Gravity and Entanglement Josh June 27, 2018 This seminar is a very - - PowerPoint PPT Presentation

Gravity and Entanglement

Josh June 27, 2018

This seminar is a very quick introduction to the role played by entanglement in quantum gravity and holography. If you want to learn more, the following three sets of lecture notes are highly recommended:

• Daniel Harlow. “Jerusalem Lectures on Black Holes and Quantum

Information”. In: Rev. Mod. Phys. 88 (2016), p. 015002. doi: 10.1103/RevModPhys.88.015002. arXiv: 1409.1231 [hep-th]

• Mark Van Raamsdonk. “Lectures on Gravity and Entanglement”. In:

Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, USA, June 1-26, 2015. 2017, pp. 297–351. doi: 10.1142/9789813149441_0005. arXiv: 1609.00026 [hep-th]

• Daniel Harlow. “TASI Lectures on the Emergence of the Bulk in

AdS/CFT”. In: (2018). arXiv: 1802.01040 [hep-th]

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• 1. Entanglement
• 2. Entanglement entropy
• 3. Entanglement entropy in QFT
• 4. Holographic entanglement entropy
• 5. Geometry from entanglement

◀ ▶ 2/24

• 1. Entanglement
• 2. Entanglement entropy
• 3. Entanglement entropy in QFT
• 4. Holographic entanglement entropy
• 5. Geometry from entanglement

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• 1. Entanglement
• 2. Entanglement entropy
• 3. Entanglement entropy in QFT
• 4. Holographic entanglement entropy
• 5. Geometry from entanglement

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• 1. Entanglement
• 2. Entanglement entropy
• 3. Entanglement entropy in QFT
• 4. Holographic entanglement entropy
• 5. Geometry from entanglement

◀ ▶ 2/24

• 1. Entanglement
• 2. Entanglement entropy
• 3. Entanglement entropy in QFT
• 4. Holographic entanglement entropy
• 5. Geometry from entanglement

◀ ▶ 2/24

Entanglement

Let HA, HB be two Hilbert spaces, with orthonormal bases {|i⟩A}, {|j⟩B} respectively, where i, j = 1, 2, . . . . The states i j i A j B comprise an orthonormal basis of the product Hilbert space

A B.

Elements of are joint states.

◀ ▶ 3/24

Let HA, HB be two Hilbert spaces, with orthonormal bases {|i⟩A}, {|j⟩B} respectively, where i, j = 1, 2, . . . . The states |i, j⟩ = |i⟩A ⊗ |j⟩B comprise an orthonormal basis of the product Hilbert space H = HA ⊗ HB. Elements of are joint states.

◀ ▶ 3/24

Let HA, HB be two Hilbert spaces, with orthonormal bases {|i⟩A}, {|j⟩B} respectively, where i, j = 1, 2, . . . . The states |i, j⟩ = |i⟩A ⊗ |j⟩B comprise an orthonormal basis of the product Hilbert space H = HA ⊗ HB. Elements of H are joint states.

◀ ▶ 3/24

The product structure of H allows for the presence of entanglement in joint states. If a state can be written as the product of two states

A A B B, i.e. A B , then there is no

entanglement. Otherwise, it is an entangled state. For example, 1 1 is not entangled, while

1 2

1 2 2 1 is.

◀ ▶ 4/24

The product structure of H allows for the presence of entanglement in joint states. If a state |ψ⟩ ∈ H can be written as the product of two states |ψA⟩ ∈ HA, |ψB⟩ ∈ HB, i.e. |ψ⟩ = |ψA⟩ ⊗ |ψB⟩, then there is no entanglement. Otherwise, it is an entangled state. For example, 1 1 is not entangled, while

1 2

1 2 2 1 is.

◀ ▶ 4/24

The product structure of H allows for the presence of entanglement in joint states. If a state |ψ⟩ ∈ H can be written as the product of two states |ψA⟩ ∈ HA, |ψB⟩ ∈ HB, i.e. |ψ⟩ = |ψA⟩ ⊗ |ψB⟩, then there is no entanglement. Otherwise, it is an entangled state. For example, 1 1 is not entangled, while

1 2

1 2 2 1 is.

