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 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 1/24 Information”. In: Rev. Mod. Phys. 88 (2016), p. 015002. doi : AdS/CFT”. In: (2018). arXiv: 1802.01040 [hep-th] ◀ ▶

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 ◀ ▶

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 ◀ ▶

1. Entanglement 2. Entanglement entropy 3. Entanglement entropy in QFT 4. Holographic entanglement entropy 5. Geometry from entanglement 2/24 ◀ ▶

Entanglement

i A j B comprise an orthonormal basis of the The states i j product Hilbert space A B . Elements of are joint states. 3/24 Let H A , H B be two Hilbert spaces, with orthonormal bases {| i ⟩ A } , {| j ⟩ B } respectively, where i , j = 1 , 2 , . . . . ◀ ▶

Elements of are joint states. 3/24 Let H A , H B 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 = H A ⊗ H B . ◀ ▶

3/24 Let H A , H B 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 = H A ⊗ H B . Elements of H are joint states. ◀ ▶

B , then there is no 4/24 is. 2 1 1 2 2 1 For example, 1 1 is not entangled, while Otherwise, it is an entangled state. entanglement. entanglement in joint states. A B , i.e. B A A can be written as the product of two states If a state The product structure of H allows for the presence of ◀ ▶

entanglement in joint states. 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 If a state | ψ ⟩ ∈ H can be written as the product of two states | ψ A ⟩ ∈ H A , | ψ B ⟩ ∈ H B , i.e. | ψ ⟩ = | ψ A ⟩ ⊗ | ψ B ⟩ , then there is no ◀ ▶

entanglement in joint states. 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 If a state | ψ ⟩ ∈ H can be written as the product of two states | ψ A ⟩ ∈ H A , | ψ B ⟩ ∈ H B , i.e. | ψ ⟩ = | ψ A ⟩ ⊗ | ψ B ⟩ , then there is no ◀ ▶

entanglement in joint states. entanglement. Otherwise, it is an entangled state. 1 4/24 The product structure of H allows for the presence of If a state | ψ ⟩ ∈ H can be written as the product of two states | ψ A ⟩ ∈ H A , | ψ B ⟩ ∈ H B , i.e. | ψ ⟩ = | ψ A ⟩ ⊗ | ψ B ⟩ , then there is no For example, | 1 , 1 ⟩ is not entangled, while 2 ( | 1 , 2 ⟩ + | 2 , 1 ⟩ ) is. √ ◀ ▶

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. 5/24 ◀ ▶

Entanglement entropy

When a state is entangled, knowledge about the part of the state This only happens for entangled states. Furthermore, the amount of 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 in H A provides information about the part of the state in H B . ◀ ▶

When a state is entangled, knowledge about the part of the state This only happens for entangled states. Furthermore, the amount of 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 in H A provides information about the part of the state in H B . ◀ ▶

When a state is entangled, knowledge about the part of the state This only happens for entangled states. Furthermore, the amount of 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 in H A provides information about the part of the state in H B . ◀ ▶

generalisation of a classical set of probabilities p i satisfying i p i 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 1, p i 0. Density matrices can account for both quantum uncertainty and statistical uncertainty. 7/24 ◀ ▶

generalisation of a classical set of probabilities p i satisfying i p i To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. Density matrices should be thought of as the quantum 1, p i 0. Density matrices can account for both quantum uncertainty and statistical uncertainty. 7/24 A density matrix on a Hilbert space H is an self-adjoint, positive semi-defjnite operator ρ ∈ L ( H ) satisfying tr ρ = 1. ◀ ▶

To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. Density matrices should be thought of as the quantum Density matrices can account for both quantum uncertainty and statistical uncertainty. 7/24 A density matrix on a Hilbert space H is an self-adjoint, positive semi-defjnite operator ρ ∈ L ( H ) satisfying tr ρ = 1. generalisation of a classical set of probabilities p i satisfying ∑ i p i = 1, p i ≥ 0. ◀ ▶

To calculate the ‘entanglement entropy’ of a state, we need to use density matrices. Density matrices should be thought of as the quantum Density matrices can account for both quantum uncertainty and statistical uncertainty. 7/24 A density matrix on a Hilbert space H is an self-adjoint, positive semi-defjnite operator ρ ∈ L ( H ) satisfying tr ρ = 1. generalisation of a classical set of probabilities p i satisfying ∑ i p i = 1, p i ≥ 0. ◀ ▶

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 p i p i (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 (1) This is the quantum mechanical generalisation of the classical Shannon entropy S i p i p i (2) 8/24 S = − tr( ρ log ρ ) . ◀ ▶

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 (1) This is the quantum mechanical generalisation of the classical Shannon entropy i (2) 8/24 S = − tr( ρ log ρ ) . ∑ S = − p i log p i . ◀ ▶