime To make full use of this, we need to understand the AdS/CFT dictionary better. QG motivation Study bulk geometry in AdS/CFT Holographic dual of EE is a geometric quantity we can use entanglement entropy to study bulk geometry.

8 Motivation Time-dependent Multi-region Summary Probing AdS/CFT AdS/CFT correspondence provides a useful framework for addressing questions in quantum gravity e.g. emergence of spacetime To make full use of this, we need to understand the AdS/CFT dictionary better. QG motivation Study bulk geometry in AdS/CFT Holographic dual of EE is a geometric quantity we can use entanglement entropy to study bulk geometry.

9 Motivation Time-dependent Multi-region Summary Probing AdS/CFT AdS/CFT correspondence provides a useful framework for addressing questions in quantum gravity e.g. emergence of spacetime To make full use of this, we need to understand the AdS/CFT dictionary better. QG motivation Study bulk geometry in AdS/CFT Holographic dual of EE is a geometric quantity we can use entanglement entropy to study bulk geometry.

10 Motivation Time-dependent Multi-region Summary Area law of entanglement entropy: S A area of A suggestive of a holographic relation... Holographic dual of entanglement entropy (for static bulk ST) = area of miminal co-dim. 2 surface S in bulk anchored at A (e.g. in 3-D bulk, given by zero energy spacelike geodesics) Ryu & Takayanagi Fursaev A UV divergence matches boundary

11 Motivation Time-dependent Multi-region Summary Area law of entanglement entropy: S A area of A suggestive of a holographic relation... Holographic dual of entanglement entropy (for static bulk ST) = area of miminal co-dim. 2 surface S in bulk anchored at A (e.g. in 3-D bulk, given by zero energy spacelike geodesics) bulk boundary A S Ryu & Takayanagi S A = Area(S) 4 G (d+1) N Fursaev UV divergence matches

12 Motivation Time-dependent Multi-region Summary Area law of entanglement entropy: S A area of A suggestive of a holographic relation... Holographic dual of entanglement entropy (for static bulk ST) = area of miminal co-dim. 2 surface S in bulk anchored at A (e.g. in 3-D bulk, given by zero energy spacelike geodesics) bulk boundary A S Ryu & Takayanagi S A = Area(S) 4 G (d+1) N Fursaev UV divergence matches

13 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications 1 Motivation & background 2 Entanglement entropy with time-dependence Candidate covariant constructions Tests & applications 3 Entanglement entropy for disconnected regions Proof for disconnected regions in dimensions Conjecture for disconnected regions in dimensions 4 Summary and future directions

14 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Difficulty in time-dependent backgrounds In Lorentzian spacetime, minimal area surface (we can decrease area of spacelike surface by wiggling in time) We could bypass this by separating time & space directions (cf. in static spacetime, work on a const. t slice) But in a general, time-dependent background, there is no natural time slicing... Strategy: Seek covariantly-defined co-dim.2 surfaces which: reduce to minimal surface S in static backgrounds end on A at boundary

15 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Difficulty in time-dependent backgrounds In Lorentzian spacetime, minimal area surface (we can decrease area of spacelike surface by wiggling in time) We could bypass this by separating time & space directions (cf. in static spacetime, work on a const. t slice) But in a general, time-dependent background, there is no natural time slicing... Strategy: Seek covariantly-defined co-dim.2 surfaces which: reduce to minimal surface S in static backgrounds end on A at boundary

16 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Difficulty in time-dependent backgrounds In Lorentzian spacetime, minimal area surface (we can decrease area of spacelike surface by wiggling in time) We could bypass this by separating time & space directions (cf. in static spacetime, work on a const. t slice) But in a general, time-dependent background, there is no natural time slicing... Strategy: Seek covariantly-defined co-dim.2 surfaces which: reduce to minimal surface S in static backgrounds end on A at boundary

17 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Difficulty in time-dependent backgrounds In Lorentzian spacetime, minimal area surface (we can decrease area of spacelike surface by wiggling in time) We could bypass this by separating time & space directions (cf. in static spacetime, work on a const. t slice) But in a general, time-dependent background, there is no natural time slicing... Strategy: Seek covariantly-defined co-dim.2 surfaces which: reduce to minimal surface S in static backgrounds end on A at boundary

18 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Set-up start with asymp. AdS bulk M and boundary M (corresp. state) pick a time t on the M (well-defined: fixed static bgd) pick a region A on M at time t (bulk co-dim.2) we want to construct a co-dim.2, covariantly defined, surface in M anchored on A We ll consider 4 candidate surfaces, W, X, Y, and Z.

