A Generalized Entanglement Entropy and Holography Kotaro Tamaoka (YITP) Based on 1809.09109 (Phys. Rev. Lett. 122, 141601) and work in progress with Yuya Kusuki (YITP) It from Qubit school/workshop, June 27, 2019
Motivation Measure for Mixed states in AdS/CFT? Ryu-Takayanagi formula γ A S ( ρ A ) = Ryu-Takayanagi ’06,… Entanglement entropy : S ( ρ A ) nice measure for pure states How is the measure for mixed states? There are many candidates in the literature … (Any nice ones in AdS/CFT ?)
Motivation Measure for Mixed states in AdS/CFT? Subregion/subregion duality Czech-Karczmarek-Nogueira-VanRaamsdonk, Wall ’12, Headrick, Hubeny-Lawrence-Rangamani ‘14… ( AB ) c ρ AB = Tr ( AB ) c | 0 ih 0 | A B ( AB ) c reduced density matrix entanglement wedge (a mixed state)
Motivation Measure for Mixed states in AdS/CFT? Subregion/subregion duality Czech-Karczmarek-Nogueira-VanRaamsdonk, Wall ’12, Headrick, Hubeny-Lawrence-Rangamani ‘14… E W A B A natural object in the bulk : Umemoto-Takayanagi ‘17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ‘17 entanglement wedge cross section
Motivation Measure for Mixed states in AdS/CFT? Subregion/subregion duality Czech-Karczmarek-Nogueira-VanRaamsdonk, Wall ’12, Headrick, Hubeny-Lawrence-Rangamani ‘14… Purification conjectures: E W E P = E W A B (min S(AA’) for ∀ purification | Ψ AA’BB’ >) Umemoto-Takayanagi ‘17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ‘17 entanglement wedge S R = 2 E W cross section (S(AA’) for TFD-like purification | Ψ AA’BB’ >) Dutta-Faulkner ‘19
This talk from CFT without purification E W the entanglement wedge should know its cross section without introducing purified states! It from an “odd” generalization of the entropy
・ Odd (Entanglement) Entropy KT ‘18 Tr( ρ T B AB ) n o − 1 S o ( ρ AB ) := lim 1 − n o n o :odd → 1 ・ partial transposition: h i A , j B | ⇢ T B AB | k A , ` B i ⌘ h i A , ` B | ⇢ AB | k A , j B i ρ T B AB can have negative eigenvalues ∃ negative eigenvalue → ∃ entanglement btw A&B Peres ‘96
Odd (Entanglement) Entropy KT ‘18 X X S o ( ρ AB ) = − | λ i | log | λ i | + | λ i | log | λ i | λ i > 0 λ i < 0 λ i : eigenvalues for ρ T B AB For pure states, the same as usual EE! S o ( ρ AB ) = S ( ρ A ) (if ρ AB is a pure state) Note: it also counts classical correlations ↔ logarithmic negativity Kudler-Flam—Ryu ‘18 n e :even → 1 log Tr( ρ T B AB ) n e ∝ E W E = lim (with back-reaction) Vidal-Werner ’02, Calabrese-Cardy-Tonni ’12
Results in 2d holographic CFT S o ( ρ AB ) S ( ρ AB ) E W ( ρ AB ) + = minimal surfaces EWCS ・ Vacuum and Thermal state KT’18 ・ Heavy Excited states ・ Quench by local heavy op. Yuya Kusuki & KT to appear Can use Replica trick Calabrese-Cardy ‘04 ,Calabrese-Cardy-Tonni ’12, Large-c + Sparse spectrum Hartman ’13, Can use Fusion (Crossing) Kernel Kusuki ’18, Collier-Gobeil-Maxfield-Perlmutter ’18, Kusuki-Miyaji ’19
A Lesson from the odd entropy e.g.) an inequality from entanglement wedge Freedman-Headrick’16, Umemoto-Takayanagi ‘17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ‘17 E W ( ρ AB ) ≥ I ( A : B ) / 2 2 E W ( ρ AB )+ S ( ρ AB ) ≥ S ( ρ A ) + S ( ρ B ) ⇔ ∵ ) S ( ρ A ) , S ( ρ B ) A B E W ≥
A Lesson from the odd entropy ? S o ( ρ AB ) − S ( ρ AB ) ≥ I ( A : B ) / 2 It rather specializes the holographic CFT! A counterexample: ρ AB = q | Ψ ih Ψ | + (1 � q ) σ A ⌦ σ B 1 p | Ψ i = 2( | 0 A i | 1 B i � | 1 A i | 0 B i ) σ = 1 2 | 0 ih 0 | + 1 2 | 1 ih 1 |
Discussion ・ More general many body systems General properties of odd EE ? / How holographic CFT is special? ・Relation to conditional & di fg erential entropy Balasubramanian-Chowdhury-Czech-de Boer-Heller '13, ... ・ From gravitational path integral c.f. Lewkowycz-Maldacena, Faulkner-Lewkowycz-Maldacena '13 …
Summary: odd entropy and holography Tr( ρ T B AB ) n o − 1 S o ( ρ AB ) := lim 1 − n o n o :odd → 1 S o ( ρ AB ) S ( ρ AB ) E W ( ρ AB ) + = minimal surfaces EWCS A B → new constraints for holographic CFT
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