◀ ▶ 4/24

The product structure of H allows for the presence of entanglement in joint states. If a state |ψ⟩ ∈ H can be written as the product of two states |ψA⟩ ∈ HA, |ψB⟩ ∈ HB, i.e. |ψ⟩ = |ψA⟩ ⊗ |ψB⟩, then there is no entanglement. Otherwise, it is an entangled state. For example, |1, 1⟩ is not entangled, while

1 √ 2 (|1, 2⟩ + |2, 1⟩) is. ◀ ▶ 4/24

This defjnition of entanglement is qualitative. Either a state is entangled, or it isn’t. It would be desirable to have a quantitative measure of entanglement, i.e. one that can tell us the degree to which a state is entangled.

◀ ▶ 5/24

This defjnition of entanglement is qualitative. Either a state is entangled, or it isn’t. It would be desirable to have a quantitative measure of entanglement, i.e. one that can tell us the degree to which a state is entangled.

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Entanglement entropy

When a state is entangled, knowledge about the part of the state in HA provides information about the part of the state in HB. This only happens for entangled states. Furthermore, the amount

• f information one can obtain is dependent on the nature of the

entanglement. We should therefore be able to quantify entanglement in terms of the amount of information available. Information is most naturally measured in terms of entropy.

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When a state is entangled, knowledge about the part of the state in HA provides information about the part of the state in HB. This only happens for entangled states. Furthermore, the amount

• f information one can obtain is dependent on the nature of the

entanglement. We should therefore be able to quantify entanglement in terms of the amount of information available. Information is most naturally measured in terms of entropy.

◀ ▶ 6/24

When a state is entangled, knowledge about the part of the state in HA provides information about the part of the state in HB. This only happens for entangled states. Furthermore, the amount

• f information one can obtain is dependent on the nature of the

entanglement. We should therefore be able to quantify entanglement in terms of the amount of information available. Information is most naturally measured in terms of entropy.

◀ ▶ 6/24

To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. A density matrix on a Hilbert space is an self-adjoint, positive semi-defjnite operator satisfying 1. Density matrices should be thought of as the quantum generalisation of a classical set of probabilities pi satisfying

i pi

1, pi 0. Density matrices can account for both quantum uncertainty and statistical uncertainty.

◀ ▶ 7/24

To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. A density matrix on a Hilbert space H is an self-adjoint, positive semi-defjnite operator ρ ∈ L(H) satisfying tr ρ = 1. Density matrices should be thought of as the quantum generalisation of a classical set of probabilities pi satisfying

i pi

1, pi 0. Density matrices can account for both quantum uncertainty and statistical uncertainty.

◀ ▶ 7/24

To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. A density matrix on a Hilbert space H is an self-adjoint, positive semi-defjnite operator ρ ∈ L(H) satisfying tr ρ = 1. Density matrices should be thought of as the quantum generalisation of a classical set of probabilities pi satisfying ∑

i pi = 1, pi ≥ 0.

Density matrices can account for both quantum uncertainty and statistical uncertainty.

◀ ▶ 7/24

To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. A density matrix on a Hilbert space H is an self-adjoint, positive semi-defjnite operator ρ ∈ L(H) satisfying tr ρ = 1. Density matrices should be thought of as the quantum generalisation of a classical set of probabilities pi satisfying ∑

i pi = 1, pi ≥ 0.

Density matrices can account for both quantum uncertainty and statistical uncertainty.

◀ ▶ 7/24

Whenever there is statistical uncertainty in a state, measuring that state will give us some information about the real world. In terms of density matrices, the amount of information we can expect to obtain from such a measurement is given by the von Neumann entropy S (1) This is the quantum mechanical generalisation of the classical Shannon entropy S

i

pi pi (2)

◀ ▶ 8/24

Whenever there is statistical uncertainty in a state, measuring that state will give us some information about the real world. In terms of density matrices, the amount of information we can expect to obtain from such a measurement is given by the von Neumann entropy S = − tr(ρ log ρ). (1) This is the quantum mechanical generalisation of the classical Shannon entropy S

i

pi pi (2)

