Subregion Duality in the Entanglement Wedge
Daniel Harlow
Harvard University

June 15, 2016

1

Subregion Duality?

Subregion Duality?

In recent years people have considered the possibility of “subregion
duality” in AdS/CFT. Bousso/Liechenauer/Freivogel/Rosenhaus/Zukowski,
Czech/Karczmarek/Nogueira/van Raamsdonk

2

Subregion Duality?

Subregion Duality?

In recent years people have considered the possibility of “subregion
duality” in AdS/CFT. Bousso/Liechenauer/Freivogel/Rosenhaus/Zukowski,
Czech/Karczmarek/Nogueira/van Raamsdonk

Given complete information about a boundary subregion A, is there asubregion of the bulk about which we have complete information?

2

Subregion Duality?

Subregion Duality?

In recent years people have considered the possibility of “subregion
duality” in AdS/CFT. Bousso/Liechenauer/Freivogel/Rosenhaus/Zukowski,
Czech/Karczmarek/Nogueira/van Raamsdonk

Given complete information about a boundary subregion A, is there asubregion of the bulk about which we have complete information?
In this talk, I will present a theorem that answers this question in theaffirmative: given a boundary subregion A, we have full access to allbulk information in the entanglement wedge of A. Dong/Harlow/Wall2

Subregion Duality?

Subregion Duality?

In recent years people have considered the possibility of “subregion
duality” in AdS/CFT. Bousso/Liechenauer/Freivogel/Rosenhaus/Zukowski,
Czech/Karczmarek/Nogueira/van Raamsdonk

Given complete information about a boundary subregion A, is there asubregion of the bulk about which we have complete information?
In this talk, I will present a theorem that answers this question in theaffirmative: given a boundary subregion A, we have full access to allbulk information in the entanglement wedge of A. Dong/Harlow/Wall
I will make use of a recent result relating bulk and boundary relative
entropies, which I will review. Jafferis/Lewkowycz/Maldacena/Suh

2

Subregion Duality?

The first reason people suspected such a duality might exist was the
AdS/Rindler Reconstruction: Hamilton/Kabat/Lifschytz/Lowe,Morrison

D[A]

x
A

Z
φ(x)

CA

=

dX K̂ (x; X )O(X ) + O(1/N).
D[A]3

Subregion Duality?

The first reason people suspected such a duality might exist was the
AdS/Rindler Reconstruction: Hamilton/Kabat/Lifschytz/Lowe,Morrison

D[A]

x
A

Z
φ(x)

CA

=

dX K̂ (x; X )O(X ) + O(1/N).
D[A]Here the region CA is the causal wedge of A: Hubeny/Rangamani
CA ≡ j + [D(A)] ∩ j − [D(A)].
3

Subregion Duality?

The construction proceeds by perturbatively solving the bulk
equations of motion as operator equations in the CFT, allowing the
CFT to “simulate” bulk effective field theory in CA .

4

Subregion Duality?

The construction proceeds by perturbatively solving the bulk
equations of motion as operator equations in the CFT, allowing the
CFT to “simulate” bulk effective field theory in CA .
In fact the construction has so far only been worked out in detail forball-shaped regions, there are some delicate PDE questions involved
in deciding whether or not this method can work for arbitrary A.

4Subregion Duality?

The construction proceeds by perturbatively solving the bulk
equations of motion as operator equations in the CFT, allowing the
CFT to “simulate” bulk effective field theory in CA .
In fact the construction has so far only been worked out in detail forball-shaped regions, there are some delicate PDE questions involved
in deciding whether or not this method can work for arbitrary A.
I want to emphasize that it is understood how to compute the 1/N
corrections, and they depend on the details of the CFT (if we take A
to be the whole boundary then the subtlety just mentioned does notarise).

4

Subregion Duality?

