Holographic entanglement entropy for time …...Holographic entanglement entropy for time dependent states and disconnected regions Veronika Hubeny Durham University INT08: From Strings

Motivation Time-dependent Multi-region Summary

Holographic entanglement entropyfor time dependent states and disconnected regions

Veronika Hubeny

Durham University

INT08: From Strings to Things,April 3, 2008

VH, M. Rangamani, T. Takayanagi, arXiv:0705.0016

VH & M. Rangamani, arXiv:0711.4118

Veronika Hubeny Holographic Entanglement Entropy


1 Motivation & background

2 Entanglement entropy with time-dependenceCandidate covariant constructionsTests & applications

3 Entanglement entropy for disconnected regionsProof for disconnected regions in 1 + 1 dimensionsConjecture for disconnected regions in 2 + 1 dimensions

4 Summary and future directions



Entanglement Entropy

Entanglement entropy of a given region A in the bdy CFTencodes the # of entangled/operative degrees of freedom.

Applications to condensed matter systems, quantum information, ...

It is the von Neumann entropy for reduced density matrix ρA:

SA = −Tr ρA log ρA

where ρA is trace of density matrix over the complement of A.

Depends on theory, state, and region A.

FT motivation

Study operative DOF under time-dependenceand for non-trivial (disconnected) regions










FT motivation











FT motivation











FT motivation




Probing AdS/CFT

AdS/CFT correspondence provides a useful frameworkfor addressing questions in quantum gravitye.g. emergence of spacetime

To make full use of this, we need to understand the AdS/CFTdictionary 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.



Probing AdS/CFT



QG motivation





Probing AdS/CFT



QG motivation





Holographic Entanglement Entropy

Area law of entanglement entropy: SA ∼ 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

boundary

AUV divergence matches








Fursaev

A

S

boundary

bulk

SA =Area(S)

4 G(d+1)N

UV divergence matches








Fursaev

A

S

boundary

bulk

SA =Area(S)

4 G(d+1)N

UV divergence matches


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







Strategy:










Strategy:










Strategy:






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.



Set-up







anchored on ∂A




Set-up







anchored on ∂A




Set-up







anchored on ∂A




Minimal surface on maximal slice, X

Construction I:

pick a region A on ∂M at time t

find maximal-area spacelike slice Σin M anchored at t on ∂M

K µµ = 0

on Σ, find mimimal-area surface X inM with ∂X = ∂A

This is natural if EE pertains to a given time slice...




Construction I:



K µµ = 0






Construction I:



K µµ = 0






Construction I:



K µµ = 0





Extremal surface, W

Construction II:


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 aspacelike geodesic anchored at ∂A

This is most natural extension of S,but no longer pertains to particular spacelike slice of bulk.



Extremal surface, W

Construction II:



with ∂W = ∂A






Extremal surface, W

Construction II:



with ∂W = ∂A






Extremal surface, W

Construction II:



with ∂W = ∂A






Light-sheet construction, Y

Construction III:


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...




Construction III:






θ+ = θ− = 0





Construction III:






θ+ = θ− = 0





Construction III:






θ+ = θ− = 0




Causal construction, Z

Construction IV:


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, findmaximal-area surface Z in M with∂Z = ∂A

Simpler to compute, but does not necessarily reduce to S for staticspacetimes... (may provide a bound for the EE)




Construction IV:










Construction IV:










Construction IV:










Construction IV:









Relations between constructions

Z 6= S for some static bulk spacetimes(since uses null structure ⇒ sensitive to gtt)⇒ 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 isgiven by area of extremal co-dim.2 bulk surface, or equivalentlysurface with vanishing null expansions, anchored at ∂A.








conjecture:









conjecture:









conjecture:




Test I: static BTZ

In 2-D CFT (finite T ), we can calculate EE explicitly. Calabrese & Cardy

For non-rotating BTZ: EE agrees w/ expression from W.

ds2 = −(r2 − r2+) dt2 +

dr2

(r2 − r2+)

+ r2 dx2

0.1 0.5 1 2

SA =LW

4 G(3)N

=c

3log

(β

π εsinh

2π h

β

)



Test II: rotating BTZ

CFT density matrix: ρ = e−β H+βΩ P

Trick for calculating EE: SA = − ∂∂n log Trρn

A

∣∣∣∣n=1

TrρnA obtained from 2-pt.fn. of twist opers w/ ∆n = c

24 (n − 1n ).

Calabrese & Cardy

Both CFT and LW/4G(3)N give

SA =c

6log

[β+ β−π2 ε2

sinh

(π∆l

β+

)sinh

(π∆l

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.



Test II: rotating BTZ

CFT density matrix: ρ = e−β H+βΩ P

Trick for calculating EE: SA = − ∂∂n log Trρn

A

∣∣∣∣n=1

TrρnA obtained from 2-pt.fn. of twist opers w/ ∆n = c

24 (n − 1n ).

Calabrese & Cardy

Both CFT and LW/4G(3)N give

SA =c

6log

[β+ β−π2 ε2

sinh

(π∆l

β+

)sinh

(π∆l

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.



