Entanglement distillation in tensor networks

Takato Mori

γA

A

Based on arXiv:2205.06633 [hep-th]
with H. Matsueda (Tohoku U) and H. Manabe (Kyoto U)
熱場の量子論とその応用, 2022年9月21日（オンライン参加）

Take-home message

Tensor network de nes a set of
reduced transition matrices
They describe entanglement distillation via geometry
The method works for arbitrary tensor networks;a systematic, quantitative study of states vs. geometryfi

in tensor networks is now possible

Outline
1. Introduction ̶ AdS/CFT, tensor networks as
toy models
2. Entanglement distillation in MERA
3. Numerical results for random MERA
4. Entanglement distillation in MPS
5. Summary
3 /36

Motivation: Understanding AdS/CFT from entanglement(d+1)-dim. AdS spacetime⇆

d-dim. quantum eld theory(CFT)

geometry[Maldacena]quantum information

Ryu-Takayanagi formula [Ryu-Takayanagi]=

SA = − trρA log ρA, ρA = trĀ ρ
Area(γA)SHEE (A) = minγA
4GN

subregionγA🤔 Beyond AdS (hyperbolic) / CFT (critical)?

A
t=0
surfacecf. [Freedman-Headrick]Can we really see EPR pairs across γA?
fi

How does the geometry arise from a wave function?

4 /36

Tensor networks as toy models of holography• Tensor networks (=variational wave function) provide a
qualitative picture[Swingle]

STN (A) ≲ min (# bond cut by γA) × log χ 〜RT formula?
γAγAMulti-scale entanglement renormalization ansatz(MERA)[Vidal]• Some proposals try to mimic holography (esp. RT formula)| Ψ⟩ = ( ⊗⟨x,y⟩ ⟨MES |xy ) ( ⊗x Urandom | 0x⟩)

Holographic state[Pastawski et al.]
CONS: Lack of expressivity; TN state

Random TN with large bond dim.
[Hayden et al.]
[Qi-Yang][Apel et al.]conformally invariant

5 /36

Entanglement distillation in holographic tensor networks[Pastawski et al.](Similar work: [Bao-Penington-Sorce-Wall], [Lin-Sun-Sun])• A holographic state (or isometric tensor networks in general)is known to geometrize entanglement distillation= extracting SA bits of EPR pairs from the state• Removing tensors, we obtain EPR pairs across the minimal∵ SA(V | Ψ⟩) = SA( | Ψ⟩)surface

j

i

Ac

Ac

=

δij
2

| i⟩ | j⟩= isometry

A

A

e.g.

=

6 /36

Entanglement distillation in holographic tensor networks[Pastawski et al.](Similar work: [Bao-Penington-Sorce-Wall], [Lin-Sun-Sun])• A holographic state (or isometric tensor networks in general)is known to geometrize entanglement distillation= extracting SA bits of EPR pairs from the state• Removing tensors, we obtain EPR pairs across the minimal∵ SA(V | Ψ⟩) = SA( | Ψ⟩)surfaceδijj TNs.
| i⟩ | j⟩=i
So far thismanipulationislimitedtoisometric
c
c
A
A
2

Geometrization of entanglement distillation in other types of TNs?
= isometry➜ Clarify the operational role of internal d.o.f. after optimization!
A

A

e.g.

=

7 /36

✔︎

✔︎

Entanglement distillation in tensor networks
The important aspect for entanglement distillation is
1. Conservation of entanglement (entropy)
Reduced transition matrix instead of reduced density
matrix2. Extracting strongly entangled pairs
Nontrivial for non-isometic TNs but can besystematically studied

8 /36

✔︎

✔︎

Entanglement distillation in tensor networks
The important aspect for entanglement distillation is
1. Conservation of entanglement (entropy)
Reduced transition matrix instead of reduced density
matrix2. Extracting strongly entangled pairs
Nontrivial for non-isometic TNs but can besystematically studied

9 /36

Multi-scale entanglement renormalization ansatz(MERA)[Vidal]• MERA has minimal bond cut surface(s) γ* = min(γA, γĀ)?

• What we want to see: A state on γ* = EPR pairs• Need to provide a method to properly de ne a state on abond cut surface γ

A := @C(A)| Ψ⟩A Ā =<latexit sha1_base64="HyIMYPN7dH0LiVtvKgpmGmZokcw=">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</latexit>

Ā := @C(Ā)<latexit sha1_base64="2ZvN9JhqJ6vIjFuDSt8C8DWd2nc=">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</latexit>

C(A)<latexit sha1_base64="o9c/i/FTd7ymK/HqG1k1ZW0A6Co=">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</latexit>

C(Ā)

A

Ā
fi

<latexit sha1_base64="mVfSNej3msmLnYbIWqIEDoZDn/I=">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</latexit>

=
=
†
10/36

[TM-Manabe-Matsueda]• Cut internal bonds across a bond cut surface γ
instead of removing tensors from
This de nes a reduced transition matrix ργ = trĀ ( | Ψ(γ)⟩⟨Φ(γ) | ) on ℋγ
we consider foliations {γ}• To relate it with entanglement distillation,(a)(a)

s.t. ∂γ = ∂A.