The construction proceeds by perturbatively solving the bulk
equations of motion as operator equations in the CFT, allowing the
CFT to “simulate” bulk effective field theory in CA .
In fact the construction has so far only been worked out in detail forball-shaped regions, there are some delicate PDE questions involved
in deciding whether or not this method can work for arbitrary A.
I want to emphasize that it is understood how to compute the 1/N
corrections, and they depend on the details of the CFT (if we take A
to be the whole boundary then the subtlety just mentioned does notarise).
Indeed the construction will fail to reproduce bulk effective field theory
unless the CFT has the appropriate properties: large N factorization
and a large gap in the spectrum of single-trace primary operators.

4Subregion Duality?

The construction proceeds by perturbatively solving the bulk
equations of motion as operator equations in the CFT, allowing the
CFT to “simulate” bulk effective field theory in CA .
In fact the construction has so far only been worked out in detail forball-shaped regions, there are some delicate PDE questions involved
in deciding whether or not this method can work for arbitrary A.
I want to emphasize that it is understood how to compute the 1/N
corrections, and they depend on the details of the CFT (if we take A
to be the whole boundary then the subtlety just mentioned does notarise).
Indeed the construction will fail to reproduce bulk effective field theory
unless the CFT has the appropriate properties: large N factorization
and a large gap in the spectrum of single-trace primary operators.
This isn’t just kinematics!

4

Subregion Duality?

A)says
Is the causal wedge the whole story? The RT formula SA = Area(γ
4G
no!

5

Subregion Duality?

A)says
Is the causal wedge the whole story? The RT formula SA = Area(γ
4G
no!

A

A

A

A

Figure 6. A reconstruction phase transition? As we increase the region A, the extremalarea codimension two surface of smallest area whose boundary is @A, shown as the solid lines,changes discontinuously. Does this mean that we can now reconstruct the point in the centeras an operator on A?

operator algebra quantum error correction requires sub-AdS scale bulk locality. Thisis a special property of those CFTs that have local holographic duals, which we havehere reformulated in the language of quantum information theory.

5Subregion Duality?

A)says
Is the causal wedge the whole story? The RT formula SA = Area(γ
4G
no!

A

A

A

A

Figure 6. A reconstruction phase transition? As we increase the region A, the extremalarea codimension
two surfaceof smallest
areaboundary is beyond@A, shown theas thecausal
solid lines,
At leastin this case,ρA hasaccesstowhoseinformation
changes discontinuously. Does this mean that we can now reconstruct the point in the center
wedge!
as an operator on A?

operator algebra quantum error correction requires sub-AdS scale bulk locality. Thisis a special property of those CFTs that have local holographic duals, which we havehere reformulated in the language of quantum information theory.

5Subregion Duality?

A)says
Is the causal wedge the whole story? The RT formula SA = Area(γ
4G
no!

A

A

A

A

Figure 6. A reconstruction phase transition? As we increase the region A, the extremalarea codimension
two surfaceof smallest
areaboundary is beyond@A, shown theas thecausal
solid lines,
At leastin this case,ρA hasaccesstowhoseinformation
changes discontinuously. Does this mean that we can now reconstruct the point in the center
wedge!
as an operator on A?
But how far can we go?

operator algebra quantum error correction requires sub-AdS scale bulk locality. Thisis a special property of those CFTs that have local holographic duals, which we havehere reformulated in the language of quantum information theory.

5Subregion Duality?

Definition: say that Ξ is an achronal surface such that ∂Ξ = A ∪ γA
(shaded blue in the figure). Then the entanglement wedge of A isdefined as
EA ≡ d(Ξ).

6Subregion Duality?

Definition: say that Ξ is an achronal surface such that ∂Ξ = A ∪ γA
(shaded blue in the figure). Then the entanglement wedge of A isdefined as
EA ≡ d(Ξ).
Basic theorem: Wall, Headrick/Hubeny/Lawrence/Rangamani
CA ⊆ EA

6

Subregion Duality?