◀ ▶ 8/24

Whenever there is statistical uncertainty in a state, measuring that state will give us some information about the real world. In terms of density matrices, the amount of information we can expect to obtain from such a measurement is given by the von Neumann entropy S = − tr(ρ log ρ). (1) This is the quantum mechanical generalisation of the classical Shannon entropy S = − ∑

i

pi log pi. (2)

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For example, given a normalised state |ψ⟩ ∈ H, one can construct an associated density matrix ρ = |ψ⟩ ⟨ψ|. Density matrices constructed in this way are pure, meaning there is no statistical uncertainty in the quantum state. This is consistent with the fact that their corresponding von Neumann entropies are zero. On the other hand, consider for example the density matrix

1 2 1 1 2 2 , where 1 2 are orthonormal.

This is a mixed density matrix. It represents a state which could be either

1 or 2 , each with probability half. The von Neumann

entropy is S 2.

◀ ▶ 9/24

For example, given a normalised state |ψ⟩ ∈ H, one can construct an associated density matrix ρ = |ψ⟩ ⟨ψ|. Density matrices constructed in this way are pure, meaning there is no statistical uncertainty in the quantum state. This is consistent with the fact that their corresponding von Neumann entropies are zero. On the other hand, consider for example the density matrix

1 2 1 1 2 2 , where 1 2 are orthonormal.

This is a mixed density matrix. It represents a state which could be either

1 or 2 , each with probability half. The von Neumann

entropy is S 2.

◀ ▶ 9/24

For example, given a normalised state |ψ⟩ ∈ H, one can construct an associated density matrix ρ = |ψ⟩ ⟨ψ|. Density matrices constructed in this way are pure, meaning there is no statistical uncertainty in the quantum state. This is consistent with the fact that their corresponding von Neumann entropies are zero. On the other hand, consider for example the density matrix ρ = 1

2(|ψ1⟩ ⟨ψ1| + |ψ2⟩ ⟨ψ2|), where |ψ1⟩ , |ψ2⟩ are orthonormal.

This is a mixed density matrix. It represents a state which could be either

1 or 2 , each with probability half. The von Neumann

entropy is S 2.

◀ ▶ 9/24

For example, given a normalised state |ψ⟩ ∈ H, one can construct an associated density matrix ρ = |ψ⟩ ⟨ψ|. Density matrices constructed in this way are pure, meaning there is no statistical uncertainty in the quantum state. This is consistent with the fact that their corresponding von Neumann entropies are zero. On the other hand, consider for example the density matrix ρ = 1

2(|ψ1⟩ ⟨ψ1| + |ψ2⟩ ⟨ψ2|), where |ψ1⟩ , |ψ2⟩ are orthonormal.

This is a mixed density matrix. It represents a state which could be either |ψ1⟩ or |ψ2⟩, each with probability half. The von Neumann entropy is S = log 2.

◀ ▶ 9/24

We need one more piece to defjne entanglement entropy. Given a density matrix

• n the product Hilbert space

A B, we can defjne the ‘reduced density matrix’ A on

the

A factor by taking the partial trace of

• ver the

B factor.

The partial trace is defjned through

A B k

k B k B (3) and the components of the reduced density matrix are i A

A j A k

i k j k (4)

◀ ▶ 10/24

We need one more piece to defjne entanglement entropy. Given a density matrix ρ on the product Hilbert space H = HA ⊗ HB, we can defjne the ‘reduced density matrix’ ρA on the HA factor by taking the partial trace of ρ over the HB factor. The partial trace is defjned through

A B k

k B k B (3) and the components of the reduced density matrix are i A

A j A k

i k j k (4)

◀ ▶ 10/24

We need one more piece to defjne entanglement entropy. Given a density matrix ρ on the product Hilbert space H = HA ⊗ HB, we can defjne the ‘reduced density matrix’ ρA on the HA factor by taking the partial trace of ρ over the HB factor. The partial trace is defjned through ρA = trB ρ = ∑

k

⟨k|B ρ |k⟩B , (3) and the components of the reduced density matrix are i A

A j A k

i k j k (4)

