Strings 2018 (OIST)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.

Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy• Bekenstein-Hawking entropy
Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛 )𝑆𝐵𝐻 =4𝐺𝑁

Strings 2018 (OIST)Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy• Bekenstein-Hawking entropy
Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛 )𝑆𝐵𝐻 =4𝐺𝑁• Holographic entanglement entropy [Ryu-Takayanagi '06]
Area(𝛾𝑅𝑇 )𝑆𝐴 = −Tr𝜌𝐴 log𝜌𝐴 =4𝐺𝑁𝑨
[Refer to Xi Dong’s Review Talk]

𝑨𝒄 = 𝑩𝜸𝑹𝑻𝛾𝑅𝑇𝑨𝒄

𝑨(time slice of bulk/boundary)Strings 2018 (OIST)Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy• Bekenstein-Hawking entropy
Area(𝛾ℎ𝑜𝑟𝑖𝑧𝑜𝑛 )𝑆𝐵𝐻 =4𝐺𝑁• Holographic entanglement entropy [Ryu-Takayanagi '06]
Area(𝛾𝑅𝑇 )𝑆𝐴 = −Tr𝜌𝐴 log𝜌𝐴 =4𝐺𝑁𝑨
[Refer to Xi Dong’s Review Talk]

𝑨𝒄 = 𝑩𝜸𝑹𝑻𝛾𝑅𝑇𝑨𝒄

𝑨(time slice of bulk/boundary)• Entanglement ≌ Geometry ?
Tensor Network, ER=EPR, Bit Threads,…
Strings 2018 (OIST)Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed

Strings 2018 (OIST)Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed
Ex.

𝑨𝒄

𝑩≠𝑨

2 disjoint subsystems

Strings 2018 (OIST)Introduction ∣ Black hole entropy ➤ Holographic entanglement entropy• Entanglement entropy is not entanglement if 𝜌𝐴𝐵 is mixed
Ex.

𝑨𝒄

𝑩≠𝑨

2 disjoint subsystems

Question

Entanglement ≌ Geometry for mixed state setups?
Strings 2018 (OIST)Entanglement entropy

𝐴𝛾𝑅𝑇𝐴𝑐Entanglement entropy ➤ Entanglement of purification

𝐴𝛾𝑅𝑇𝐴𝑐𝐴

Σ𝐸𝑊𝐶𝑆 𝐵

Proposal ∣ Holographic entanglement of purification• Entanglement of purification [Terhal-Horodecki-Leung-DiVincenzo '02]
𝐸𝑃 𝜌𝐴𝐵 ≔min 𝑆 ′Ψ 𝐴𝐴′ 𝐵𝐵′ 𝐴𝐴(𝜌𝐴𝐴′ ≔ Tr𝐵𝐵′ [ Ψ Ψ 𝐴𝐴′ 𝐵𝐵′ ])• A measure of total correlation (entanglement + classical one)

Strings 2018 (OIST)Proposal ∣ Holographic entanglement of purification• Entanglement of purification [Terhal-Horodecki-Leung-DiVincenzo '02]
𝐸𝑃 𝜌𝐴𝐵 ≔min 𝑆 ′Ψ 𝐴𝐴′ 𝐵𝐵′ 𝐴𝐴(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 𝜌𝐴𝐵Area(Σ𝐴𝐵 )≔ min
Σ𝐴𝐵4𝐺𝑁𝐴

Σ𝐴𝐵 𝐵• Minimal cross-section of entanglement wedge

Strings 2018 (OIST)Proposal ∣ Holographic entanglement of purification• Entanglement of purification [Terhal-Horodecki-Leung-DiVincenzo '02]
𝐸𝑃 𝜌𝐴𝐵 ≔min 𝑆 ′Ψ 𝐴𝐴′ 𝐵𝐵′ 𝐴𝐴(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 𝜌𝐴𝐵Area(Σ𝐴𝐵 )≔ minΣ𝐴𝐵4𝐺𝑁𝐴

Σ𝐴𝐵 𝐵• Minimal cross-section of entanglement wedge• Conjecture

𝑬𝑷 = 𝑬𝑾Strings 2018 (OIST)Proposal ∣ Evidence• Agreements of information-theoretic properties
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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# 𝜌𝐴𝐵1𝐸# 𝜌𝐴𝐵 ≥ 𝐼(𝐴: 𝐵)/2, 𝐸# 𝜌𝐴𝐵1𝐵2 ≥ 𝐼 𝐴: 𝐵1 /2 + 𝐼(𝐴: 𝐵2 )/2 …

Strings 2018 (OIST)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# 𝜌𝐴𝐵1𝐸# 𝜌𝐴𝐵 ≥ 𝐼(𝐴: 𝐵)/2, 𝐸# 𝜌𝐴𝐵1𝐵2 ≥ 𝐼 𝐴: 𝐵1 /2 + 𝐼(𝐴: 𝐵2 )/2 …