Definition: say that Ξ is an achronal surface such that ∂Ξ = A ∪ γA
(shaded blue in the figure). Then the entanglement wedge of A isdefined as
EA ≡ d(Ξ).
Basic theorem: Wall, Headrick/Hubeny/Lawrence/Rangamani
CA ⊆ EA
Strongest possible conjecture: we can in fact reconstruct bulk
operators throughout EA as CFT operators on A. This is the
Entanglement Wedge Reconstruction Conjecture. CKNvR,W,HHLR

6

Subregion Duality?

Definition: say that Ξ is an achronal surface such that ∂Ξ = A ∪ γA
(shaded blue in the figure). Then the entanglement wedge of A isdefined as
EA ≡ d(Ξ).
Basic theorem: Wall, Headrick/Hubeny/Lawrence/Rangamani
CA ⊆ EA
Strongest possible conjecture: we can in fact reconstruct bulk
operators throughout EA as CFT operators on A. This is the
Entanglement Wedge Reconstruction Conjecture. CKNvR,W,HHLR
Evidence for this conjecture was given in the context of tensor
network models by HArlow/Pastawski/Preskill/Yoshida, Hayden/Nezami/Qi/Thomas/Walter/Yang.

6Subregion Duality?

Definition: say that Ξ is an achronal surface such that ∂Ξ = A ∪ γA
(shaded blue in the figure). Then the entanglement wedge of A isdefined as
EA ≡ d(Ξ).
Basic theorem: Wall, Headrick/Hubeny/Lawrence/Rangamani
CA ⊆ EA
Strongest possible conjecture: we can in fact reconstruct bulk
operators throughout EA as CFT operators on A. This is the
Entanglement Wedge Reconstruction Conjecture. CKNvR,W,HHLR
Evidence for this conjecture was given in the context of tensor
network models by HArlow/Pastawski/Preskill/Yoshida, Hayden/Nezami/Qi/Thomas/Walter/Yang.
Today we will prove it!
6Quantum Error Correction

Quantum Error Correction

Recall that the HKLL procedure was perturbative in 1/N, and thislimits not just its precision but also its regime of validity.

7

Quantum Error Correction

Quantum Error Correction

Recall that the HKLL procedure was perturbative in 1/N, and thislimits not just its precision but also its regime of validity.
Indeed there must be states where any particular bulk operator
reconstruction fails! Almheiri/Dong/Harlow

7Quantum Error Correction

Quantum Error Correction

Recall that the HKLL procedure was perturbative in 1/N, and thislimits not just its precision but also its regime of validity.
Indeed there must be states where any particular bulk operator
reconstruction fails! Almheiri/Dong/Harlow
Roughly speaking, bulk operators can be swallowed behind black holehorizons, and in such states they do not need to have effective field
theory interpretations. This is the hack that AdS/CFT employs toallow a lower-dimensional theory to be equivalent to ahigher-dimensional one.

7

Quantum Error Correction

With Almheiri and Dong, we formalized this restriction as the idea ofa code subspace, within which bulk effective field theory gives a good
approximation to what is going on.

8Quantum Error Correction

With Almheiri and Dong, we formalized this restriction as the idea ofa code subspace, within which bulk effective field theory gives a good
approximation to what is going on.
One good example is obtained by taking the set of states in N = 4
SYM theory on S3 × R whose energies are at most N 1/4 , and thentaking the image of these states under conformal transformations: a“no black hole subspace”.

8Quantum Error Correction

With Almheiri and Dong, we formalized this restriction as the idea ofa code subspace, within which bulk effective field theory gives a good
approximation to what is going on.
One good example is obtained by taking the set of states in N = 4
SYM theory on S3 × R whose energies are at most N 1/4 , and thentaking the image of these states under conformal transformations: a“no black hole subspace”.
More general subspaces are also possible, but we are then only able toreconstruct bulk fields which are at best “not too far” behind blackhole horizons.

8Quantum Error Correction

Using this language, we can formalize the entanglement wedge
reconstruction conjecture:9

Quantum Error Correction

Using this language, we can formalize the entanglement wedge
reconstruction conjecture:
Say that HCFT = HA ⊗ HA is the Hilbert space of a holographic
CFT, with a code subspace Hcode = Ha ⊗ Ha .

