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)
Strings 2018 (OIST)
• Bekenstein-Hawking entropy
Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy
𝑆𝐵𝐻 =Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛)
4𝐺𝑁
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]
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]
Strings 2018 (OIST)
• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed
Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy
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.
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.
Entanglement entropy
𝐴 𝐴𝑐 𝛾𝑅𝑇
Entanglement entropy ➤ Entanglement of purification
𝐴 𝐴𝑐 𝐴 𝐵 𝛾𝑅𝑇 Σ𝐸𝑊𝐶𝑆
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)
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
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
• 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 …
• 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
• 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
• 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′]
• 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′]
• 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′]
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]
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