◀ ▶ 10/24

We need one more piece to defjne entanglement entropy. Given a density matrix ρ on the product Hilbert space H = HA ⊗ HB, we can defjne the ‘reduced density matrix’ ρA on the HA factor by taking the partial trace of ρ over the HB factor. The partial trace is defjned through ρA = trB ρ = ∑

k

⟨k|B ρ |k⟩B , (3) and the components of the reduced density matrix are ⟨i|A ρA |j⟩A = ∑

k

⟨i, k| ρ |j, k⟩ . (4)

◀ ▶ 10/24

The reduced density matrix ρA is the density matrix we would use if we were completely ignorant of the state in B. In general, even if the original density matrix is pure, the reduced density matrix

A may be mixed. In fact, this occurs exactly when

contains entanglement. The entanglement entropy SA of a density matrix in the factor

A is defjned as the von Neumann entropy of A.

SA

A A A

(5)

◀ ▶ 11/24

The reduced density matrix ρA is the density matrix we would use if we were completely ignorant of the state in B. In general, even if the original density matrix ρ is pure, the reduced density matrix ρA may be mixed. In fact, this occurs exactly when ρ contains entanglement. The entanglement entropy SA of a density matrix in the factor

A is defjned as the von Neumann entropy of A.

SA

A A A

(5)

◀ ▶ 11/24

The reduced density matrix ρA is the density matrix we would use if we were completely ignorant of the state in B. In general, even if the original density matrix ρ is pure, the reduced density matrix ρA may be mixed. In fact, this occurs exactly when ρ contains entanglement. The entanglement entropy SA of a density matrix ρ in the factor HA is defjned as the von Neumann entropy of ρA. SA = − trA ρA log ρA (5)

◀ ▶ 11/24

Here are some examples:

• , where

1 1 . Then

A

1 A 1 A, and SA 0.

• , where

1 2 1 2

2 1 . Then

A 1 2 1 A 1 A

2 A 2 A , and SA 2.

• 1

2 1 2

1 2 2 1 2 1 . Then

A 1 2 1 A 1 A

2 A 2 A , and SA 2.

◀ ▶ 12/24

Here are some examples:

• ρ = |ψ⟩ ⟨ψ|, where |ψ⟩ = |1, 1⟩. Then ρA = |1⟩A ⟨1|A, and

SA = 0.

• , where

1 2 1 2

2 1 . Then

A 1 2 1 A 1 A

2 A 2 A , and SA 2.

• 1

2 1 2

1 2 2 1 2 1 . Then

A 1 2 1 A 1 A

2 A 2 A , and SA 2.

◀ ▶ 12/24

Here are some examples:

• ρ = |ψ⟩ ⟨ψ|, where |ψ⟩ = |1, 1⟩. Then ρA = |1⟩A ⟨1|A, and

SA = 0.

• ρ = |ψ⟩ ⟨ψ|, where |ψ⟩ =

1 √ 2(|1, 2⟩ + |2, 1⟩). Then

ρA = 1

2(|1⟩A ⟨1|A + |2⟩A ⟨2|A), and SA = log 2.

• 1

2 1 2

1 2 2 1 2 1 . Then

A 1 2 1 A 1 A

2 A 2 A , and SA 2.

◀ ▶ 12/24

Here are some examples:

• ρ = |ψ⟩ ⟨ψ|, where |ψ⟩ = |1, 1⟩. Then ρA = |1⟩A ⟨1|A, and

SA = 0.

• ρ = |ψ⟩ ⟨ψ|, where |ψ⟩ =

1 √ 2(|1, 2⟩ + |2, 1⟩). Then

ρA = 1

2(|1⟩A ⟨1|A + |2⟩A ⟨2|A), and SA = log 2.

• ρ = 1

2(|1, 2⟩ ⟨1, 2| + |2, 1⟩ ⟨2, 1|). Then

ρA = 1

2(|1⟩A ⟨1|A + |2⟩A ⟨2|A), and SA = log 2. ◀ ▶ 12/24

Entanglement entropy in QFT

Consider now a general QFT. The Hilbert space H of such a theory consists of states living on a spatial surface Σ. Σ A B Suppose we decompose into two complementary subregions A and B A. Under reasonable locality assumptions, the full Hilbert space decomposes as

A B, where A contains the degrees of

freedom living in A, and

B contains the degrees of freedom living

in B. (Gauge symmetries can make this more complicated.)