9Quantum Error Correction

Using this language, we can formalize the entanglement wedge
reconstruction conjecture:
Say that HCFT = HA ⊗ HA is the Hilbert space of a holographic
CFT, with a code subspace Hcode = Ha ⊗ Ha .
Then for any operator Oa on Ha we have an operator OA on HA suche ∈ Hcode we have:that for all |ψie = Oa |ψie
OA |ψie = O † |ψieO † |ψiAa9

Quantum Error Correction

In proving this conjecture we can make use of a theorem from ADH,
which shows that such an OA can exist if and only if:e hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.

(1)

10

Quantum Error Correction

In proving this conjecture we can make use of a theorem from ADH,
which shows that such an OA can exist if and only if:e hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.

(1)

The proof of this theorem is constructive, albeit somewhat uselessly
so! I’ll sketch it at the end if we have time.

10Quantum Error Correction

In proving this conjecture we can make use of a theorem from ADH,
which shows that such an OA can exist if and only if:e hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.

(1)

The proof of this theorem is constructive, albeit somewhat uselessly
so! I’ll sketch it at the end if we have time.
Claim: (1) follows from the quantum corrected RT formula, which inthis language says that for all ρ on Hcode , we have:
Ryu/Takayanagi,Faulkner/Lewkowycz/Maldacena

S(ρA ) = Tr(ρa Aloc ) + S(ρa ).

10

Quantum Error Correction

In proving this conjecture we can make use of a theorem from ADH,
which shows that such an OA can exist if and only if:e hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.

(1)

The proof of this theorem is constructive, albeit somewhat uselessly
so! I’ll sketch it at the end if we have time.
Claim: (1) follows from the quantum corrected RT formula, which inthis language says that for all ρ on Hcode , we have:
Ryu/Takayanagi,Faulkner/Lewkowycz/Maldacena

S(ρA ) = Tr(ρa Aloc ) + S(ρa ).
Here Aloc is some operator integrated on γA , which to leading orderA)
in G is Area(γ.
4G
10Quantum Error Correction

You might worry about the assumption of bulk factorization, this isindeed subtle, see Donnelly,Casini/Huerta/Rosabal, but it seems likely that
including enough UV degrees of freedom near γA we can justify it
Harlow. And actually we don’t need to assume it if we work algebraically.

11

Quantum Error Correction

You might worry about the assumption of bulk factorization, this isindeed subtle, see Donnelly,Casini/Huerta/Rosabal, but it seems likely that
including enough UV degrees of freedom near γA we can justify it
Harlow. And actually we don’t need to assume it if we work algebraically.
For simplicity I will only make the argument to O(G 0 ): using a
conjectural extension of RT to higher orders by Engelhardt/Wall, we werealso able to give an argument for entanglement wedge reconstruction
at higher orders in G . There are some subtleties in this extension
which are not yet ironed out, although see Dong/Lewkowycz.

11Quantum Error Correction

You might worry about the assumption of bulk factorization, this isindeed subtle, see Donnelly,Casini/Huerta/Rosabal, but it seems likely that
including enough UV degrees of freedom near γA we can justify it
Harlow. And actually we don’t need to assume it if we work algebraically.
For simplicity I will only make the argument to O(G 0 ): using a
conjectural extension of RT to higher orders by Engelhardt/Wall, we werealso able to give an argument for entanglement wedge reconstruction
at higher orders in G . There are some subtleties in this extension
which are not yet ironed out, although see Dong/Lewkowycz.
In fact this theorem also has a converse: any quantum error
correcting code obeys a version of the RT formula! Harlow In general it
involves the algebraic definition of entropy, and gives a “completely
boundary” picture of how the formula works, to be contrasted withthe “completely bulk” derivation of Lewkowycz/Maldacena,Faulkner/Lewkowycz/Maldacena.

11

JLMS

Relative entropy in the bulk and boundary

I’ll first need a result from JLMS, which I’ll derive for you here.

