Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST)...

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Holographic Entanglement of Purification Koji Umemoto (YITP, Kyoto U.) w/ Tadashi Takayanagi (YITP) Nat. Phys. 14, 573 (2018). Yang Zhou (Fudan U.) arXiv:1805.02625. Strings 2018 (OIST)

Transcript of Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST)...

Page 1: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Holographic Entanglement of Purification

Koji Umemoto (YITP, Kyoto U.)

w/ Tadashi Takayanagi (YITP) Nat. Phys. 14, 573 (2018).

Yang Zhou (Fudan U.) arXiv:1805.02625.

Strings 2018 (OIST)

Page 2: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Bekenstein-Hawking entropy

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy

𝑆𝐵𝐻 =Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛)

4𝐺𝑁

Page 3: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Bekenstein-Hawking entropy

𝑆𝐵𝐻 =Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛)

4𝐺𝑁

• Holographic entanglement entropy

𝑆𝐴 = −Tr𝜌𝐴log𝜌𝐴 =Area(𝛾𝑅𝑇)

4𝐺𝑁

𝑨𝒄 𝑨

𝜸𝑹𝑻

𝑨

𝛾𝑅𝑇

𝑨𝒄 = 𝑩

(time slice of bulk/boundary)

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy

[Refer to Xi Dong’s Review Talk]

[Ryu-Takayanagi '06]

Page 4: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Entanglement ≌ Geometry ?

Tensor Network, ER=EPR, Bit Threads,…

• Bekenstein-Hawking entropy

𝑆𝐵𝐻 =Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛)

4𝐺𝑁

• Holographic entanglement entropy

𝑆𝐴 = −Tr𝜌𝐴log𝜌𝐴 =Area(𝛾𝑅𝑇)

4𝐺𝑁

𝑨𝒄 𝑨

𝜸𝑹𝑻

𝑨

𝛾𝑅𝑇

𝑨𝒄 = 𝑩

(time slice of bulk/boundary)

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy

[Ryu-Takayanagi '06]

[Refer to Xi Dong’s Review Talk]

Page 5: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy

Page 6: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy

2 disjoint subsystems

• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed

𝑨 𝑩 ≠ 𝑨𝒄

Ex.

Page 7: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

2 disjoint subsystems

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy

Question

Entanglement ≌ Geometry for mixed state setups?

• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed

𝑨 𝑩 ≠ 𝑨𝒄

Ex.

Page 8: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Entanglement entropy

𝐴 𝐴𝑐 𝛾𝑅𝑇

Page 9: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Entanglement entropy ➤ Entanglement of purification

𝐴 𝐴𝑐 𝐴 𝐵 𝛾𝑅𝑇 Σ𝐸𝑊𝐶𝑆

Page 10: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Entanglement of purification [Terhal-Horodecki-Leung-DiVincenzo '02]

𝐸𝑃 𝜌𝐴𝐵 ≔ minΨ 𝐴𝐴′𝐵𝐵′

𝑆𝐴𝐴′ (𝜌𝐴𝐴′ ≔ Tr𝐵𝐵′[ Ψ Ψ 𝐴𝐴′𝐵𝐵′])

Proposal ∣ Holographic entanglement of purification

• A measure of total correlation (entanglement + classical one)

Page 11: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Entanglement of purification [Terhal-Horodecki-Leung-DiVincenzo '02]

𝐸𝑃 𝜌𝐴𝐵 ≔ minΨ 𝐴𝐴′𝐵𝐵′

𝑆𝐴𝐴′ (𝜌𝐴𝐴′ ≔ Tr𝐵𝐵′[ Ψ Ψ 𝐴𝐴′𝐵𝐵′])

Proposal ∣ Holographic entanglement of purification

• A measure of total correlation (entanglement + classical one)

• Holographic dual: Entanglement wedge cross-section [Takayanagi-KU, Nguyen-Devakul-Halbasch-Zaletel-Swingle '17]

𝐸𝑊 𝜌𝐴𝐵 ≔ minΣ𝐴𝐵

Area(Σ𝐴𝐵)

4𝐺𝑁 𝐴 𝐵 Σ𝐴𝐵

• Minimal cross-section of entanglement wedge

Page 12: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Strings 2018 (OIST)

• Entanglement of purification [Terhal-Horodecki-Leung-DiVincenzo '02]