Entanglement entropy during collapse

Example of extremal surfaces for collapse ( Vaidya-AdS3 ):

v0 ! 0.1 v0 ! 0.5 v0 ! 1

v0 ! "2 v0 ! "1 v0 ! 0



‘Second Law’ for Entanglement entropy

Entanglement entropy increases in time-dependent background(satisfying the null energy condition)

-2 -1 1 2 3 4v!

-0.3

-0.2

-0.1

0.1

0.2

0.3

L


Motivation Time-dependent Multi-region Summary 1 + 1 dimensions 2 + 1 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 twointervals?

Naive guess: x ≡ S(X ) = s12 + s34

Doesn’t work (cf. p2 → p3: in general s12 + s34 6= s14)



Entanglement entropy for disconnected regions

Answer from CFT: (cf. Calabrese & Cardy)

x = s12 + s34 + s23 + s14 − s13 − s24 .

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 pi → pj

Useful tool:

Strong subadditivity (cf. Lieb & Ruskai)

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)



Geometric proof of strong subadditivity (cf. Headrick & Takayanagi)


consider S(A) + S(B)





compare with S(A ∪B) + S(A ∩B)





S(A) + S(B) ≥ S(A ∪B) + S(A ∩B)





S(A) + S(B) ≥ S(A ∪B) + S(A ∩B)

compare S(A) + S(B) with S(A\B) + S(B\A)





S(A) + S(B) ≥ S(A ∪B) + S(A ∩B)

S(A) + S(B) ≥ S(A\B) + S(B\A)



Argument for two disconnected intervals in 1 + 1 dim

Holographic EE as bulk minimal surface implies:

s14 + s23 ≤ s13 + s24

s12 + s34 ≤ s13 + s24

s13 ≤ s12 + s23

s13 ≤ s14 + s34

s24 ≤ s12 + s14

s24 ≤ s23 + s34





s14 + s23 ≤ s13 + s24

s12 + s34 ≤ s13 + s24

sij ≤ sik + sjkfor i , j , k = 1, 2, 3, 4(12 redundant relationsreduce to 4)s13 ≤ s12 + s23

s13 ≤ s14 + s34

s24 ≤ s12 + s14

s24 ≤ s23 + s34





s14 + s23 ≤ s13 + s24

s12 + s34 ≤ s13 + s24

s13 ≤ s12 + s23

s13 ≤ s14 + s34

s24 ≤ s12 + s14

s24 ≤ s23 + s34




Strong subadditivity implies:

x ≤ s12 + s34

x ≤ s14 + s23

s12 + s14 − s13 ≤ xs14 + s34 − s24 ≤ xs12 + s23 − s24 ≤ xs23 + s34 − s13 ≤ x

where x ≡ S(X )



Argument I for two disconnected intervals in 1 + 1 dim

Altogether, we have 6 relations among sij and 6 constraints on x :

s14 + s23 ≤ s13 + s24

s12 + s34 ≤ s13 + s24

s13 ≤ s12 + s23

s13 ≤ s14 + s34

s24 ≤ s12 + s14

s24 ≤ s23 + s34

x ≤ s12 + s34

x ≤ s14 + s23


Note: ∃ natural pairing between equations.Use these to get coincident upper & lower bounds on x , assumingtwo of the inequalities can be saturated:

s12 + s23 + s34 + s14− s13− s24 ≤ x ≤ s12 + s23 + s34 + s14− s13− s24

⇒ x = s12 + s34 + s23 + s14 − s13 − s24 . details



Argument I for two disconnected intervals in 1 + 1 dim

Altogether, we have 6 relations among sij and 6 constraints on x :

s14 + s23 ≤ s13 + s24

s12 + s34 ≤ s13 + s24

s13 ≤ s12 + s23

s13 ≤ s14 + s34

s24 ≤ s12 + s14

s24 ≤ s23 + s34

x ≤ s12 + s34

x ≤ s14 + s23


Note: ∃ natural pairing between equations.Use these to get coincident upper & lower bounds on x , assumingtwo of the inequalities can be saturated:

s12 + s23 + s34 + s14− s13− s24 ≤ x ≤ s12 + s23 + s34 + s14− s13− s24

⇒ x = s12 + s34 + s23 + s14 − s13 − s24 . details



Argument II for two disconnected intervals in 1 + 1 dim

Assume a linear ansatz:

x = c12 s12 + c13 s13 + c14 s14 + c23 s23 + c24 s24 + c34 s34

for some (universal) constants cij .Now consider special limits where we know x explicitly:

p1 → p2 ⇒ s12 → 0, s13 → s23, s14 → s24, and x → s34

c34 = 1, c13 + c23 = 0, c14 + c24 = 0.

p2 → p3 ⇒ s14 → 0, s12 → s13, s24 → s34, and x → s14

c14 = 1, c12 + c13 = 0, c24 + c34 = 0.

p3 → p4 ⇒ s34 → 0, s13 → s14, s23 → s24, and x → s12

c12 = 1, c13 + c14 = 0, c23 + c24 = 0.