12

JLMS

Relative entropy in the bulk and boundary

I’ll first need a result from JLMS, which I’ll derive for you here.
Definitions:S(ρ) = −Tr(ρ log ρ)Kρ = − log ρS(ρ|σ) = Tr(ρ log ρ) − Tr(ρ log σ) = −S(ρ) + TrρKσ .

12

JLMS

Relative entropy in the bulk and boundary

I’ll first need a result from JLMS, which I’ll derive for you here.
Definitions:S(ρ) = −Tr(ρ log ρ)Kρ = − log ρS(ρ|σ) = Tr(ρ log ρ) − Tr(ρ log σ) = −S(ρ) + TrρKσ .
These are related by the “first law of entanglement”:
S(ρ + δρ) − S(ρ) = Tr(δρKσ ) + O(δρ2 ).

12

JLMS

We can apply this “first law” to both sides of the RT formula
S(σA + δρA ) = Tr ((σa + δρa )Aloc ) + S(σa + δρa ),

13

JLMS

We can apply this “first law” to both sides of the RT formula
S(σA + δρA ) = Tr ((σa + δρa )Aloc ) + S(σa + δρa ),to find
Tr (δρA KσA ) = Tr (δρa (Aloc + Kσa )) .

13

JLMS

We can apply this “first law” to both sides of the RT formula
S(σA + δρA ) = Tr ((σa + δρa )Aloc ) + S(σa + δρa ),to find
Tr (δρA KσA ) = Tr (δρa (Aloc + Kσa )) .
Both sides are linear in δρ, so we can integrate to find
Tr (ρA KσA ) = Tr (ρa (Aloc + Kσa ))∀ρ, σ on Hcode .

13

JLMS

We can apply this “first law” to both sides of the RT formula
S(σA + δρA ) = Tr ((σa + δρa )Aloc ) + S(σa + δρa ),to find
Tr (δρA KσA ) = Tr (δρa (Aloc + Kσa )) .
Both sides are linear in δρ, so we can integrate to find
Tr (ρA KσA ) = Tr (ρa (Aloc + Kσa ))∀ρ, σ on Hcode .

This then implies that
S(ρA |σA ) = −S(ρA ) + Tr (ρA KσA )= −S(ρa ) − Tr(ρa Aloc ) + Tr(ρa Aloc ) + Tr(ρa Kσa )= S(ρa |σa ).

13

JLMS

We can apply this “first law” to both sides of the RT formula
S(σA + δρA ) = Tr ((σa + δρa )Aloc ) + S(σa + δρa ),to find
Tr (δρA KσA ) = Tr (δρa (Aloc + Kσa )) .
Both sides are linear in δρ, so we can integrate to find
Tr (ρA KσA ) = Tr (ρa (Aloc + Kσa ))∀ρ, σ on Hcode .

This then implies that
S(ρA |σA ) = −S(ρA ) + Tr (ρA KσA )= −S(ρa ) − Tr(ρa Aloc ) + Tr(ρa Aloc ) + Tr(ρa Kσa )= S(ρa |σa ).

So in particular, ρa = σa ⇔ ρA = σA .
13

A Reconstruction Theorem

A Reconstruction Theorem
Now to finish the argument, we just need to show that indeed we havee ∀Oa , hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.

14

A Reconstruction Theorem

A Reconstruction Theorem
Now to finish the argument, we just need to show that indeed we havee ∀Oa , hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.
Without loss of generality we can take Oa to be hermitian. Then the statesee|ψ(λ)i≡ e iλOa |ψihave identical ρa .