19 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Set-up start with asymp. AdS bulk M and boundary M (corresp. state) pick a time t on the M (well-defined: fixed static bgd) pick a region A on M at time t (bulk co-dim.2) we want to construct a co-dim.2, covariantly defined, surface in M anchored on A We ll consider 4 candidate surfaces, W, X, Y, and Z.

20 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Set-up start with asymp. AdS bulk M and boundary M (corresp. state) pick a time t on the M (well-defined: fixed static bgd) pick a region A on M at time t (bulk co-dim.2) we want to construct a co-dim.2, covariantly defined, surface in M anchored on A We ll consider 4 candidate surfaces, W, X, Y, and Z.

21 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Set-up start with asymp. AdS bulk M and boundary M (corresp. state) pick a time t on the M (well-defined: fixed static bgd) pick a region A on M at time t (bulk co-dim.2) we want to construct a co-dim.2, covariantly defined, surface in M anchored on A We ll consider 4 candidate surfaces, W, X, Y, and Z.

22 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Minimal surface on maximal slice, X Construction I: pick a region A on M at time t find maximal-area spacelike slice Σ in M anchored at t on M K µ µ = 0 on Σ, find mimimal-area surface X in M with X = A This is natural if EE pertains to a given time slice...

23 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Minimal surface on maximal slice, X Construction I: pick a region A on M at time t find maximal-area spacelike slice Σ in M anchored at t on M K µ µ = 0 on Σ, find mimimal-area surface X in M with X = A This is natural if EE pertains to a given time slice...

24 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Minimal surface on maximal slice, X Construction I: pick a region A on M at time t find maximal-area spacelike slice Σ in M anchored at t on M K µ µ = 0 on Σ, find mimimal-area surface X in M with X = A This is natural if EE pertains to a given time slice...

25 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Minimal surface on maximal slice, X Construction I: pick a region A on M at time t find maximal-area spacelike slice Σ in M anchored at t on M K µ µ = 0 on Σ, find mimimal-area surface X in M with X = A This is natural if EE pertains to a given time slice...

26 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Extremal surface, W Construction II: pick a region A on M at time t find an extremal surface W in M with W = A (if there are several solutions, pick the one with mimimal-area) if bulk is 3-dimensional, W is a spacelike geodesic anchored at A This is most natural extension of S, but no longer pertains to particular spacelike slice of bulk.

27 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Extremal surface, W Construction II: pick a region A on M at time t find an extremal surface W in M with W = A (if there are several solutions, pick the one with mimimal-area) if bulk is 3-dimensional, W is a spacelike geodesic anchored at A This is most natural extension of S, but no longer pertains to particular spacelike slice of bulk.

28 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Extremal surface, W Construction II: pick a region A on M at time t find an extremal surface W in M with W = A (if there are several solutions, pick the one with mimimal-area) if bulk is 3-dimensional, W is a spacelike geodesic anchored at A This is most natural extension of S, but no longer pertains to particular spacelike slice of bulk.

29 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Extremal surface, W Construction II: pick a region A on M at time t find an extremal surface W in M with W = A (if there are several solutions, pick the one with mimimal-area) if bulk is 3-dimensional, W is a spacelike geodesic anchored at A This is most natural extension of S, but no longer pertains to particular spacelike slice of bulk.

30 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Light-sheet construction, Y Construction III: pick a region A on M at time t consider light sheets in M (non-positive null expansions θ ± ) find the surface Y in M with Y = A from which both expansions vanish θ + = θ = 0 Turns out: equivalent to the extremal surface W; Cf. Bousso s Covariant Entropy bound...

31 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Light-sheet construction, Y Construction III: pick a region A on M at time t consider light sheets in M (non-positive null expansions θ ± ) find the surface Y in M with Y = A from which both expansions vanish θ + = θ = 0 Turns out: equivalent to the extremal surface W; Cf. Bousso s Covariant Entropy bound...

32 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Light-sheet construction, Y Construction III: pick a region A on M at time t consider light sheets in M (non-positive null expansions θ ± ) find the surface Y in M with Y = A from which both expansions vanish θ + = θ = 0 Turns out: equivalent to the extremal surface W; Cf. Bousso s Covariant Entropy bound...

