Entanglement Entropy, QFT and Holography Or what some string - - PowerPoint PPT Presentation

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Entanglement Entropy, QFT and Holography Or what some string - - PowerPoint PPT Presentation

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Entanglement Entropy, QFT and Holography Entanglement Entropy, QFT and Holography Or what some string theorists do nowadays? Julio Virrueta Stony Brook University April 5, 2017 Entanglement Entropy, QFT and Holography Outline Introduction

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Entanglement Entropy, QFT and Holography

Entanglement Entropy, QFT and Holography

Or what some string theorists do nowadays? Julio Virrueta

Stony Brook University

April 5, 2017

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Entanglement Entropy, QFT and Holography

Outline

1

Introduction

2

Entanglement in QM

3

Entanglement in QFT

4

Entanglement in String Theory

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Entanglement Entropy, QFT and Holography Introduction

Introduction

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Entanglement Entropy, QFT and Holography Introduction

Since it’s first proposal, Entanglement Entropy has been a topic

  • f interest in several areas of physics:

Validity test for quantum mechanics: Bell inequalities. Many-body quantum mechanics: Tensor networks and phase transitions. Quantum Information and Computing: Practically the whole field. QFT and String Theory

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Entanglement Entropy, QFT and Holography Entanglement in QM

Entanglement in QM

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Entanglement Entropy, QFT and Holography Entanglement in QM

Basics of QM States and Hilbert space: |ψ = α |↑ + β |↓ ∈ H Probabilities: P(↑) = |α|2 P(↓) = |β|2 (Einstein: this is bullshit...hidden variables)

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Entanglement Entropy, QFT and Holography Entanglement in QM

Multi-Body QM H = HA ⊗ HB: |ψAB =

  • ij

ωij |φiA ⊗ |χjB For instance: |ψAB = 1 √ 2 (|↑, ↓ ± |↓, ↑)

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Entanglement Entropy, QFT and Holography Entanglement in QM

EPR paradox If we measure the state on A and observe it to be in the state |↑ the state on B will be, with probability 1, in the state |↓, |ψAB is said to be entangled. Entangled state are such that cannot be written as: |ψAB = |ψA ⊗ |ψB (Einstein: You see? Bullshit)

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Entanglement Entropy, QFT and Holography Entanglement in QM

Bell’s theorem No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. Entanglement is the quintessential QM phenomena! (Eintein’s attemp to show QM is wrong... show QM is the only choice)

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Entanglement Entropy, QFT and Holography Entanglement in QM

Density Matrix and Mix States For state |ψ: ρ = |ψ ψ| A generalization, mixed states: ρ =

  • i

piρi =

  • i

pi |ψi ψi| How mixed is a state? SV N = −Trρ log ρ = −

  • i

pi log pi

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Entanglement Entropy, QFT and Holography Entanglement in QM

Example: binary state with p1 = p and p2 = 1 − p: SV N = −p log p − (1 − p) log(1 − p)

0.2 0.4 0.6 0.8 1.0 p 0.1 0.2 0.3 0.4 0.5 0.6 0.7 SVN

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Entanglement Entropy, QFT and Holography Entanglement in QM

Entanglement Entropy Suppose H = HA ⊗ HB and ρ = |ψ ψ| Define the reduced density matrix: ρA = TrBρ |ψ = |φA ⊗ |χB ⇔ ρA = |φA φA|

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Entanglement Entropy, QFT and Holography Entanglement in QM

Entangled ⇔ Mixed Define the Entanglement Entropy as: S(A) = −TrρA log ρA Some properties: S(A) = S(−A) Strong Subadditivity: S(ABC) + S(B) ≤ S(AB) + S(BC)

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Entanglement Entropy, QFT and Holography Entanglement in QM

If you like this and want to know more about it: http://www.theory.caltech.edu/people/preskill/ph229/ Steven Weinberg. Lectures on quantum mechanics. 2013.

  • A. Einstein, B. Podolsky, and N. Rosen. Can

quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47:777–780, May 1935.

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Entanglement Entropy, QFT and Holography

Intermedio

(QFT and String Theory to follow)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Entanglement in QFT (Con dibujitos)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Density matrix in QFT Real scalar field theory, ˆ φ( x, t). The Hilbert space is spanned by {|α(x)} ˆ φ(x) |α(x) = α(x) |α(x)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Ground state functional: Φ(α) = 0|α ∼ φ(

x,0)=α( x) φ( x,−∞)=0

Dφ(x)e−SE[φ] x t α( x)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Ground state density matrix: ρ = |0 0| and matrix elements: ρ(α′, α) = α′|0 0|α = Φ(α′)∗Φ(α) t α( x) α′( x)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

For H = HA ⊗ H ¯

A, α = αA × β ¯ A:

ρA(α′

A, αA) = TrBρ ∼

  • DβΦ(α′)∗Φ(α)

