Butterflies from Information Metric hep-th/1507.07555 Phys. Rev. Lett 115 (2015) with Numasawa, Shiba, Takayanagi, Watanabe “Distance between Quantum States and Gauge-Gravity Duality” hep-th/1607.01467 (to appear in JHEP) “Butterflies from Information Metric” Masamichi Miyaji Yukawa Institute for Theoretical Physics, Kyoto University
We give explicit example of this using gravity dual of Fisher information. Motivation and Outline Fisher information Ryu-Takayanagi formula relates bulk geometry to boundary quantum imformation. First law of entanglement = Einstein equation, Connectivity ~ Entanglement, dual of Renyi and relative entropy, AdS/ Tensor network, etc Fisher information metric should have holographic dual ! Scrambling Semiclassically, scrambling can be related to butterfly effect. We find fast scrambling is equivalent to sensitive dependence of the system on external environment.
1. Gravity dual of Fisher Information Metric hep-th/1507.07555 Phys. Rev. Lett 115 (2015) with Numasawa, Shiba, Takayanagi, Watanabe “Distance between Quantum States and Gauge-Gravity Duality”
Quantum Information Theory : Fisher Information Metric F ( λ , λ + δλ ) = | h Ψ λ | Ψ λ + δλ i | = 1 � G λλ · ( δλ ) 2 + O ( δλ 3 ) Fisher Information Metric Fidelity Order parameter [Quan, Song, Liu, Zanardi, Sun][Zanardi, Paunkovic] Order parameter for quantum phase transitions. Fidelity of vacuum states of decays at critical points. | Ψ λ i H 0 + λ V Critical point F ( λ , λ + δλ ) 1 λ c λ Estimation theory Reciprocal of Fisher Information gives lower bound of variance of unbiased estimator of parameter λ . (Cramer-Rao bound) For any linear operator with , h Ψ λ | ˆ λ | Ψ λ i = λ 1 h Ψ λ | (ˆ λ � λ ) 2 | Ψ λ i � holds. G λλ
Holographic dual of Fisher Information metric [M.M, Numasawa, Shiba, Takayanagi, Watanabe ] Proposal Fisher information metric of states with Hamiltonian H CF T + λ · V | h Ψ λ + δλ ( t 0 ) | Ψ λ ( t 0 ) i | = 1 � G λλ ( δλ ) 2 + O (( δλ ) 3 ) for marginal deformation V is given by the volume AdS Σ d t = t 0 Z G λλ = n d √ g Σ d where is extremal volume surface. Σ d
Remarks • Able to capture nontrivial time dependent Fisher information. • Not the distance between vacuum and excited states. [Lashkari,Raamsdonk] [Lashkari, Lin, Ooguri, Stoica, Raamsdonk] • Motivated by AdS/tensor network picture. • Consistent with nontrivial time dependent example. • Maximal volume was first used in the context of complexity . [Susskind, Stanfords]
Motivation from AdS/Tensor network proposal AdS/Tensor network Tensor network: Contraction of tensors can express complicated states. Entanglement entropy in Tensor network and AdS/CFT look alike. Proposal: MERA tensor network = Bulk in AdS/CFT [Swingle] Motivation from Tensor network Marginal deformation H CF T + ( λ + δλ ) · V H CF T + λ · V modifies each tensors of tensor network uniformly. One can assume those tensors contribute to Fisher Σ information metric uniformly. G λλ ∝ Vol( Σ )
Example Two sided BH Dual to Thermofield double state of CFT | Ψ T F D ( β , λ , t ) i Almost identical time evolution of Fisher information can be confirmed in 2d CFT computation. Two sided BH u Asymptotic behavior: Exact Match! v | Ψ T F D ( β , t ) i Only small deviation! t Maximal | Ψ T F D ( β ) i volume surface d = 2
