A Generalized Entanglement Entropy and Holography Kotaro Tamaoka - - PowerPoint PPT Presentation

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

How is the measure for mixed states? There are many candidates in the literature … (Any nice ones in AdS/CFT ?)

Ryu-Takayanagi ’06,…

Measure for Mixed states in AdS/CFT?

Ryu-Takayanagi formula

Motivation

Entanglement entropy : nice measure for pure states

γA S(ρA) =

S(ρA)

ρAB = Tr(AB)c |0ih0|

entanglement wedge

Czech-Karczmarek-Nogueira-VanRaamsdonk, Wall ’12, Headrick, Hubeny-Lawrence-Rangamani ‘14…

Subregion/subregion duality reduced density matrix (a mixed state) B A

(AB)c (AB)c

Measure for Mixed states in AdS/CFT?

Motivation

Czech-Karczmarek-Nogueira-VanRaamsdonk, Wall ’12, Headrick, Hubeny-Lawrence-Rangamani ‘14…

Subregion/subregion duality A natural object in the bulk :

Umemoto-Takayanagi ‘17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ‘17

entanglement wedge cross section

EW

B A

Measure for Mixed states in AdS/CFT?

Motivation

Czech-Karczmarek-Nogueira-VanRaamsdonk, Wall ’12, Headrick, Hubeny-Lawrence-Rangamani ‘14…

Subregion/subregion duality

EW

B A

EP = EW

SR = 2EW

Purification conjectures:

Umemoto-Takayanagi ‘17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ‘17

(S(AA’) for TFD-like purification |ΨAA’BB’>)

Dutta-Faulkner ‘19

(min S(AA’) for ∀ purification |ΨAA’BB’>)

Measure for Mixed states in AdS/CFT?

Motivation

entanglement wedge cross section

from CFT without purification

the entanglement wedge should know its cross section without introducing purified states!

This talk

It from an “odd” generalization of the entropy

EW

So(ρAB) := lim

no:odd→1

Tr(ρTB

AB)no − 1

1 − no Odd (Entanglement) Entropy

KT ‘18

hiA, jB| ⇢TB

AB |kA, `Bi ⌘ hiA, `B| ⇢AB |kA, jBi

・partial transposition: ∃ negative eigenvalue → ∃ entanglement btw A&B

ρTB

AB can have negative eigenvalues

・

Peres ‘96

Odd (Entanglement) Entropy So(ρAB) = S(ρA) (if ρAB is a pure state)

For pure states, the same as usual EE!

So(ρAB) = − X

λi>0

|λi| log |λi| + X

λi<0

|λi| log |λi|

λi : eigenvalues for ρTB

AB

↔ logarithmic negativity

E = lim

ne:even→1 log Tr(ρTB AB)ne ∝ EW

(with back-reaction)

Kudler-Flam—Ryu ‘18

Note: it also counts classical correlations

KT ‘18

Vidal-Werner ’02, Calabrese-Cardy-Tonni ’12

So(ρAB) S(ρAB) EW (ρAB) + =

minimal surfaces EWCS ・Vacuum and Thermal state ・Heavy Excited states ・Quench by local heavy op.

KT’18 Yuya Kusuki & KT to appear

Results in 2d holographic CFT

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

A B

≥

⇔

e.g.) an inequality from entanglement wedge ∵)

EW (ρAB)≥ I(A : B)/2 2EW (ρAB)+S(ρAB) ≥ S(ρA) + S(ρB)

S(ρA), S(ρB)

EW

Freedman-Headrick’16, Umemoto-Takayanagi ‘17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ‘17

It rather specializes the holographic CFT!

A counterexample:

So(ρAB) − S(ρAB) ≥ I(A : B)/2

?

ρAB = q |ΨihΨ| + (1 q)σA ⌦ σB

σ = 1 2 |0ih0| + 1 2 |1ih1|

|Ψi = 1 p 2(|0Ai |1Bi |1Ai |0Bi)

A Lesson from the odd entropy

Discussion

・More general many body systems ・Relation to conditional & difgerential entropy ・From gravitational path integral

c.f. Lewkowycz-Maldacena, Faulkner-Lewkowycz-Maldacena '13

Balasubramanian-Chowdhury-Czech-de Boer-Heller '13, ...

General properties of odd EE ? / How holographic CFT is special?

…

So(ρAB) := lim

no:odd→1

Tr(ρTB

AB)no − 1

1 − no Summary: odd entropy and holography So(ρAB) S(ρAB) EW (ρAB) + =

minimal surfaces EWCS

→ new constraints for holographic CFT

B A