Motivation Time-dependent Multi-region Summary Holographic entanglement entropy for 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
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Holographic entanglement entropy for time …...Holographic entanglement entropy for time dependent states and disconnected regions Veronika Hubeny Durham University INT08: From Strings
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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
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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.
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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.
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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.
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
A
S
boundary
bulk
SA =Area(S)
4 G(d+1)N
UV divergence matches
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
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
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.
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.
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.
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.
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
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:
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
Veronika Hubeny Holographic Entanglement Entropy
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 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
Veronika Hubeny Holographic Entanglement Entropy
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 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
Veronika Hubeny Holographic Entanglement Entropy
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 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
Veronika Hubeny Holographic Entanglement Entropy
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 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
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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)
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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)
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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)
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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)
Veronika Hubeny Holographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary
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
Veronika Hubeny Holographic Entanglement Entropy
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.