33 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Light-sheet construction, Y Construction III: pick a region A on M at time t consider light sheets in M (non-positive null expansions θ ± ) find the surface Y in M with Y = A from which both expansions vanish θ + = θ = 0 Turns out: equivalent to the extremal surface W; Cf. Bousso s Covariant Entropy bound...

34 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Causal construction, Z Construction IV: pick a region A on M at time t find the domain of dependence D ± [A] in M find the causal wedge of D ± [A] in M and consider its boundary in M inside this co-dim.1 surface, find maximal-area surface Z in M with Z = A Simpler to compute, but does not necessarily reduce to S for static spacetimes... (may provide a bound for the EE)

35 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Causal construction, Z Construction IV: pick a region A on M at time t find the domain of dependence D ± [A] in M find the causal wedge of D ± [A] in M and consider its boundary in M inside this co-dim.1 surface, find maximal-area surface Z in M with Z = A Simpler to compute, but does not necessarily reduce to S for static spacetimes... (may provide a bound for the EE)

36 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Causal construction, Z Construction IV: pick a region A on M at time t find the domain of dependence D ± [A] in M find the causal wedge of D ± [A] in M and consider its boundary in M inside this co-dim.1 surface, find maximal-area surface Z in M with Z = A Simpler to compute, but does not necessarily reduce to S for static spacetimes... (may provide a bound for the EE)

37 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Causal construction, Z Construction IV: pick a region A on M at time t find the domain of dependence D ± [A] in M find the causal wedge of D ± [A] in M and consider its boundary in M inside this co-dim.1 surface, find maximal-area surface Z in M with Z = A Simpler to compute, but does not necessarily reduce to S for static spacetimes... (may provide a bound for the EE)

38 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Causal construction, Z Construction IV: pick a region A on M at time t find the domain of dependence D ± [A] in M find the causal wedge of D ± [A] in M and consider its boundary in M inside this co-dim.1 surface, find maximal-area surface Z in M with Z = A Simpler to compute, but does not necessarily reduce to S for static spacetimes... (may provide a bound for the EE)

39 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Relations between constructions Z S for some static bulk spacetimes (since uses null structure sensitive to g tt ) not a viable candidate for holographic EE W = X = Y = S for static bulk spacetime a-priori all viable candidates for EE W = Y in general (motivated via partition fn.: Lorentzian GKP-W relation) X = Y (only) if Σ is totally geodesic submanifold conjecture: Holographic dual of EE in general time-dependent spacetime is given by area of extremal co-dim.2 bulk surface, or equivalently surface with vanishing null expansions, anchored at A.

40 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Relations between constructions Z S for some static bulk spacetimes (since uses null structure sensitive to g tt ) not a viable candidate for holographic EE W = X = Y = S for static bulk spacetime a-priori all viable candidates for EE W = Y in general (motivated via partition fn.: Lorentzian GKP-W relation) X = Y (only) if Σ is totally geodesic submanifold conjecture: Holographic dual of EE in general time-dependent spacetime is given by area of extremal co-dim.2 bulk surface, or equivalently surface with vanishing null expansions, anchored at A.

41 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Relations between constructions Z S for some static bulk spacetimes (since uses null structure sensitive to g tt ) not a viable candidate for holographic EE W = X = Y = S for static bulk spacetime a-priori all viable candidates for EE W = Y in general (motivated via partition fn.: Lorentzian GKP-W relation) X = Y (only) if Σ is totally geodesic submanifold conjecture: Holographic dual of EE in general time-dependent spacetime is given by area of extremal co-dim.2 bulk surface, or equivalently surface with vanishing null expansions, anchored at A.

42 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Relations between constructions Z S for some static bulk spacetimes (since uses null structure sensitive to g tt ) not a viable candidate for holographic EE W = X = Y = S for static bulk spacetime a-priori all viable candidates for EE W = Y in general (motivated via partition fn.: Lorentzian GKP-W relation) X = Y (only) if Σ is totally geodesic submanifold conjecture: Holographic dual of EE in general time-dependent spacetime is given by area of extremal co-dim.2 bulk surface, or equivalently surface with vanishing null expansions, anchored at A.