◀ ▶ 13/24

Consider now a general QFT. The Hilbert space H of such a theory consists of states living on a spatial surface Σ. Σ A B Suppose we decompose Σ into two complementary subregions A and B = A. Under reasonable locality assumptions, the full Hilbert space decomposes as

A B, where A contains the degrees of

freedom living in A, and

B contains the degrees of freedom living

in B. (Gauge symmetries can make this more complicated.)

◀ ▶ 13/24

Consider now a general QFT. The Hilbert space H of such a theory consists of states living on a spatial surface Σ. Σ A B Suppose we decompose Σ into two complementary subregions A and B = A. Under reasonable locality assumptions, the full Hilbert space decomposes as H = HA ⊗ HB, where HA contains the degrees of freedom living in A, and HB contains the degrees of freedom living in B. (Gauge symmetries can make this more complicated.)

◀ ▶ 13/24

Consider now a general QFT. The Hilbert space H of such a theory consists of states living on a spatial surface Σ. Σ A B Suppose we decompose Σ into two complementary subregions A and B = A. Under reasonable locality assumptions, the full Hilbert space decomposes as H = HA ⊗ HB, where HA contains the degrees of freedom living in A, and HB contains the degrees of freedom living in B. (Gauge symmetries can make this more complicated.)

◀ ▶ 13/24

If we are given a density matrix ρ on the full Hilbert space H, we can as before compute the reduced density matrix ρA on the factor HA, and then fjnd the entanglement entropy SA = − trA ρA log ρA. We can similarly associate an entanglement entropy SA with every possible subregion A . SA tells us how much information we can expect to learn about the state outside of A, after measuring the state inside of A.

◀ ▶ 14/24

If we are given a density matrix ρ on the full Hilbert space H, we can as before compute the reduced density matrix ρA on the factor HA, and then fjnd the entanglement entropy SA = − trA ρA log ρA. We can similarly associate an entanglement entropy SA with every possible subregion A ⊂ Σ. SA tells us how much information we can expect to learn about the state outside of A, after measuring the state inside of A.

◀ ▶ 14/24

If we are given a density matrix ρ on the full Hilbert space H, we can as before compute the reduced density matrix ρA on the factor HA, and then fjnd the entanglement entropy SA = − trA ρA log ρA. We can similarly associate an entanglement entropy SA with every possible subregion A ⊂ Σ. SA tells us how much information we can expect to learn about the state outside of A, after measuring the state inside of A.

◀ ▶ 14/24

The calculation of SA is generally hard, but there are some special cases for which it has been done. For example, in a 2D CFT with central charge c, consider the vacuum state 0 . Let A be a subregion of length L. Then SA c 3 L (6) where 1 is a UV cutofg. It is characteristic of local QFTs that the vacuum state is highly entangled.

◀ ▶ 15/24

The calculation of SA is generally hard, but there are some special cases for which it has been done. For example, in a 2D CFT with central charge c, consider the vacuum state ρ = |0⟩ ⟨0|. Let A be a subregion of length L. Then SA = c 3 log (L ϵ ) , (6) where 1

ϵ is a UV cutofg.

It is characteristic of local QFTs that the vacuum state is highly entangled.

◀ ▶ 15/24

The calculation of SA is generally hard, but there are some special cases for which it has been done. For example, in a 2D CFT with central charge c, consider the vacuum state ρ = |0⟩ ⟨0|. Let A be a subregion of length L. Then SA = c 3 log (L ϵ ) , (6) where 1

ϵ is a UV cutofg.

It is characteristic of local QFTs that the vacuum state is highly entangled.

◀ ▶ 15/24

Holographic entanglement entropy

Entanglement entropy has very interesting properties in quantum gravity. The most well-known of these is given by the Ryu-Takayanagi conjecture in the context of AdS/CFT. All we will need to know about AdS/CFT is that it provides a

• ne-to-one mapping between degrees of freedom living in a CFT
• n the boundary of a spacetime, and those living in the

gravitational bulk.

