Si Signa gnatur ure o of qua quantum cha um chaos i in n op - - PowerPoint PPT Presentation
Si Signa gnatur ure o of qua quantum cha um chaos i in n op operator or entanglement t in 2d 2d CF CFTs MASAHIRO NOZAKI( iTHEMS Riken and UCB) MASAHIRO NOZAKI( iTHEMS Riken and UCB) A postdoc A faculty in China Si Signa gnatur
MASAHIRO NOZAKI( iTHEMS Riken and UCB)
MASAHIRO NOZAKI( iTHEMS Riken and UCB)
A postdoc A faculty in China
MASAHIRO NOZAKI( iTHEMS Riken and UCB)
Based on the collaboration with
Shinsei Ryu, Laimei Nie, Mao Tian Tan, Jonah Kudler-Flam, Eric Mascot, and Masaki Tezuka
arXiv:1812.00013 [hep-th] arXiv:19xx.xxxxx [hep-th]
Ψ1
0,
Ψ2
0,
, Ψ3
0,
Ψ1
0,
Ψ2
0,
, Ψ3
0,
Ψ1
0,
Ψ2
0,
, Ψ3
0,
Ψ1
0,
Ψ2
0,
, Ψ3
0,
Ψ1
0,
Ψ2
0,
, Ψ3
0,
t→∞ U(t)
late
t→∞ U(t)
late
late
Ψ1
0,
Ψ2
0,
, Ψ3
0,
t→∞ U(t)
late
t→∞ U(t)
late
late
Ψ1
0,
Ψ2
0,
, Ψ3
0,
t→∞ U(t)
late
t→∞ U(t)
late
late
Ψ1
0,
Ψ2
0,
, Ψ3
0,
t→∞ U(t)
late
t→∞ U(t)
late
late
In Indep epen enden ent
es, lo locally ally.
Ex.
States can be approximated by thermal state, locally . ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − ⟨Ψ(t)| O |Ψ(t)⟩ → tr
Ex.
States can be approximated by thermal state, locally .
⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − ⟨Ψ(t)| O |Ψ(t)⟩ → tr
Ex.
States can be approximated by thermal state, locally .
⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − ⟨Ψ(t)| O |Ψ(t)⟩ → tr
Ex.
States can be approximated by thermal state, locally .
⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − ⟨Ψ(t)| O |Ψ(t)⟩ → tr
Scrambling effect depends on time evolution operator.
Scrambling effect depends on time evolution operator. I would like to know how the scrambling effect depends on the unitary channels.
Scrambling effect depends on time evolution operator.
Scrambling effect depends on time evolution operator.
Space
input B A time
How much information sent from A to B due to
Space
input B A time
Space
input B A time
Space
input A B
Unitary channel: Channel-state dual map:
Unitary channel: Channel-state dual map:
Dual state: Hilbert space:
Unitary channel: Channel-state dual map:
A regulator for normalization. Dual state: Hilbert space:
Unitary channel: Channel-state dual map:
Only this depends on the initial state. Dual state: Hilbert space:
Unitary channel: Channel-state dual map:
Dual state: Hilbert space:
Our initial state
Energy eigenstate
⎪ ⎩ |Initial⟩ = N
1
Ca |a⟩ ≈ N
Cae−H |a⟩
Unitary channel: Channel-state dual map:
Dual state: Hilbert space:
Our initial state
Energy eigenstate
⎪ ⎩ |Initial⟩ = N
1
Ca |a⟩ ≈ N
Cae−H |a⟩
Unitary channel: Channel-state dual map:
Dual state: Hilbert space:
Space
input A
B A B
Space
input A
B A B
Correlation between input and output subsystems Correlation between A and B in .
Space
input A
B A B
Correlation between input and output subsystems Correlation between A and B in . Measured by mutual information .
Space
input A
B A B
Correlation between input and output subsystems Correlation between A and B in . Measured by , .
Space
input
B A
Space
input
B A
Holographic channel
For disjoint or late-time case, for any B,
⟩ , E(A, B) = 0
Space
input
B A
Holographic channel
Holographic c ch channel shows a signature of scr crambling.
For disjoint or late-time case, for any B,
⟩ , E(A, B) = 0
is the whole of output system.
Output system Input system Time
t
1 2
We have computed tri-partite operator mutual information:
I(A, B) = SA + SB − SA∪B
is the whole of output system.