43 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Test I: static BTZ In 2-D CFT (finite T ), we can calculate EE explicitly. For non-rotating BTZ: EE agrees w/ expression from W. ds 2 = (r 2 r 2 +) dt 2 + dr 2 (r 2 r 2 + ) + r 2 dx 2 Calabrese & Cardy S A = L W 4 G (3) N = c ( β 3 log π ε sinh 2 π h ) β

44 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Test II: rotating BTZ CFT density matrix: ρ = e β H+β Ω P Trick for calculating EE: S A = n log Trρn A n=1 Trρ n A obtained from 2-pt.fn. of twist opers w/ n = c 24 (n 1 n ). Both CFT and L W /4G (3) N give S A = c 6 log [ β+ β π 2 ε 2 ( ) ( )] π l π l sinh sinh β + BUT X does not coincide with W: t is Killing but not hypersurface orthogonal while Σ is at const.t, geods. W are not. Conclusion: W = Y gives agreement w/ EE, X does not. β Calabrese & Cardy

45 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Test II: rotating BTZ CFT density matrix: ρ = e β H+β Ω P Trick for calculating EE: S A = n log Trρn A n=1 Trρ n A obtained from 2-pt.fn. of twist opers w/ n = c 24 (n 1 n ). Both CFT and L W /4G (3) N give S A = c 6 log [ β+ β π 2 ε 2 ( ) ( )] π l π l sinh sinh β + BUT X does not coincide with W: t is Killing but not hypersurface orthogonal while Σ is at const.t, geods. W are not. Conclusion: W = Y gives agreement w/ EE, X does not. β Calabrese & Cardy

46 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Entanglement entropy during collapse Example of extremal surfaces for collapse ( Vaidya-AdS 3 ): v0 2 v0 1 v0 0 v0 0.1 v0 0.5 v0 1

47 Motivation Time-dependent Multi-region Summary Obstacle Constructions Tests Applications Second Law for Entanglement entropy Entanglement entropy increases in time-dependent background (satisfying the null energy condition) L v

48 Motivation Time-dependent Multi-region Summary dimensions dimensions 1 Motivation & background 2 Entanglement entropy with time-dependence Candidate covariant constructions Tests & applications 3 Entanglement entropy for disconnected regions Proof for disconnected regions in dimensions Conjecture for disconnected regions in dimensions 4 Summary and future directions

49 Motivation Time-dependent Multi-region Summary dimensions dimensions Entanglement entropy for disconnected regions Question: Given EE for any interval A in some state of a 1+1 dim QFT, what is the EE of a disconnected region X composed of two intervals? Naive guess: x S(X ) = s 12 + s 34 Doesn t work (cf. p 2 p 3 : in general s 12 + s 34 s 14 )

50 Motivation Time-dependent Multi-region Summary dimensions dimensions Entanglement entropy for disconnected regions Answer from CFT: x = s 12 + s 34 + s 23 + s 14 s 13 s 24. We will prove this using the holographic dual, via: 1 coincident upper & lower bounds on x from geometry 2 ansatz for x & consistency of limits p i p j (cf. Calabrese & Cardy) Useful tool: Strong subadditivity Given two overlapping boundary regions A and B, (cf. Lieb & Ruskai) S(A) + S(B) S(A B) + S(A B) S(A) + S(B) S(A\B) + S(B\A)

51 Motivation Time-dependent Multi-region Summary dimensions dimensions Geometric proof of strong subadditivity (cf. Headrick & Takayanagi) Given two overlapping boundary regions A and B, consider S(A) + S(B)

52 Motivation Time-dependent Multi-region Summary dimensions dimensions Geometric proof of strong subadditivity (cf. Headrick & Takayanagi) Given two overlapping boundary regions A and B, compare with S(A B) + S(A B)

53 Motivation Time-dependent Multi-region Summary dimensions dimensions Geometric proof of strong subadditivity (cf. Headrick & Takayanagi) Given two overlapping boundary regions A and B, S(A) + S(B) S(A B) + S(A B)

54 Motivation Time-dependent Multi-region Summary dimensions dimensions Geometric proof of strong subadditivity (cf. Headrick & Takayanagi) Given two overlapping boundary regions A and B, S(A) + S(B) S(A B) + S(A B) compare S(A) + S(B) with S(A\B) + S(B\A)