◀ ▶ 16/24

Entanglement entropy has very interesting properties in quantum gravity. The most well-known of these is given by the Ryu-Takayanagi conjecture in the context of AdS/CFT. All we will need to know about AdS/CFT is that it provides a

• ne-to-one mapping between degrees of freedom living in a CFT
• n the boundary of a spacetime, and those living in the

gravitational bulk.

◀ ▶ 16/24

Entanglement entropy has very interesting properties in quantum gravity. The most well-known of these is given by the Ryu-Takayanagi conjecture in the context of AdS/CFT. All we will need to know about AdS/CFT is that it provides a

• ne-to-one mapping between degrees of freedom living in a CFT
• n the boundary of a spacetime, and those living in the

gravitational bulk.

◀ ▶ 16/24

ρ A A A density matrix ρ in AdS/CFT can be interpreted as living on the boundary. Let A be a subregion of the boundary, and let A be its complement. The Hilbert space of the CFT decomposes into two factors containing the degrees of freedom present in A and A respectively. Given a density matrix for the whole system,

• ne can thus obtain a reduced density matrix

for the degrees of freedom in A by taking the partial trace over

A: A A ◀ ▶ 17/24

ρ A A A density matrix ρ in AdS/CFT can be interpreted as living on the boundary. Let A be a subregion of the boundary, and let A be its complement. The Hilbert space of the CFT decomposes into two factors containing the degrees of freedom present in A and A respectively. Given a density matrix for the whole system,

• ne can thus obtain a reduced density matrix

for the degrees of freedom in A by taking the partial trace over

A: A A ◀ ▶ 17/24

ρ A A A density matrix ρ in AdS/CFT can be interpreted as living on the boundary. Let A be a subregion of the boundary, and let A be its complement. The Hilbert space of the CFT decomposes into two factors containing the degrees of freedom present in A and A respectively. Given a density matrix for the whole system,

• ne can thus obtain a reduced density matrix

for the degrees of freedom in A by taking the partial trace over

A: A A ◀ ▶ 17/24

ρ A A A density matrix ρ in AdS/CFT can be interpreted as living on the boundary. Let A be a subregion of the boundary, and let A be its complement. The Hilbert space of the CFT decomposes into two factors containing the degrees of freedom present in A and A respectively. Given a density matrix ρ for the whole system,

• ne can thus obtain a reduced density matrix

for the degrees of freedom in A by taking the partial trace over HA: ρA = trA ρ.

◀ ▶ 17/24

A A ρ T The claim of Ryu and Takayanagi is that (at leading order in ℏ) the associated entanglement entropy should be given by one quarter of the area of a certain surface in the bulk. Precisely, let T be a surface in the bulk which shares its boundary with A, T

• A. Let

Tmin be the surface with this property whose area is minimal, and let

min be the area of

Tmin. Then the entanglement entropy at leading

• rder is given by

SA

min

4

◀ ▶ 18/24

A A ρ T The claim of Ryu and Takayanagi is that (at leading order in ℏ) the associated entanglement entropy should be given by one quarter of the area of a certain surface in the bulk. Precisely, let T be a surface in the bulk which shares its boundary with A, ∂T = ∂A. Let Tmin be the surface with this property whose area is minimal, and let Amin be the area of Tmin. Then the entanglement entropy at leading

• rder is given by

SA

min

4

◀ ▶ 18/24

A A ρ T The claim of Ryu and Takayanagi is that (at leading order in ℏ) the associated entanglement entropy should be given by one quarter of the area of a certain surface in the bulk. Precisely, let T be a surface in the bulk which shares its boundary with A, ∂T = ∂A. Let Tmin be the surface with this property whose area is minimal, and let Amin be the area of Tmin. Then the entanglement entropy at leading

• rder is given by

SA = Amin 4 .

◀ ▶ 18/24

SA = Amin 4 . Many checks of the Ryu-Takayanagi conjecture have been carried

• ut, and it has always been found to be reliable.

There exist also some non-rigorous proofs of its validity. The Ryu-Takayanagi conjecture applies to holographic theories where the bulk theory of gravity is general relativity, but there are generalisations to higher derivative theories of gravity. In these cases the area is replaced by some other geometric quantity.

