Butterflies from Information Metric hep-th/1507.07555 Phys. Rev. - - PowerPoint PPT Presentation
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
hep-th/1607.01467 (to appear in JHEP) “Butterflies from Information Metric”
Masamichi Miyaji
Yukawa Institute for Theoretical Physics, Kyoto University
hep-th/1507.07555
“Distance between Quantum States and Gauge-Gravity Duality”
Motivation and Outline
Fisher information metric should have holographic dual !
First law of entanglement = Einstein equation, Connectivity ~ Entanglement, dual of Renyi and relative entropy, AdS/ Tensor network, etc
Fisher information
Semiclassically, scrambling can be related to butterfly effect.
Scrambling
We find fast scrambling is equivalent to sensitive dependence of the system on external environment.
We give explicit example of this using gravity dual of Fisher information.
Ryu-Takayanagi formula relates bulk geometry to boundary quantum imformation.
hep-th/1507.07555
“Distance between Quantum States and Gauge-Gravity Duality”
Quantum Information Theory : Fisher Information Metric
F(λ, λ + δλ) = |hΨλ|Ψλ+δλi| = 1 Gλλ · (δλ)2 + O(δλ3)
Fidelity Fisher Information Metric
|Ψλi
H0 + λV
λc
F(λ, λ + δλ)
λ
1
Order parameter for quantum phase transitions. Fidelity of vacuum states of decays at critical points.
Critical point
Order parameter [Quan, Song, Liu, Zanardi, Sun][Zanardi, Paunkovic]
Reciprocal of Fisher Information gives lower bound of variance of unbiased estimator of parameter λ. (Cramer-Rao bound)
Estimation theory For any linear operator with ,
holds.
hΨλ|ˆ λ|Ψλi = λ
hΨλ|(ˆ λ λ)2|Ψλi 1 Gλλ
Holographic dual of Fisher Information metric
|hΨλ+δλ(t0)|Ψλ(t0)i| = 1 Gλλ(δλ)2 + O((δλ)3)
Gλλ = nd Z
Σd
√g
Fisher information metric of states with Hamiltonian for marginal deformation V is given by the volume
where is extremal volume surface.
Σd
[M.M, Numasawa, Shiba, Takayanagi, Watanabe ]
HCF T + λ · V
AdS
t = t0
Proposal
Σd
Remarks
[Susskind, Stanfords]
[Lashkari,Raamsdonk] [Lashkari, Lin, Ooguri, Stoica, Raamsdonk]
information.
Motivation from AdS/Tensor network proposal
HCF T + (λ + δλ) · V HCF T + λ · V
Marginal deformation modifies each tensors of tensor network uniformly. One can assume those tensors contribute to Fisher information metric uniformly.
Gλλ ∝ Vol(Σ)
Motivation from Tensor network
AdS/Tensor network
Tensor network: Contraction of tensors can express complicated states. Entanglement entropy in Tensor network and AdS/CFT look alike.
[Swingle]
Proposal: MERA tensor network = Bulk in AdS/CFT
Two sided BH
|ΨT F D(β)i
|ΨT F D(β, t)i
u
v
t
Almost identical time evolution of Fisher information can be confirmed in 2d CFT computation. Two sided BH
d = 2
Dual to Thermofield double state of CFT |ΨT F D(β, λ, t)i
Example
Only small deviation!
Maximal volume surface
Asymptotic behavior: Exact Match!
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 Ac
[Sekino, Susskind]
Dual CFT of Einstein gravity is fastest scrambler among large N
Scrambling
{q(t), p(0)}2
P = ( δq(t)
δq(0))2
Semiclassically,
Scrambling of excitation W is characterized by the growth of
for all local Vs. t
h[W(t), V ]2iβ
δq(0)
δq(t)
I want to describe scrambling by butterfly effect in fully quantum mechanical way.
[Larkin, Ovchinnikov] Kitaev, Maldacena, Shenker, Stanfords, Reberts, etc…
∝ 1 N 2 eλLt
∝ eλt
For holographic CFT,
λL = 2π β = 2πkBT ~
Quantum Butterfly Effect
[Peres, Jalabert, Pastawski, Jacquod, Silvestrov, Beenakker, Cerruti, Wisniacki, Cucchietti, Gorin, Prosen, Zurek, Seligman, etc…]
|Ψλi |Ψλ+δλi
t
Hλ+δλ Hλ
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.
Hλ+δλ
Hλ
|hΨλ+δλ|eiHλ+δλte−iHλt|Ψλi|
= 1 − Gλλ(δλ)2 + O((δλ)3) Fisher information metric
Themofield double state
|ΨT F D(β, λ, tw)W i = e−iHλtwWeiHλtw|ΨT F D(β, λ, tw)i
We perturb TFD state at by acting on . t = −tw
W
HL
TFD state at t=0 is
|ΨT F D(β)i = 1 Z(β) X
i
e− βEi
2 |Eii ⌦ |Eii ∈ HL ⊗ HR
Thermofield double state e−iHλtwWeiHλtwρ(β)e−iHλtwWeiHλtw
Tracing out right Hilbert space gives |hΨT F D(β, λ + δλ, tw)W |ΨT F D(β, λ, tw)W i|
We wil consider = 1 − Gλλ(δλ)2 + O((δλ)3)
W
1 2 Z tw dt1dt2Tr h e−βH ZW (β, λ) h W(−tw), V (−t1) i · h W(−tw), V (−t2) i†i + i 2 Z
β 2
dtE Z tw dtTr h e−βH ZW (β, λ)eHtEV e−HtEh h W(−tw), V (−t) i , W(−tw) ii
−1 2( Z tw dtTr h e−βH ZW (β, λ) h W(−tw), V (−t) i W(−tw) i )2
All of the terms contain commutator
h W(−tw), V (−t) i
GW :c
λλ
= Re
h
i
GW
λλ = G(0) λλ + GW :c λλ
Fisher Information metric of TFD state
Fisher information metric
Trivial, time independent part.
Chaotic part of Fisher information metric
At late time,
GW :c
λλ
∝ 1 N 2 eλLt
u
v
α
Maximal volume brane
Shock wave geometry
u
v
Two sided BH
G(W :c)
λλ
= nd∆Vol
G(0)
λλ = ndVol
Numerical result shows
α = E 4M e
2π β tw
GW :c
λλ
grows proportional to
GW :c
λλ
= f0α + O(α2)
f0 > 0
f0 ∼ O(N 0)
2∆(Vol) R2
R = 2π β
Holographic computation
2d CFT
Conclusion
We proposed volume of extremal volume surface as gravity dual of Fisher information metric. Growth of volume of extremal volume surface in shock wave geometry is consistent with butterfly effect.
We find equivalence between fast scrambling and sensitivity to external environment.