55 Motivation Time-dependent Multi-region Summary dimensions dimensions Geometric proof of strong subadditivity (cf. Headrick & Takayanagi) Given two overlapping boundary regions A and B, S(A) + S(B) S(A B) + S(A B) S(A) + S(B) S(A\B) + S(B\A)

56 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument for two disconnected intervals in dim Holographic EE as bulk minimal surface implies: s 14 + s 23 s 13 + s 24 s 12 + s 34 s 13 + s 24 s 13 s 12 + s 23 s 13 s 14 + s 34 s 24 s 12 + s 14 s 24 s 23 + s 34

57 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument for two disconnected intervals in dim Holographic EE as bulk minimal surface implies: s 14 + s 23 s 13 + s 24 s 12 + s 34 s 13 + s 24 s ij s ik + s jk for i, j, k = 1, 2, 3, 4 (12 redundant relations reduce to 4) s 13 s 12 + s 23 s 13 s 14 + s 34 s 24 s 12 + s 14 s 24 s 23 + s 34

58 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument for two disconnected intervals in dim Holographic EE as bulk minimal surface implies: s 14 + s 23 s 13 + s 24 s 12 + s 34 s 13 + s 24 s 13 s 12 + s 23 s 13 s 14 + s 34 s 24 s 12 + s 14 s 24 s 23 + s 34

59 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument for two disconnected intervals in dim Strong subadditivity implies: x s 12 + s 34 x s 14 + s 23 s 12 + s 14 s 13 x s 14 + s 34 s 24 x s 12 + s 23 s 24 x s 23 + s 34 s 13 x where x S(X )

60 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument I for two disconnected intervals in dim Altogether, we have 6 relations among s ij and 6 constraints on x: s 14 + s 23 s 13 + s 24 s 12 + s 34 s 13 + s 24 s 13 s 12 + s 23 s 13 s 14 + s 34 s 24 s 12 + s 14 s 24 s 23 + s 34 x s 12 + s 34 x s 14 + s 23 s 14 + s 34 s 24 x s 12 + s 23 s 24 x s 23 + s 34 s 13 x s 12 + s 14 s 13 x Note: natural pairing between equations. Use these to get coincident upper & lower bounds on x, assuming two of the inequalities can be saturated: s 12 + s 23 + s 34 + s 14 s 13 s 24 x s 12 + s 23 + s 34 + s 14 s 13 s 24 x = s 12 + s 34 + s 23 + s 14 s 13 s 24. details

61 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument I for two disconnected intervals in dim Altogether, we have 6 relations among s ij and 6 constraints on x: s 14 + s 23 s 13 + s 24 s 12 + s 34 s 13 + s 24 s 13 s 12 + s 23 s 13 s 14 + s 34 s 24 s 12 + s 14 s 24 s 23 + s 34 x s 12 + s 34 x s 14 + s 23 s 14 + s 34 s 24 x s 12 + s 23 s 24 x s 23 + s 34 s 13 x s 12 + s 14 s 13 x Note: natural pairing between equations. Use these to get coincident upper & lower bounds on x, assuming two of the inequalities can be saturated: s 12 + s 23 + s 34 + s 14 s 13 s 24 x s 12 + s 23 + s 34 + s 14 s 13 s 24 x = s 12 + s 34 + s 23 + s 14 s 13 s 24. details

62 Motivation Time-dependent Multi-region Summary dimensions dimensions Argument II for two disconnected intervals in dim Assume a linear ansatz: x = c 12 s 12 + c 13 s 13 + c 14 s 14 + c 23 s 23 + c 24 s 24 + c 34 s 34 for some (universal) constants c ij. Now consider special limits where we know x explicitly: p 1 p 2 s 12 0, s 13 s 23, s 14 s 24, and x s 34 c 34 = 1, c 13 + c 23 = 0, c 14 + c 24 = 0. p 2 p 3 s 14 0, s 12 s 13, s 24 s 34, and x s 14 c 14 = 1, c 12 + c 13 = 0, c 24 + c 34 = 0. p 3 p 4 s 34 0, s 13 s 14, s 23 s 24, and x s 12 c 12 = 1, c 13 + c 14 = 0, c 23 + c 24 = 0. c 12 = c 14 = c 23 = c 34 = c 13 = c 24 = 1 x = s 12 + s 34 + s 23 + s 14 s 13 s 24. QED