◀ ▶ 19/24

SA = Amin 4 . Many checks of the Ryu-Takayanagi conjecture have been carried

• ut, and it has always been found to be reliable.

There exist also some non-rigorous proofs of its validity. The Ryu-Takayanagi conjecture applies to holographic theories where the bulk theory of gravity is general relativity, but there are generalisations to higher derivative theories of gravity. In these cases the area is replaced by some other geometric quantity.

◀ ▶ 19/24

SA = Amin 4 . Many checks of the Ryu-Takayanagi conjecture have been carried

• ut, and it has always been found to be reliable.

There exist also some non-rigorous proofs of its validity. The Ryu-Takayanagi conjecture applies to holographic theories where the bulk theory of gravity is general relativity, but there are generalisations to higher derivative theories of gravity. In these cases the area is replaced by some other geometric quantity.

◀ ▶ 19/24

It can be shown that the Ryu-Takayanagi formula implies the Bekenstein-Hawking formula. Thus the Ryu-Takayanagi conjecture is a vast generalisation of black hole entropy.

◀ ▶ 20/24

It can be shown that the Ryu-Takayanagi formula implies the Bekenstein-Hawking formula. Thus the Ryu-Takayanagi conjecture is a vast generalisation of black hole entropy.

◀ ▶ 20/24

Spacetime from entanglement

Up to this point we have been assuming the existence of geometry and locality, and stating properties about the entanglement of states consistent with such assumptions. But the Ryu-Takayanagi conjecture implies that much of the bulk geometry is encoded in the entanglement structure of the density matrix . Entanglement is an absolutely fundamental feature of a quantum

• theory. Geometry is not.

This has lead to the fascinating suggestion that, in quantum gravity, geometry does not exist a priori, but emerges from the entanglement structure of the quantum state of the universe.

◀ ▶ 21/24

Up to this point we have been assuming the existence of geometry and locality, and stating properties about the entanglement of states consistent with such assumptions. But the Ryu-Takayanagi conjecture implies that much of the bulk geometry is encoded in the entanglement structure of the density matrix ρ. Entanglement is an absolutely fundamental feature of a quantum

• theory. Geometry is not.

This has lead to the fascinating suggestion that, in quantum gravity, geometry does not exist a priori, but emerges from the entanglement structure of the quantum state of the universe.

◀ ▶ 21/24

Up to this point we have been assuming the existence of geometry and locality, and stating properties about the entanglement of states consistent with such assumptions. But the Ryu-Takayanagi conjecture implies that much of the bulk geometry is encoded in the entanglement structure of the density matrix ρ. Entanglement is an absolutely fundamental feature of a quantum

• theory. Geometry is not.

This has lead to the fascinating suggestion that, in quantum gravity, geometry does not exist a priori, but emerges from the entanglement structure of the quantum state of the universe.

◀ ▶ 21/24

Up to this point we have been assuming the existence of geometry and locality, and stating properties about the entanglement of states consistent with such assumptions. But the Ryu-Takayanagi conjecture implies that much of the bulk geometry is encoded in the entanglement structure of the density matrix ρ. Entanglement is an absolutely fundamental feature of a quantum

• theory. Geometry is not.

This has lead to the fascinating suggestion that, in quantum gravity, geometry does not exist a priori, but emerges from the entanglement structure of the quantum state of the universe.

◀ ▶ 21/24

The most succinct statement of this idea is provided by the conceptual equation ER = EPR. (7) ER stands for Einstein-Rosen, and refers to the wormhole connecting the two sides of a black hole. EPR stands for Einstein-Podolsky-Rosen, and refers to two far separated particles set up in an entangled state. This equation is meant to portray the idea that setting up the EPR pair is equivalent to creating a wormhole connecting them. In other words, by modifying the entanglement structure of the state of universe, we have modifjed the geometry of spacetime.