Output system Input system Time
t
1 2
We have computed tri-partite operator logarithmic negativity:
is the whole of output system.
Output system Input system Time
t
1 2
,E3(A, B1, B2) = 0
is the whole of output system.
Output system Input system Time
t
1 2
This can be interpreted in terms of the relativistic propagation of local objects (quasi-particles) . ,E3(A, B1, B2) = 0
is the whole of output system.
Output system Input system Time
t
1 2
Late time: E
≡ E E − E E3(A, B1, B2) → −2 × 3 4S(1/2)
A
= −2EA, ¯
A
is the whole of output system.
Output system Input system Time
t
1 2
Late time: E
≡ E E − E E3(A, B1, B2) → −2 × 3 4S(1/2)
A
= −2EA, ¯
A
0, ¯ A 0, ¯ A
is the whole of output system.
Output system Input system Time
t
1 2
Late time: E
≡ E E − E E3(A, B1, B2) → −2 × 3 4S(1/2)
A
= −2EA, ¯
A
Lower bound
is the whole of output system.
Output system Input system Time
t
1 2
Late time: E
≡ E E − E E3(A, B1, B2) → −2 × 3 4S(1/2)
A
= −2EA, ¯
A
Lower bound We expect QFT-channels with strong scrambling ability to satisfy this lower bound, eventually.
is the whole of output system.
Output system Input system Time
t
1 2
Late time: E
≡ E E − E E3(A, B1, B2) → −2 × 3 4S(1/2)
A
= −2EA, ¯
A
Scrambling Unscrambling
Free fermion channel Compact boson channel
Scrambling channel (maximally scramble)
Bipartite operator mutual information in the replica trick
A B
Setup:
State:
I(A, B) = SA + SB − SA∪B = lim
n→1
A
+ S(n)
B
− S(n)
A∪B
→
∪
n→1
1 1 − n [log trA (ρA)n + log trB (ρB)n − log trA∪B (ρA∪B)n]
A B
Setup:
B A A B
Bipartite operator mutual information in the replica trick
B A A B
B A A B
: Twist operator in A : Twist operator in B
B A A B
: Twist operator in A : Twist operator in B
Independent of channels
B A A B
: Twist operator in A : Twist operator in B
Depend on channels!!
→
− I(A, B) = lim
n→1
1 1 − n [log trA (ρA)n + log trB (ρB)n − log trA∪B (ρA∪B)n]
Bipartite operator mutual information in the replica trick
Bipartite operator mutual information in the replica trick
E EA,B = lim
ne→1 log
A∪B
ne
Bipartite operator mutual information in the replica trick
E EA,B = lim
ne→1 log
A∪B
ne
B A
Bipartite operator mutual information in the replica trick
E EA,B = lim
ne→1 log
A∪B
ne
B A
→
ne¯
σA
ne¯
σB
neσB ne
We consider the following setups to extract properties of free fermion channel:
A B A B A B
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B
Red cu curve : Purple cu curve : Blue cu curve :
Slopes and bumps shows properties of free fermion ch
channel are interpreted in terms of the relativistic c propagation of quasi-particl cles.
A B B1 B2
A B B1 B2
A B B1 B2
E3(A, B1, B2) ) =
I(A, B1, B2) = 0
The time evolution of operator mutual information (logarithmic negativity) and tripartite operator mutual information (logarithmic negativity) for free fermion channel can be interpreted in terms of the relativistic propagation of local
the input subsystem size .
A
# of particles in B.
A B
Red cu curve :
A B
Purple cu curve :
A B
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B A B
A B
Red cu curve : Purple cu curve : Blue cu curve :
A B A B A B
propagation of local objects, too.
1 2
propagation of local objects, too.
1 2
@time =0
propagation of local objects, too.
1 2
@time =0
propagation of local objects, too.
1 2
@time =0
propagation of local objects, too.
1 2
@time =0
propagation of local objects, too.
1 2
@time = t
propagation of local objects, too.
1 2
@time = t
propagation of local objects, too.
1 2
@time = t
propagation of local objects, too.
1 2
@time = t
propagation of local objects, too.