• Tensor network description
𝐴′ 𝐵′𝐸𝑃 (𝜌𝐴𝐵 ) = min𝐴𝜓 𝐴𝐴′ 𝐵𝐵′𝑆𝐴𝐴′𝐴′ 𝐵′𝐴′ 𝐵′

𝐵=𝐵

𝐴= 𝐸𝑊 (𝜌𝐴𝐵 )𝐴′ 𝐵′Strings 2018 (OIST)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# 𝜌𝐴𝐵1𝐸# 𝜌𝐴𝐵 ≥ 𝐼(𝐴: 𝐵)/2, 𝐸# 𝜌𝐴𝐵1𝐵2 ≥ 𝐼 𝐴: 𝐵1 /2 + 𝐼(𝐴: 𝐵2 )/2 …

• Tensor network description𝐴′ 𝐵′

𝐸𝑃 (𝜌𝐴𝐵 ) = min

𝐴𝑆𝐴𝐴′𝜓 𝐴𝐴′ 𝐵𝐵′𝐴′ 𝐵′

𝐵=𝐴′ 𝐵′

𝐵

𝐴= 𝐸𝑊 (𝜌𝐴𝐵 )𝐴′ 𝐵′• Numerical results in toy models
Ex. Phase Transition of 𝐸𝑊 /𝐸𝑃[Bhattacharyya-Takayanagi-KU '18]
Strings 2018 (OIST)Generalization ∣ Multipartite setup• Multipartite entanglement of purification [Bao-Halpern, Zhou-KU '18]
Δ𝑊 𝜌𝐴𝐵𝐶

1≔min[𝑆𝐴𝐴′ + 𝑆𝐵𝐵′ + 𝑆𝐶𝐶 ′ ]2 𝜓 𝐴𝐴′𝐵𝐵′𝐶𝐶′• New measure of multipartite total correlation• We proved various information-theoretic properties of Δ𝑃

Strings 2018 (OIST)Generalization ∣ Multipartite setup• Multipartite entanglement of purification [Bao-Halpern, Zhou-KU '18]
Δ𝑊 𝜌𝐴𝐵𝐶

1≔min[𝑆𝐴𝐴′ + 𝑆𝐵𝐵′ + 𝑆𝐶𝐶 ′ ]2 𝜓 𝐴𝐴′𝐵𝐵′𝐶𝐶′• New measure of multipartite total correlation• We proved various information-theoretic properties of Δ𝑃• Holographic dual proposal [Zhou-KU '18]
Δ𝑊 𝜌𝐴𝐵𝐶

1Area(Σ𝐴𝐵𝐶 )≔ min
2 Σ𝐴𝐵𝐶
4𝐺𝑁𝐶• Inspired by tensor network• satisfies all properties of Δ𝑃 we proved

𝐶𝚺𝑨𝑩𝑪 B

𝚺𝑨𝑩𝑪𝐴

𝐴

BStrings 2018 (OIST)Generalization ∣ Multipartite setup• Multipartite entanglement of purification [Bao-Halpern, Zhou-KU '18]Δ𝑊 𝜌𝐴𝐵𝐶

1≔min
[𝑆𝐴𝐴′ + 𝑆𝐵𝐵′ + 𝑆𝐶𝐶 ′ ]2 𝜓 𝐴𝐴′𝐵𝐵′𝐶𝐶′• New measure of multipartite total correlation• We proved various information-theoretic properties of Δ𝑃• Holographic dual proposal [Zhou-KU '18]
Δ𝑊 𝜌𝐴𝐵𝐶

1Area(Σ𝐴𝐵𝐶 )≔ min
2 Σ𝐴𝐵𝐶
4𝐺𝑁

𝐶• Inspired by tensor network• satisfies all properties of Δ𝑃 we proved• Generalized conjecture𝐶𝚺𝑨𝑩𝑪 B𝚺𝑨𝑩𝑪𝐴𝐴B

𝚫𝑷 = 𝚫𝑾
Strings 2018 (OIST)Concluding remarks• Squashed entanglement in holography• Monogamy of holographic mutual information [Hayden-Headrick-Maloney '11]
• Multipartite strong superadditivity of multipartite holographic MI• Operational meanings of 𝑬𝑷• EPR pairs and LOq - Tensor network? Bit threads?

• Conditional entanglement of purification [Bao-Halpern '17, '18]
• Strong superadditivity of 𝑬𝑾• Future directions• CFT calculation of 𝐸𝑃 [Hirai-Tamaoka-Yokoya '18]
• Operational interpretation of Δ𝑃• Structure of quantum states saturating 𝐸𝑠𝑞 ≤ 𝐼/2 …
Thank you for your attention
Strings 2018 (OIST)