∼ φ(

x,0+)=α′

A(

x); x∈A φ( x,0−)=αA( x); x∈A

Dφ(x)e−SE[φ]

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Entanglement Entropy, QFT and Holography Entanglement in QFT

t αA( x) α′

A(

x) With compact time: t αA( x) α′

A(

x)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

The trace is: TrρA ∼

  • DαρA(α, α) ∼
  • Dφe−SE[φ]

t

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Entanglement and R´ enyi entropies: Evaluating S(A) = −TrρA log ρA is well... a pain in the ass. Instead: Sn(A) = 1 1 − n log Trρn

A

lim

n→1 Sn(A) = S(A)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Trρn

A ∼

  • (Dα1α2...αn) ρA(α1, α2)...ρA(αn, α1)

= Z(n) Z(1)n

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Entanglement Entropy, QFT and Holography Entanglement in QFT

Entanglement entropy: S(A) = lim

n→1

1 1 − n log Z(n) Z(1)n (1)

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Entanglement Entropy, QFT and Holography Entanglement in QFT

If you liked EE in QFT: Pasquale Calabrese and John L. Cardy. Entanglement entropy and quantum field theory. J.Stat.Mech., 0406:P06002, 2004. Pasquale Calabrese and John Cardy. Entanglement entropy and conformal field theory. J.Phys., A42:504005, 2009.

  • H. Casini and M. Huerta. Entanglement entropy in free

quantum field theory. J.Phys., A42:504007, 2009.

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Entanglement in String Theory (M´ as dibujitos...)

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

AdS/CFT correspondence Striking String Theory result: (Quantum) Gravity in d + 1 = Super Yang-Mills in d

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

String Theory QFT Type IIB Strings in AdS5 × S5 N = 4 SYM with SU(N) AdS isometries SO(4, 2) Conformal symmetry SO(4, 2) S5 isometries SO(6) R-symmetry SU(4) 32 Killing spinors 32 super-charges of N = 4

  • L

ls

4 λ = Ng2

Y M

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

AdS geometry Locus: −X2

−1 − X2 0 +

X2 = −L2 Coordinate patches: ds2 = L4 z2

  • dz2 − dt2 + d

x2 ds2 = − r2 L2 + 1

  • dt2 +

dr2

r2 L2 + 1 + r2dΩ2

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Boundary conditions and sources Field equations:

  • − + m2

φ(z, x) = 0 m2 = ∆(∆ − d) Boundary conditions: lim

z→0 φ(z, x) = z∆φ0(x)

Correlation functions: e

  • ddxφ0(x)O∆(x)CFT = ZSUGRA[φ0]
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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Holographic Entanglement Recall S(A) = lim

n→1

1 1 − n log Z(n) Z(1)n (2)

Z(n) Z(1)n =

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Vacuum CFT ⇒ AdS geometry ⇒ R = −d(d+1)

L2

Green surface ⇒ 2π(n−1) angle deficit ⇒ R = 4π(1−n)δ(γA) Gravitational action: S = − 1 16πGN

  • dd+1x√g (R + Λ) = −1 − n

4GN

  • γA

ddx√g + ... = −1 − n 4GN Area(γA) + ...

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Z(n) Z(1)n = ZSUGRA[γA] = exp 1 − n 4GN Area(γA)

  • ⇒ S(A) = lim

n→1

1 1 − n 1 − n 4GN Area(γA)

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

S(A) = 1 4GN Area(γA) z γA A ¯ A CFTd AdSd+1

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

If you remember something from this talk, let’s that be: Entanglement ⇔ Minimal Surface Or even better: QFT Information ⇔ Geometrical information

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Current problem and research opportunities: EE for dynamical spacetime (time dependent entanglement) Higher Curvature corrections EE for gauge and chiral theories Beyond EE: Geometrization of QFT (Kinematic Space) AdS/MERA

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

If you like... AdS/CFT

  • O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y.
  • Oz. Large N Field Theories, String Theory and Gravity.

Phys.Rept.323:183-386,2000 Edward Witten. Anti-de Sitter space and holography. Adv.Theor.Math.Phys., 2:253–291, 1998. Eric D’Hoker, Daniel Z. Freedman. Supersymmetric Gauge Theories and the AdS/CFT Correspondence. UCLA/02/TEP/3, MIT-CTP-3242

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Entanglement Entropy, QFT and Holography Entanglement in String Theory

Or Holographic EE Shinsei Ryu and Tadashi Takayanagi. Aspects of Holographic Entanglement Entropy. JHEP, 0608:045, 2006. Veronika E. Hubeny. Extremal surfaces as bulk probes in AdS/CFT. JHEP, 1207:093, 2012 Matthew Headrick and Tadashi Takayanagi. A Holographic proof of the strong subadditivity of entanglement entropy. 10.1103/PhysRevD.76.106013

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Entanglement Entropy, QFT and Holography

Gracias

Note: When the speaker is a overly large mexican, courtesy demands to offer him a beer.