⇒ c12 = c14 = c23 = c34 = −c13 = −c24 = 1

⇒ x = s12 + s34 + s23 + s14 − s13 − s24 . QED



Multiply disconnected intervals in 1 + 1 dim

Consider X given by m intervals specified by n = 2m endpoints pi .We have shown that for n = 4,

x = s12 + s34 + s23 + s14 − s13 − s24 .

We can easily generalise this to

x =n∑

i ,j=1j>i

(−1)i+j+1 sij .

In the process we obtain higher-level ‘subadditivity’-type relations,e.g. for 6 endpoints pi < pj < pk < pl < pm < pn,

sik + sjm + sln ≥ sin + sjk + slm



Higher dimensions

Natural question:

Can we generalize this geometric construction for entanglemententropy of disconnected regions to higher dimensions?

Note:

@ QFT methods, but:

holographic entanglement entropy as given by area ofextremal (minimal) surface holds in all dimensions

⇒ strong subadditivity holds in all dimensions

For translationally invariant (effectively 1-D) systems, thearguments are identical to the 1 + 1 dim. case

We offer a conjecture for 2 + 1 dim. QFT



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) = s12 + s34 + s23 + s14 − s13 − s24

however, case (b) can be ‘resolved’ into (a) or (c).This motivates a conjecture for the general cases.



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) = s12 + s34 + s23 + s14 − s13 − s24

however, case (b) can be ‘resolved’ into (a) or (c).This motivates a conjecture for the general cases.



Conjecture for 2-dimensional regions

Since we need at least 6 minimal surfaces in the bulk, let usintroduce auxiliary curves C5 and C6 to lift (a) and (c) toconfigurations which admit 6 bulk surfaces:

Conjecture: Since x cannot depend on our choice of the auxiliarycurves C5 and C6, we minimize over all such configurations:

x(a) = infC5,C6 s12 − s1536 + s1546 + s2536 − s2546 + s34x(c) = infC5,C6 s1526 − s1536 + s14 + s23 − s2546 + s3546



Support for the conjecture

x(a) = infC5,C6 s12 − s1536 + s1546 + s2536 − s2546 + s34x(c) = infC5,C6 s1526 − s1536 + s14 + s23 − s2546 + s3546

for limiting case (b), s12, s13, s14, s23, s24, s34 are necessaryto specify x(b); hence x(a,c) comprise the minimal ansatz

infimum to satisfy subadditivity relations as C5 → C6...

UV divergence ∝ C1 + C2 + C3 + C4 as required

correct limiting behaviour as (a) → (b) and (c) → (b)

correct single-region limits



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 geometriese.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



Summary (part I)








Summary (part I)








Summary (part I)








Summary (part II)

For a static state in 1 + 1 QFT, we derive the expression forentanglement entropy for disconnected regions:

x =n∑

i ,j=1j>i

(−1)i+j+1 sij .

This reproduces and generalizes the expression obtained byCFT methods.

Geometrical picture immediately yields higher ordergeneneralizations of strong subadditivity.

We presented conjecture & supporting evidencefor EE of disconnected regions in 2 + 1 QFT(no direct comparison w/ QFT results yet)



Summary (part II)


x =n∑

i ,j=1j>i

(−1)i+j+1 sij .






Summary (part II)


x =n∑

i ,j=1j>i

(−1)i+j+1 sij .






Summary (part II)


x =n∑

i ,j=1j>i

(−1)i+j+1 sij .






Future directions

General proof for strong subadditivity in time-dependent cases

Proof of conjecture for disconnected regions in 2 + 1

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


Details of Vaidya-AdS

Use to describe collapse to a black hole in AdS; 3-dim metric:

ds2 = −f (r , v) dv2 + 2 dv dr + r2 dx2

with f (r , v) ≡ r2 −m(v) interpolating between AdS andSchw-AdS.

Energy-momentum tensor Tµν = Rµν − 12 R gµν + Λ gµν :

Tvv =1

2 r

dm(v)

dv

Null energy condition holds ⇐⇒ mass accretes: m′(v) ≥ 0(NEC: Tµνk

µkν ≥ 0 for any null vector kµ)back


Details of coincident bounds argument

Assume x is given by a unique expression in terms of the sijwhich satisfy the geometric constraints (for any state, etc.)

Now use an extreme case of ‘allowed’ sij :

s14 + s23 = s13 + s24 and s13 = s12 + s23

(one can check these are mutually consistent)

Then ∀ constants a1 and a2,s14 + s34 − s24 + (s12 + s23 − s13) a1 ≤ x ≤

s12 + s34 + (s14 + s23 − s13 − s24) a2

Now set a1 = a2 = 1.⇒ x = s12 + s34 + s23 + s14 − s13 − s24

Any choice of saturating pair of eqs will yield the same answer.

back


Documents

Holographic entanglement entropy for time …...Holographic entanglement entropy for time dependent states and disconnected regions Veronika Hubeny Durham University INT08: From Strings