63 Motivation Time-dependent Multi-region Summary dimensions dimensions Multiply disconnected intervals in dim Consider X given by m intervals specified by n = 2m endpoints p i. We have shown that for n = 4, x = s 12 + s 34 + s 23 + s 14 s 13 s 24. We can easily generalise this to x = n ( 1) i+j+1 s ij. i,j=1 j>i In the process we obtain higher-level subadditivity -type relations, e.g. for 6 endpoints p i < p j < p k < p l < p m < p n, s ik + s jm + s ln s in + s jk + s lm

64 Motivation Time-dependent Multi-region Summary dimensions dimensions Higher dimensions Natural question: Can we generalize this geometric construction for entanglement entropy of disconnected regions to higher dimensions? Note: QFT methods, but: holographic entanglement entropy as given by area of extremal (minimal) surface holds in all dimensions strong subadditivity holds in all dimensions For translationally invariant (effectively 1-D) systems, the arguments are identical to the dim. case We offer a conjecture for dim. QFT

65 Motivation Time-dependent Multi-region Summary dimensions dimensions Types of 2-dimensional regions cases (a) and (c) are generic, but they naturally define only three bulk surfaces case (b) is special intermediate case; but it naturally defines six bulk surfaces we can apply previous arguments here: x (b) = s 12 + s 34 + s 23 + s 14 s 13 s 24 however, case (b) can be resolved into (a) or (c). This motivates a conjecture for the general cases.

66 Motivation Time-dependent Multi-region Summary dimensions dimensions Types of 2-dimensional regions cases (a) and (c) are generic, but they naturally define only three bulk surfaces case (b) is special intermediate case; but it naturally defines six bulk surfaces we can apply previous arguments here: x (b) = s 12 + s 34 + s 23 + s 14 s 13 s 24 however, case (b) can be resolved into (a) or (c). This motivates a conjecture for the general cases.

67 Motivation Time-dependent Multi-region Summary dimensions dimensions Conjecture for 2-dimensional regions Since we need at least 6 minimal surfaces in the bulk, let us introduce auxiliary curves C 5 and C 6 to lift (a) and (c) to configurations which admit 6 bulk surfaces: Conjecture: Since x cannot depend on our choice of the auxiliary curves C 5 and C 6, we minimize over all such configurations: x (a) = inf C5,C 6 {s 12 s s s 2536 s s 34 } x (c) = inf C5,C 6 {s 1526 s s 14 + s 23 s s 3546 }

68 Motivation Time-dependent Multi-region Summary dimensions dimensions Support for the conjecture x (a) = inf C5,C 6 {s 12 s s s 2536 s s 34 } x (c) = inf C5,C 6 {s 1526 s s 14 + s 23 s s 3546 } for limiting case (b), {s 12, s 13, s 14, s 23, s 24, s 34 } are necessary to specify x (b) ; hence x (a,c) comprise the minimal ansatz infimum to satisfy subadditivity relations as C 5 C 6... UV divergence C 1 + C 2 + C 3 + C 4 as required correct limiting behaviour as (a) (b) and (c) (b) correct single-region limits

69 Motivation Time-dependent Multi-region Summary Summary (part I) Holographic dual of entanglement entropy (for static bulk ST) = area of miminal surface in bulk anchored at A For time-dependent bulk, use covariant generalization: = area of extremal surface in bulk anchored at A also given by surface with vanishing null expansions Can be used to study EE for time-dependent bulk geometries e.g. EE increases in Vaidya satisfying energy conditions Since well-defined in time-dependent system, can analyze quantum systems far from equilibrium, zero-temperature quantum phase transitions,... study condensed matter systems using AdS/CFT

70 Motivation Time-dependent Multi-region Summary Summary (part I) Holographic dual of entanglement entropy (for static bulk ST) = area of miminal surface in bulk anchored at A For time-dependent bulk, use covariant generalization: = area of extremal surface in bulk anchored at A also given by surface with vanishing null expansions Can be used to study EE for time-dependent bulk geometries e.g. EE increases in Vaidya satisfying energy conditions Since well-defined in time-dependent system, can analyze quantum systems far from equilibrium, zero-temperature quantum phase transitions,... study condensed matter systems using AdS/CFT