2. Butterfly effect from Fisher information hep-th/1607.01467 (to appear in JHEP) “Butterflies from Information Metric”
Scrambling Local excitation is added to thermal equilibrium. Excitation is scrambled. = Local measurement can not extract information of the original excitation. A c A • Black hole information paradox [Hayden, Preskill] • Fast scrambling conjecture [Sekino, Susskind] Dual CFT of Einstein gravity is fastest scrambler among large N theories. [Shenker, Stanford, Maldacena]
Scrambling Scrambling of excitation W is characterized by the growth of 1 h [ W ( t ) , V ] 2 i β N 2 e λ L t ∝ For holographic CFT, for all local Vs. Kitaev, Maldacena, Shenker, Stanfords, Reberts, etc… λ L = 2 π β = 2 π k B T ~ Semiclassically, P = ( δ q ( t ) ∝ e λ t { q ( t ) , p (0) } 2 δ q (0)) 2 [Larkin, Ovchinnikov] δ q ( t ) I want to describe scrambling by butterfly effect in fully quantum mechanical way. t δ q (0)
Quantum Butterfly Effect Inner product between two states with identical Hamiltonian will be conserved. Inner product between two states with different Hamiltonians and decays rapidly in chaotic systems. H λ H λ + δλ [Peres, Jalabert, Pastawski, Jacquod, Silvestrov, Beenakker, Cerruti, Wisniacki, Cucchietti, Gorin, Prosen, Zurek, Seligman, etc…] | h Ψ λ + δλ | e iH λ + δλ t e − iH λ t | Ψ λ i | = 1 − G λλ ( δλ ) 2 + O (( δλ ) 3 ) H λ Fisher information metric H λ + δλ | Ψ λ i t | Ψ λ + δλ i
Themofield double state Thermofield double state 1 e − β Ei X 2 | E i i ⌦ | E i i ∈ H L ⊗ H R | Ψ T F D ( β ) i = Z ( β ) i We perturb TFD state at by acting on . H L W t = − t w TFD state at t=0 is | Ψ T F D ( β , λ , t w ) W i = e − iH λ t w We iH λ t w | Ψ T F D ( β , λ , t w ) i Tracing out right Hilbert space gives e − iH λ t w We iH λ t w ρ ( β ) e − iH λ t w We iH λ t w We wil consider | h Ψ T F D ( β , λ + δλ , t w ) W | Ψ T F D ( β , λ , t w ) W i | = 1 − G λλ ( δλ ) 2 + O (( δλ ) 3 ) W
Fisher Information metric of TFD state Fisher information metric λλ = G (0) λλ + G W : c G W λλ Trivial, time independent part. Chaotic part of Fisher information metric Z t w e − β H 1 i † i h h h i h G W : c dt 1 dt 2 Tr W ( − t w ) , V ( − t 1 ) W ( − t w ) , V ( − t 2 ) = Re · λλ 2 Z W ( β , λ ) 0 Z t w β e − β H + i Z i 2 h Z W ( β , λ ) e Ht E V e − Ht E h h i ii dt Tr W ( − t w ) , V ( − t ) , W ( − t w ) dt E 2 0 0 Z t w e − β H − 1 h h i i ) 2 2( dt Tr W ( − t w ) , V ( − t ) W ( − t w ) Z W ( β , λ ) 0 h i All of the terms contain commutator W ( − t w ) , V ( − t ) 1 At late time, G W : c N 2 e λ L t ∝ λλ
Holographic computation Shock wave geometry Two sided BH u v v u α Maximal volume brane G (0) G ( W : c ) λλ = n d Vol = n d ∆ Vol λλ α = E 2 π grows proportional to Numerical result shows G W : c β t w 4 M e λλ 2 ∆ (Vol) 2d CFT R 2 R = 2 π β G W : c = f 0 α + O ( α 2 ) λλ f 0 ∼ O ( N 0 ) f 0 > 0
Conclusion We proposed volume of extremal volume surface as gravity dual of Fisher information metric. We find equivalence between fast scrambling and sensitivity to external environment. Growth of volume of extremal volume surface in shock wave geometry is consistent with butterfly effect.
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