1 2
@time = t
We consider the following setups to extract properties of compact boson and holographic channels by comparing them to free fermion channel:
A B A B A B
We consider the following setups to extract properties of compact boson and holographic channels by comparing them to free fermion channel:
A B A B A B
Holographic channel:
A B
:Blue :Red :Purple :Black dash Red Black Blue Purple
Free fermion channel: Compact boson channel:
Holographic channel:
A B
:Blue :Red :Purple :Black dash Red Black Blue Purple
Free fermion channel: Compact boson channel:
Qu Quasi-par particle pi pictur ure w works w well.
FF channel:
CB channel:
Holographic channel:
A B
:Blue :Red :Purple :Black dash
Ho Hologra raphic channel el does esn’t show w a platea eau.
Red Blue Purple Black
FF channel:
CB channel:
Holographic channel:
A B
:Blue :Red :Purple :Black dash
Thi This pr prope perty i is be beyond t nd the he par particle i interpr pretat ation. n.
Red Blue Purple Black
A B
Red Black Blue Purple
A B
Red Black Blue Purple
Monotonically y decreasing On Once ce quantum information le leak aks from B, , it keeps to leak k be before al all i inf nformat ation l n leak aks.
A B
Red Black Blue Purple
Monotonically y decreasing On Once ce quantum information le leak aks from B, , it keeps to leak k be before al all i inf nformat ation l n leak aks. If If quantum inform rmation is carri rried ed by y local objects,
A B Th The slo lope chan anges at .
Red Blue Purple Black
A B
A B Quantum information keeps s to go out from the left boundary y wi with . At At t=0, right-mo moving ng signal gnal appe appear ars at at the he righ ght bo boundar undary of A. . It Its speed eed is is the e lig light’s.
A B A B
A B A B Wh When en the e rig ight-moving g si sign gnal hits s the righ ght bo bounda undary of B, the the inform rmati tion n starts rts to go
B wi with .
A B A B de decrease ses s tw twice faster tha than n tha that t in n . in
A B A B Al All the the informati tion n of A A go goes o
ut f from m B B @ @ t= t=l +s/2.
A B
Red Blue Purple
Fo For , lin linearly rly decr creases .
Black
@ @ .
A B
Al All the informa mation of A A goes ou
t fr from the left ft boundary of f B be before the he signa nal arrives at the he ri right boundary. Once ce whole in informatio ion has as gone, the sig ignal al di disappe ppears.
A B Al All the the informati tion n in n A A goes
e left bou
B be before the the si signa gnal arrives s at t the the ri right t bo bounda
he signa nal di disa sappe ppears. s. Ea Early time:
La Late ti time: :
:Blue :Red :Purple :Black dash
A B
If A and B are disjoint( ),
Holographic channel: Free fermion channel: Compact boson channel:
FF channel:
CB channel:
Holographic channel:
:Blue :Red :Purple :Black dash
A B
If A and B are disjoint( ),
Time evolution can be interpreted in terms of particles, almost.
FF channel:
CB channel:
Holographic channel:
:Blue :Red :Purple :Black dash
A B
If A and B are disjoint( ),
FF channel:
CB channel:
Holographic channel:
:Blue :Red :Purple :Black dash
A B
If A and B are disjoint( ),
Information Scrambling.
Space
input
B A
Bumps
Bumps Can be interpreted in terms of particles
Space
input
B A
@
@
Here, you can get information locally.
@
@
Here, you can get information locally.
@
@
Here, you can get information locally.
@
Here, you can get information locally.
@
@
Here, you can get information locally.
@
Here, you can get information locally.
Space
input
B A
For disjoint or late-time case, for any B
Space
input
B A
For disjoint or late-time case, for any B
Beyond the quasi-particle model
@
Everywhere , you can’t get information locally at late time .
Space
input
B A
For disjoint or late-time case, for any B
Beyond the quasi-particle model We cannot mine the information in A from B locally.
Space
input
B A
Beyond the quasi-particle model We cannot mine the information in A from B locally.
Signature of information scrambling
For disjoint or late-time case, for any B
Space
input
B A
Beyond the quasi-particle model We cannot mine the information in A from B locally.
Signature of information scrambling
For disjoint or late-time case, for any B
(It might be different from usual one… Sorry.)
We cannot mine any information about A locally, but we can mine the information from the whole of output system.
We cannot mine any information about A locally, but we can mine the information from the whole of output system.