𝐸𝑃 𝜌𝐴𝐵 ≔ minΨ 𝐴𝐴′𝐵𝐵′

𝑆𝐴𝐴′ (𝜌𝐴𝐴′ ≔ Tr𝐵𝐵′[ Ψ Ψ 𝐴𝐴′𝐵𝐵′])

• A measure of total correlation (entanglement + classical one)

• Holographic dual: Entanglement wedge cross-section [Takayanagi-KU, Nguyen-Devakul-Halbasch-Zaletel-Swingle '17]

𝐸𝑊 𝜌𝐴𝐵 ≔ minΣ𝐴𝐵

Area(Σ𝐴𝐵)

4𝐺𝑁 𝐴 𝐵 Σ𝐴𝐵

• Conjecture 𝑬𝑷 = 𝑬𝑾

Proposal ∣ Holographic entanglement of purification

• Minimal cross-section of entanglement wedge

Page 13: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

• Agreements of information-theoretic properties

Strings 2018 (OIST)

Assumption: (at least) one minimal purification has a classical gravity dual.

Proposal ∣ Evidence

For pure states: 𝐸# |𝜓⟩ 𝐴𝐵 = 𝑆𝐴 Faithfulness: 𝐸# 𝜌𝐴𝐵 = 0 if and only if 𝜌𝐴𝐵 = 𝜌𝐴 ⊗ 𝜌𝐵 Less than entanglement entropies: 𝐸# 𝜌𝐴𝐵 ≤ min[𝑆𝐴, 𝑆𝐵]

Never increases by trace out: 𝐸# 𝜌𝐴𝐵1𝐵2≥ 𝐸# 𝜌𝐴𝐵1

𝐸# 𝜌𝐴𝐵 ≥ 𝐼(𝐴: 𝐵)/2, 𝐸# 𝜌𝐴𝐵1𝐵2 ≥ 𝐼 𝐴: 𝐵1 /2 + 𝐼(𝐴: 𝐵2)/2 …

Page 14: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

• Agreements of information-theoretic properties For pure states: 𝐸# |𝜓⟩ 𝐴𝐵 = 𝑆𝐴 Faithfulness: 𝐸# 𝜌𝐴𝐵 = 0 if and only if 𝜌𝐴𝐵 = 𝜌𝐴 ⊗ 𝜌𝐵 Less than entanglement entropies: 𝐸# 𝜌𝐴𝐵 ≤ min[𝑆𝐴, 𝑆𝐵]

Never increases by trace out: 𝐸# 𝜌𝐴𝐵1𝐵2≥ 𝐸# 𝜌𝐴𝐵1

𝐸# 𝜌𝐴𝐵 ≥ 𝐼(𝐴: 𝐵)/2, 𝐸# 𝜌𝐴𝐵1𝐵2 ≥ 𝐼 𝐴: 𝐵1 /2 + 𝐼(𝐴: 𝐵2)/2 …

• Tensor network description

Strings 2018 (OIST)

min𝜓 𝐴𝐴′𝐵𝐵′

= Assumption: (at least) one minimal purification has a classical gravity dual.

= 𝐸𝑊 (𝜌𝐴𝐵) 𝐴 𝐵

𝑆𝐴𝐴′

𝐴′

𝐴′ 𝐵′

𝐵′

𝐴 𝐵 𝐴′

𝐴′ 𝐵′

𝐵′

𝐸𝑃(𝜌𝐴𝐵) =

Proposal ∣ Evidence

Page 15: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

• Agreements of information-theoretic properties For pure states: 𝐸# |𝜓⟩ 𝐴𝐵 = 𝑆𝐴 Faithfulness: 𝐸# 𝜌𝐴𝐵 = 0 if and only if 𝜌𝐴𝐵 = 𝜌𝐴 ⊗ 𝜌𝐵 Less than entanglement entropies: 𝐸# 𝜌𝐴𝐵 ≤ min[𝑆𝐴, 𝑆𝐵]

Never increases by trace out: 𝐸# 𝜌𝐴𝐵1𝐵2≥ 𝐸# 𝜌𝐴𝐵1

𝐸# 𝜌𝐴𝐵 ≥ 𝐼(𝐴: 𝐵)/2, 𝐸# 𝜌𝐴𝐵1𝐵2 ≥ 𝐼 𝐴: 𝐵1 /2 + 𝐼(𝐴: 𝐵2)/2 …

• Tensor network description

Strings 2018 (OIST)

min𝜓 𝐴𝐴′𝐵𝐵′

= Assumption: (at least) one minimal purification has a classical gravity dual.

