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 1 Entanglement in QM 2 Entanglement in QFT 3 Entanglement in String Theory 4
Entanglement Entropy, QFT and Holography Introduction Introduction
Entanglement Entropy, QFT and Holography Introduction Since it’s first proposal, Entanglement Entropy has been a topic of 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
Entanglement Entropy, QFT and Holography Entanglement in QM Entanglement in QM
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)
Entanglement Entropy, QFT and Holography Entanglement in QM Multi-Body QM H = H A ⊗ H B : � | ψ � AB = ω ij | φ i � A ⊗ | χ j � B ij For instance: | ψ � AB = 1 √ 2 ( |↑ , ↓� ± |↓ , ↑� )
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 |↓� , | ψ � AB is said to be entangled . Entangled state are such that cannot be written as: | ψ � AB = | ψ � A ⊗ | ψ � B (Einstein: You see? Bullshit)
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)
Entanglement Entropy, QFT and Holography Entanglement in QM Density Matrix and Mix States For state | ψ � : ρ = | ψ � � ψ | A generalization, mixed states : � � ρ = p i ρ i = p i | ψ i � � ψ i | i i How mixed is a state? � S V N = − Tr ρ log ρ = − p i log p i i
Entanglement Entropy, QFT and Holography Entanglement in QM Example: binary state with p 1 = p and p 2 = 1 − p : S V N = − p log p − (1 − p ) log(1 − p ) S VN 0.7 0.6 0.5 0.4 0.3 0.2 0.1 p 0.2 0.4 0.6 0.8 1.0
Entanglement Entropy, QFT and Holography Entanglement in QM Entanglement Entropy Suppose H = H A ⊗ H B and ρ = | ψ � � ψ | Define the reduced density matrix : ρ A = Tr B ρ | ψ � = | φ A � ⊗ | χ B � ⇔ ρ A = | φ A � � φ A |
Entanglement Entropy, QFT and Holography Entanglement in QM Entangled ⇔ Mixed Define the Entanglement Entropy as: S ( A ) = − Tr ρ A log ρ A Some properties: S ( A ) = S ( − A ) Strong Subadditivity: S ( ABC ) + S ( B ) ≤ S ( AB ) + S ( BC )
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.
Entanglement Entropy, QFT and Holography Intermedio (QFT and String Theory to follow)
Entanglement Entropy, QFT and Holography Entanglement in QFT Entanglement in QFT (Con dibujitos)
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 ) �
Entanglement Entropy, QFT and Holography Entanglement in QFT Ground state functional: � φ ( � x, 0)= α ( � x ) Dφ ( x ) e − S E [ φ ] Φ( α ) = � 0 | α � ∼ φ ( � x, −∞ )=0 t α ( � x ) x 0
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 ) 0 α ( � x )
Entanglement Entropy, QFT and Holography Entanglement in QFT For H = H A ⊗ H ¯ A , α = α A × β ¯ A : � ρ A ( α ′ Dβ Φ( α ′ ) ∗ Φ( α ) A , α A ) = Tr B ρ ∼ � φ ( � x, 0 + )= α ′ A ( � x ); � x ∈ A Dφ ( x ) e − S E [ φ ] ∼ φ ( � x, 0 − )= α A ( � x ); � x ∈ A
Entanglement Entropy, QFT and Holography Entanglement in QFT t α ′ A ( � x ) 0 α A ( � x ) With compact time: α A ( � x ) t α ′ A ( � x )
Entanglement Entropy, QFT and Holography Entanglement in QFT The trace is: � � Dφe − S E [ φ ] Tr ρ A ∼ Dαρ A ( α, α ) ∼ t
Entanglement Entropy, QFT and Holography Entanglement in QFT Entanglement and R´ enyi entropies: Evaluating S ( A ) = − Tr ρ A log ρ A is well... a pain in the ass. Instead: 1 1 − n log Tr ρ n S n ( A ) = A n → 1 S n ( A ) = S ( A ) lim
Entanglement Entropy, QFT and Holography Entanglement in QFT � Tr ρ n A ∼ ( Dα 1 α 2 ...α n ) ρ A ( α 1 , α 2 ) ...ρ A ( α n , α 1 ) = Z ( n ) Z (1) n
Entanglement Entropy, QFT and Holography Entanglement in QFT Entanglement entropy: 1 − n log Z ( n ) 1 S ( A ) = lim (1) Z (1) n n → 1
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.
Entanglement Entropy, QFT and Holography Entanglement in String Theory Entanglement in String Theory (M´ as dibujitos...)
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
Entanglement Entropy, QFT and Holography Entanglement in String Theory String Theory QFT Type IIB Strings in AdS 5 × S 5 N = 4 SYM with SU ( N ) AdS isometries SO (4 , 2) Conformal symmetry SO (4 , 2) S 5 isometries SO (6) R-symmetry SU (4) 32 super-charges of N = 4 32 Killing spinors � 4 � L λ = Ng 2 Y M l s
Entanglement Entropy, QFT and Holography Entanglement in String Theory AdS geometry Locus: X 2 = − L 2 0 + � − X 2 − 1 − X 2 Coordinate patches: ds 2 = L 4 dz 2 − dt 2 + d� x 2 � � z 2 � r 2 dr 2 � ds 2 = − dt 2 + L 2 + 1 + r 2 d Ω 2 L 2 + 1 r 2
Entanglement Entropy, QFT and Holography Entanglement in String Theory
Entanglement Entropy, QFT and Holography Entanglement in String Theory Boundary conditions and sources Field equations: � − � + m 2 � φ ( z, x ) = 0 m 2 = ∆(∆ − d ) Boundary conditions: z → 0 φ ( z, x ) = z ∆ φ 0 ( x ) lim Correlation functions: d d xφ 0 ( x ) O ∆ ( x ) � CFT = Z SUGRA [ φ 0 ] � � e
Entanglement Entropy, QFT and Holography Entanglement in String Theory Holographic Entanglement Recall 1 − n log Z ( n ) 1 S ( A ) = lim (2) Z (1) n n → 1 Z ( n ) Z (1) n =
Entanglement Entropy, QFT and Holography Entanglement in String Theory Vacuum CFT ⇒ AdS geometry ⇒ R = − d ( d +1) L 2 Green surface ⇒ 2 π ( n − 1) angle deficit ⇒ R = 4 π (1 − n ) δ ( γ A ) Gravitational action: 1 d d +1 x √ g ( R + Λ) = − 1 − n d d x √ g + ... � � S = − 16 πG N 4 G N γ A = − 1 − n Area ( γ A ) + ... 4 G N
Entanglement Entropy, QFT and Holography Entanglement in String Theory Z ( n ) � 1 − n � Z (1) n = Z SUGRA [ γ A ] = exp Area ( γ A ) 4 G N 1 − n 1 ⇒ S ( A ) = lim Area ( γ A ) 1 − n 4 G N n → 1
Entanglement Entropy, QFT and Holography Entanglement in String Theory 1 S ( A ) = Area ( γ A ) 4 G N CFT d γ A A ¯ A AdS d +1 z
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
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
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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