71 Motivation Time-dependent Multi-region Summary Summary (part I) Holographic dual of entanglement entropy (for static bulk ST) = area of miminal surface in bulk anchored at A For time-dependent bulk, use covariant generalization: = area of extremal surface in bulk anchored at A also given by surface with vanishing null expansions Can be used to study EE for time-dependent bulk geometries e.g. EE increases in Vaidya satisfying energy conditions Since well-defined in time-dependent system, can analyze quantum systems far from equilibrium, zero-temperature quantum phase transitions,... study condensed matter systems using AdS/CFT

72 Motivation Time-dependent Multi-region Summary Summary (part I) Holographic dual of entanglement entropy (for static bulk ST) = area of miminal surface in bulk anchored at A For time-dependent bulk, use covariant generalization: = area of extremal surface in bulk anchored at A also given by surface with vanishing null expansions Can be used to study EE for time-dependent bulk geometries e.g. EE increases in Vaidya satisfying energy conditions Since well-defined in time-dependent system, can analyze quantum systems far from equilibrium, zero-temperature quantum phase transitions,... study condensed matter systems using AdS/CFT

73 Motivation Time-dependent Multi-region Summary Summary (part II) For a static state in QFT, we derive the expression for entanglement entropy for disconnected regions: x = n ( 1) i+j+1 s ij. i,j=1 j>i This reproduces and generalizes the expression obtained by CFT methods. Geometrical picture immediately yields higher order geneneralizations of strong subadditivity. We presented conjecture & supporting evidence for EE of disconnected regions in QFT (no direct comparison w/ QFT results yet)

74 Motivation Time-dependent Multi-region Summary Summary (part II) For a static state in QFT, we derive the expression for entanglement entropy for disconnected regions: x = n ( 1) i+j+1 s ij. i,j=1 j>i This reproduces and generalizes the expression obtained by CFT methods. Geometrical picture immediately yields higher order geneneralizations of strong subadditivity. We presented conjecture & supporting evidence for EE of disconnected regions in QFT (no direct comparison w/ QFT results yet)

75 Motivation Time-dependent Multi-region Summary Summary (part II) For a static state in QFT, we derive the expression for entanglement entropy for disconnected regions: x = n ( 1) i+j+1 s ij. i,j=1 j>i This reproduces and generalizes the expression obtained by CFT methods. Geometrical picture immediately yields higher order geneneralizations of strong subadditivity. We presented conjecture & supporting evidence for EE of disconnected regions in QFT (no direct comparison w/ QFT results yet)

76 Motivation Time-dependent Multi-region Summary Summary (part II) For a static state in QFT, we derive the expression for entanglement entropy for disconnected regions: x = n ( 1) i+j+1 s ij. i,j=1 j>i This reproduces and generalizes the expression obtained by CFT methods. Geometrical picture immediately yields higher order geneneralizations of strong subadditivity. We presented conjecture & supporting evidence for EE of disconnected regions in QFT (no direct comparison w/ QFT results yet)

77 Motivation Time-dependent Multi-region Summary Future directions General proof for strong subadditivity in time-dependent cases Proof of conjecture for disconnected regions in Generalizations to higher dimensions, multiple regions Relations between covariant constructions, physical interpretation of X, Z Second Law for entanglement entropy? Full bulk metric extraction? Study more general asymptopia

78 Details of Vaidya-AdS Use to describe collapse to a black hole in AdS; 3-dim metric: ds 2 = f (r, v) dv dv dr + r 2 dx 2 with f (r, v) r 2 m(v) interpolating between AdS and Schw-AdS. Energy-momentum tensor T µν = R µν 1 2 R g µν + Λ g µν : T vv = 1 2 r dm(v) dv Null energy condition holds mass accretes: m (v) 0 (NEC: T µν k µ k ν 0 for any null vector k µ ) back

79 Details of coincident bounds argument Assume x is given by a unique expression in terms of the s ij which satisfy the geometric constraints (for any state, etc.) Now use an extreme case of allowed s ij : s 14 + s 23 = s 13 + s 24 and s 13 = s 12 + s 23 (one can check these are mutually consistent) Then constants a 1 and a 2, s 14 + s 34 s 24 + (s 12 + s 23 s 13 ) a 1 x s 12 + s 34 + (s 14 + s 23 s 13 s 24 ) a 2 Now set a 1 = a 2 = 1. x = s 12 + s 34 + s 23 + s 14 s 13 s 24 Any choice of saturating pair of eqs will yield the same answer. back