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The most succinct statement of this idea is provided by the conceptual equation ER = EPR. (7) ER stands for Einstein-Rosen, and refers to the wormhole connecting the two sides of a black hole. EPR stands for Einstein-Podolsky-Rosen, and refers to two far separated particles set up in an entangled state. This equation is meant to portray the idea that setting up the EPR pair is equivalent to creating a wormhole connecting them. In other words, by modifying the entanglement structure of the state of universe, we have modifjed the geometry of spacetime.

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The most succinct statement of this idea is provided by the conceptual equation ER = EPR. (7) ER stands for Einstein-Rosen, and refers to the wormhole connecting the two sides of a black hole. EPR stands for Einstein-Podolsky-Rosen, and refers to two far separated particles set up in an entangled state. This equation is meant to portray the idea that setting up the EPR pair is equivalent to creating a wormhole connecting them. In other words, by modifying the entanglement structure of the state of universe, we have modifjed the geometry of spacetime.

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The most succinct statement of this idea is provided by the conceptual equation ER = EPR. (7) ER stands for Einstein-Rosen, and refers to the wormhole connecting the two sides of a black hole. EPR stands for Einstein-Podolsky-Rosen, and refers to two far separated particles set up in an entangled state. This equation is meant to portray the idea that setting up the EPR pair is equivalent to creating a wormhole connecting them. In other words, by modifying the entanglement structure of the state of universe, we have modifjed the geometry of spacetime.

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The most succinct statement of this idea is provided by the conceptual equation ER = EPR. (7) ER stands for Einstein-Rosen, and refers to the wormhole connecting the two sides of a black hole. EPR stands for Einstein-Podolsky-Rosen, and refers to two far separated particles set up in an entangled state. This equation is meant to portray the idea that setting up the EPR pair is equivalent to creating a wormhole connecting them. In other words, by modifying the entanglement structure of the state of universe, we have modifjed the geometry of spacetime.

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The idea of emergent geometry allows us to reason about gravitational physics by using mathematical facts about

• entanglement. For example:
• The ‘fjrst law of entanglement’ implies the linearised Einstein

equations.

• ‘Relative entropy inequalities’ allow one to defjne a

well-behaved completely general notion of perturbative gravitational energy.

• ‘Strong subadditivity’ is related to the null energy condition.

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The idea of emergent geometry allows us to reason about gravitational physics by using mathematical facts about

• entanglement. For example:
• The ‘fjrst law of entanglement’ implies the linearised Einstein

equations.

• ‘Relative entropy inequalities’ allow one to defjne a

well-behaved completely general notion of perturbative gravitational energy.

• ‘Strong subadditivity’ is related to the null energy condition.

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The idea of emergent geometry allows us to reason about gravitational physics by using mathematical facts about

• entanglement. For example:
• The ‘fjrst law of entanglement’ implies the linearised Einstein

equations.

• ‘Relative entropy inequalities’ allow one to defjne a

well-behaved completely general notion of perturbative gravitational energy.

• ‘Strong subadditivity’ is related to the null energy condition.

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The idea of emergent geometry allows us to reason about gravitational physics by using mathematical facts about

• entanglement. For example:
• The ‘fjrst law of entanglement’ implies the linearised Einstein

equations.

• ‘Relative entropy inequalities’ allow one to defjne a

well-behaved completely general notion of perturbative gravitational energy.

• ‘Strong subadditivity’ is related to the null energy condition.

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This idea is still very much in the conceptual stages, but has seen a rapid increase in attention and progress in the last few years. Attempts to take it to its logical conclusion involve phrases like ‘tensor networks’ and ‘quantum error correction’. These sound very interesting, but I haven’t yet been able to fully understand how they work. See those lecture notes for more on them.

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This idea is still very much in the conceptual stages, but has seen a rapid increase in attention and progress in the last few years. Attempts to take it to its logical conclusion involve phrases like ‘tensor networks’ and ‘quantum error correction’. These sound very interesting, but I haven’t yet been able to fully understand how they work. See those lecture notes for more on them.

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This idea is still very much in the conceptual stages, but has seen a rapid increase in attention and progress in the last few years. Attempts to take it to its logical conclusion involve phrases like ‘tensor networks’ and ‘quantum error correction’. These sound very interesting, but I haven’t yet been able to fully understand how they work. See those lecture notes for more on them.

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The end

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