14

A Reconstruction Theorem

A Reconstruction Theorem
Now to finish the argument, we just need to show that indeed we havee ∀Oa , hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.
Without loss of generality we can take Oa to be hermitian. Then the statesee|ψ(λ)i≡ e iλOa |ψihave identical ρa .
But by JLMS this means they must have identical ρA , so apparentlye = hψ|Xe |ψi.
e −iλOa X e iλOa |ψi
e
hψ|e
A
A

14

A Reconstruction Theorem

A Reconstruction Theorem
Now to finish the argument, we just need to show that indeed we havee ∀Oa , hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.
Without loss of generality we can take Oa to be hermitian. Then the statesee|ψ(λ)i≡ e iλOa |ψihave identical ρa .
But by JLMS this means they must have identical ρA , so apparentlye = hψ|Xe |ψi.
e −iλOa X e iλOa |ψi
e
hψ|e
A
A
Linearizing in λ, we indeed find
e
e
hψ|[X
A , Oa ]|ψi = 0.
14

A Reconstruction Theorem

A Reconstruction Theorem
Now to finish the argument, we just need to show that indeed we havee ∀Oa , hψ|[Xee∀XA , ∀|ψi,
A , Oa ]|ψi = 0.
Without loss of generality we can take Oa to be hermitian. Then the statesee|ψ(λ)i≡ e iλOa |ψihave identical ρa .
But by JLMS this means they must have identical ρA , so apparentlye = hψ|Xe |ψi.
e −iλOa X e iλOa |ψi
e
hψ|e
A
A
Linearizing in λ, we indeed find
e
e
hψ|[X
A , Oa ]|ψi = 0.
Woohoo!
14

A Reconstruction Theorem

What else to do?
Understand the higher-order corrections better, specifically insofar as
they relate to backreaction.

15

A Reconstruction Theorem

What else to do?
Understand the higher-order corrections better, specifically insofar as
they relate to backreaction.
Algebraic reformulation (done!)15

A Reconstruction Theorem

What else to do?
Understand the higher-order corrections better, specifically insofar as
they relate to backreaction.
Algebraic reformulation (done!)
This improves on HKLL both in scope (the whole entanglement
wedge), and in that it avoids the nasty PDE issues I mentioned. But
HKLL gives a bulk picture of what is going on, which is sorely lacking
here.

15

A Reconstruction Theorem

16Appendix

Proof of the ADH theorem
Given an operator O on Hcode , how do we know thate hψ|[Xee∀XA , ∀|ψi,
A , O]|ψi = 0
is sufficient (it is clearly necessary) for the existence of an OA ?

17

Appendix

Proof of the ADH theorem
Given an operator O on Hcode , how do we know thate hψ|[Xee∀XA , ∀|ψi,
A , O]|ψi = 0
is sufficient (it is clearly necessary) for the existence of an OA ?
The basic idea is to consider the stateX|φi ∝|iiR |eiiAA .
i17

Appendix

Proof of the ADH theorem
Given an operator O on Hcode , how do we know thate hψ|[Xee∀XA , ∀|ψi,
A , O]|ψi = 0
is sufficient (it is clearly necessary) for the existence of an OA ?
The basic idea is to consider the stateX|φi ∝|iiR |eiiAA .
i

We can clearly mirror the operator O onto R, but can we then mirror itback onto A?

17

Appendix

Proof of the ADH theorem
Given an operator O on Hcode , how do we know thate hψ|[Xee∀XA , ∀|ψi,
A , O]|ψi = 0
is sufficient (it is clearly necessary) for the existence of an OA ?
The basic idea is to consider the stateX|φi ∝|iiR |eiiAA .
i

We can clearly mirror the operator O onto R, but can we then mirror itback onto A?
What we need for this to work is that OR preserves the Schmidt basis of|φi if we decompose it as RA and A, or in other words for [OR , ρRA ] = 0.
17

Appendix

Proof of the ADH theorem
Given an operator O on Hcode , how do we know thate hψ|[Xee∀XA , ∀|ψi,
A , O]|ψi = 0
is sufficient (it is clearly necessary) for the existence of an OA ?
The basic idea is to consider the stateX|φi ∝|iiR |eiiAA .
i

We can clearly mirror the operator O onto R, but can we then mirror itback onto A?
What we need for this to work is that OR preserves the Schmidt basis of|φi if we decompose it as RA and A, or in other words for [OR , ρRA ] = 0.
But this is precisely what the commutator condition ensures!
17