= 𝐸𝑊 (𝜌𝐴𝐵) 𝐴 𝐵

𝑆𝐴𝐴′

𝐴′

𝐴′ 𝐵′

𝐵′

𝐴 𝐵 𝐴′

𝐴′ 𝐵′

𝐵′

𝐸𝑃(𝜌𝐴𝐵) =

Ex. Phase Transition of 𝐸𝑊/𝐸𝑃

• Numerical results in toy models

[Bhattacharyya-Takayanagi-KU '18]

Proposal ∣ Evidence

Page 16: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

• Multipartite entanglement of purification [Bao-Halpern, Zhou-KU '18]

• New measure of multipartite total correlation • We proved various information-theoretic properties of Δ𝑃

Strings 2018 (OIST)

Generalization ∣ Multipartite setup

Δ𝑊 𝜌𝐴𝐵𝐶 ≔1

2min

𝜓 𝐴𝐴′𝐵𝐵′𝐶𝐶′ [𝑆𝐴𝐴′ + 𝑆𝐵𝐵′ + 𝑆𝐶𝐶′]

Page 17: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

• Multipartite entanglement of purification [Bao-Halpern, Zhou-KU '18]

• New measure of multipartite total correlation • We proved various information-theoretic properties of Δ𝑃

• Holographic dual proposal

Δ𝑊 𝜌𝐴𝐵𝐶 ≔1

2minΣ𝐴𝐵𝐶

Area(Σ𝐴𝐵𝐶)

4𝐺𝑁

• Inspired by tensor network • satisfies all properties of Δ𝑃 we proved

Strings 2018 (OIST)

[Zhou-KU '18]

Generalization ∣ Multipartite setup

B

𝐴

𝐶

𝚺𝑨𝑩𝑪 B

𝐴

𝐶

𝚺𝑨𝑩𝑪

Δ𝑊 𝜌𝐴𝐵𝐶 ≔1

2min

𝜓 𝐴𝐴′𝐵𝐵′𝐶𝐶′ [𝑆𝐴𝐴′ + 𝑆𝐵𝐵′ + 𝑆𝐶𝐶′]

Page 18: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

• Multipartite entanglement of purification [Bao-Halpern, Zhou-KU '18]

• New measure of multipartite total correlation • We proved various information-theoretic properties of Δ𝑃

• Generalized conjecture 𝚫𝑷 = 𝚫𝑾

Strings 2018 (OIST)

Generalization ∣ Multipartite setup

• Holographic dual proposal

Δ𝑊 𝜌𝐴𝐵𝐶 ≔1

2minΣ𝐴𝐵𝐶

Area(Σ𝐴𝐵𝐶)

4𝐺𝑁

• Inspired by tensor network • satisfies all properties of Δ𝑃 we proved

[Zhou-KU '18]

B

𝐴

𝐶

𝚺𝑨𝑩𝑪 B

𝐴

𝐶

𝚺𝑨𝑩𝑪

Δ𝑊 𝜌𝐴𝐵𝐶 ≔1

2min

𝜓 𝐴𝐴′𝐵𝐵′𝐶𝐶′ [𝑆𝐴𝐴′ + 𝑆𝐵𝐵′ + 𝑆𝐶𝐶′]

Page 19: Holographic Entanglement of Purification · 2018-07-09 · Strings 2018 (OIST) •Bekenstein-Hawking entropy 𝑆 𝐻= Area(𝛾ℎ𝑖𝑧 â á) 4𝐺𝑁 •Holographic entanglement

Concluding remarks

• Squashed entanglement in holography • Monogamy of holographic mutual information • Multipartite strong superadditivity of multipartite holographic MI

• Strong superadditivity of 𝑬𝑾

[Hayden-Headrick-Maloney '11]

• Conditional entanglement of purification [Bao-Halpern '17, '18]

• Operational meanings of 𝑬𝑷 • EPR pairs and LOq - Tensor network? Bit threads?

Thank you for your attention Strings 2018 (OIST)

• Future directions

• CFT calculation of 𝐸𝑃 • Operational interpretation of Δ𝑃 • Structure of quantum states saturating 𝐸𝑠𝑞 ≤ 𝐼/2 …

[Hirai-Tamaoka-Yokoya '18]