B A B1
0 E(A, B1) = 0
We cannot mine any information about A locally, but we can mine the information from the whole of output system.
B A B1 B A
0 E(A, B1) = 0
0 E(A, B) ̸= 0
Tripartite information (Tripartite logarithmic negativity) is useful quantity in order to treat this phenomenon, quantitatively.
We cannot mine any information about A locally, but we can mine the information from the whole of output system.
[Hosur-Qi-Roberts-Yoshida’16]
Tripartite operator mutual information
Output system Input subsystem
B B B
1 2
A
is the whole of output system.
B B 1 and are the halves of output system. B
A is a subsystem in input system.
2
the information of A from the whole of output system B.
Output system Input subsystem
B B B
1 2
A
is the whole of output system.
B B 1 and are the halves of output system. B
A is a subsystem in input system.
2
B A
Tripartite operator mutual information
the information of A from the whole of output system B.
Output system Input subsystem
B B B
1 2
A
is the whole of output system.
B B 1 and are the halves of output system. B
A is a subsystem in input system.
2
B A B 2 A B 1
the information of A from the subsystems.
Tripartite operator mutual information
the information of A from the whole of output system B.
Output system Input subsystem
B B B
1 2
A
is the whole of output system.
B B 1 and are the halves of output system. B
A is a subsystem in input system.
2
B A B 2 A B 1
the information of A from the subsystems.
Tripartite operator mutual information
Tripartite operator logarithmic negativity
the information of A from the whole of output system B.
Output system Input subsystem
B B B
1 2
A
is the whole of output system.
B B 1 and are the halves of output system. B
A is a subsystem in input system.
2
B A B 2 A B 1
the information of A from the subsystems.
∼
E3(A, B1, B2) = E(A, B1) + E(A, B2) − E(A, B)
Tripartite operator mutual information (logarithmic negativity)
If information mined from subsystems and is smaller than the information from whole of output system ,
E B1 B2 B
Tripartite operator mutual information (logarithmic negativity)
If information mined from subsystems and is smaller than the information from whole of output system ,
) B1
1 B2
I(A, B1) + I(A, B2) ) E(A, B1) + E(A, B2)
B
Tripartite operator mutual information (logarithmic negativity)
If information mined from subsystems and is smaller than the information from whole of output system ,
) B1
1 B2
I(A, B1) + I(A, B2) ) E(A, B1) + E(A, B2) B − B − A, B E I(A, B) E ) E(A, B) − < 0 − < 0
Tripartite operator mutual information (logarithmic negativity)
If information mined from subsystems and is smaller than the information from whole of output system ,
) B1
1 B2
I(A, B1) + I(A, B2) ) E(A, B1) + E(A, B2) B − B − A, B E I(A, B) E ) E(A, B) − < 0 − < 0
Tripartite operator mutual information (logarithmic negativity)
If information mined from subsystems and is smaller than the information from whole of output system ,
) B1
1 B2
I(A, B1) + I(A, B2) ) E(A, B1) + E(A, B2) B − B − A, B E I(A, B) E ) E(A, B) − < 0 − < 0
Some information is hidden in whole of output system due to information scrambling effect.
This quantity can quantify the effect of information scrambling.
Output system Input system Time
t
1 2
Setup:
10 20 30 40
10 20 30 40
@ late time,
Constant
At late time,
We cannot mine information locally, but we can mine the information about A from the whole of output system B.
Constant
At late time,
We cannot mine information locally, but we can mine the information about A from the whole of output system B.
Constant
At late time,
We cannot mine information locally, but we can mine the information about A from the whole of output system B.
Constant
At late time,
We cannot mine information locally, but we can mine the information about A from the whole of output system B.
All information sent from A is scrambled.
Constant
At late time,
We cannot mine information locally, but we can mine the information about A from the whole of output system B.
They measure how much information is scrambled.
10 20 30 40
10 20 30 40
@ late time,
In In the lo low energ rgy lim limit it,
these ki kinks can be negligible.
.
Bipartite operator mutual information (logarithmic negativity)
t = t1
@
Free fermion and Compact boson channels: Holographic channel:
t = t1
@
Tripartite operator mutual information (Tripartite operator logarithmic negativity)
Free fermion channels:
1 2
Quasi-particles
Holographic channel:
All initial information is scrambled.
E E3(A, B1, B2) → −2E(A, ¯ A)