∈ ℋγ ⊗ ℋĀ

| Ψ(γ)⟩ =

γ

ργ =

(b)
(b)

UU1
1

⟨Φ(γ) | =≡ ⟨Ψ | Mγfi➡︎Entanglement distillation in MERA

UU22

∈ ℋ*γ ⊗ ℋ*Ā

11 /36

Entanglement distillation in MERA
[TM-Manabe-Matsueda]• Changing the location of foliations γ, we obtain a family of states
on various bond cut surfaces (e.g. ρA for γ = A, ργ* for γ = γ*)• Entanglement conservation w.r.t. ∀γ : S(ργ) = S(ρA), where S(ργ) is
the pseudo entropy [Nakata-Takayanagi et al.] S(ργ) = − trργ log ργ
(a)

(b)

(c)

γA
ρA =

(d)

ργ′ =

ργ =

γ′

γ

ργA =

(e)

(f)

γĀ

ργĀ =

ρĀ =
12/36

⬅︎

Entanglement distillation in MERA
[TM-Manabe-Matsueda]• Changing the location of foliations γ, we obtain a family of states
on various bond cut surfaces (e.g. ρA for γ = A, ργ* for γ = γ*)• Entanglement conservation w.r.t. ∀γ : S(ργ) = S(ρA), where S(ργ) is
the pseudo entropy S(ργ) = − trργ log ργThis is owing to the common eigenvalue distribution
between ργ and ρA ; the same entanglement spectrum!

• Furthermore, when Mγ is isometric, we can show ⟨Φ(γ) | = ⟨Ψ(γ) |⇒ For isometric TNs, we obtain EPR pairs across the minimalbond cut surface (i.e. ργ* ∝ 1) when γ = γ*
13/36

Entanglement distillation in MERA
[TM-Manabe-Matsueda]• To go beyond the isometric case, we need to de ne a state from a
reduced transition matrix.
We use the puri cation technique (a.k.a. channel-state duality)| ργ1/2⟩ ≡whereand | EPRγ⟩ = (dim ℋγ)−1/2

dim ℋγ∑| i⟩ ⊗ | i⟩.

i=1This will be regarded as a geometrically distilled state up to γ by TN.
Now we can make a quantitative comparison w.r.t. EPR pairs!
Trace distance from the EPR pair: Dγ ≡1−⟨EPRγ | ργ1/2⟩22
γ(related to Rényi-1/2 entropy ⟨EPRγ | ργ1/2⟩
fi

fi

𝒩

𝒩

𝒩
➡︎

† 1/2 1/2=tr(ργ[ γ ργ )]−1/2

γdim ℋγ (ργ1/2 ⊗ 1) | EPRγ⟩,=dim ℋγe S1/2)2

14/36

Random MERA

• Random TN is expected to reproduce RT formula in thelarge bond dimension limit [Hayden et al.]

🤔 Why is the minimal bond cut surface is special?
(RT formula only tells us about entanglement entropy)🤔 The large bond dimension limit is essential?

| 0⟩= HaarγA=†(All tensors are Haar random)

=

15/36

Numerical results for random MERA
[TM-Manabe-Matsueda]

Choice of foliations:
(a)

(b)

(c)

γA
ρA =

(d)

ργ′ =

ργ =

γ′

γ

ργA =

(e)

(f)

γĀ

ργĀ =

ρĀ =

16/36

Numerical results for random MERA
[TM-Manabe-Matsueda]

We compared the (averaged) trace distance Dγ ≡1 − ⟨EPRγ | ργ1/2⟩

2

between | ργ1/2⟩ and the EPR pair | EPRγ⟩.

l=6

L = 8, l = 4

L = 16l=8

isometric Mγ(incl. γ = A)
non-isometric MγMγ* (γ is minimal
surface)bond dimension

17/36

Entanglement distillation in MPS[TM-Manabe-Matsueda]

Similarly consider pushing the foliation towards the minimalbond cut surface in matrix product states (MPS)
Let us focus on the MPS in a mixed canonical form

18/36

Entanglement distillation in MPS[TM-Manabe-Matsueda]

Note:
An MPS in a mixed canonical form is an analog of MERA
(regarding its structure)

=

σ

γA

V

̂
Σ

γĀ

W

⟷

σ
V′

†

W′

19/36

Entanglement distillation in MPS[TM-Manabe-Matsueda]

We can consider the following foliations γ = γ(τ), τ = 0,1,2,3(τ ∼ distance from the minimal bond cut surface)

20/36

Entanglement distillation in MPS[TM-Manabe-Matsueda]

The reduced transition matrix and the trace distance areDγ(τ) =

e S1/2
1 − τ+1 ,χτ = 0,1,2

Dγ(3) = Dγ(2)The distillation by pushing the foliation equals removing
redundant tensors.
The entanglement spectrum σ remains unchanged.

21/36

Summary• By pushing towards the minimal surface γ*, strongly entangled pairs
are geometrically distilled in tensor networks while retaining theentanglement spectrum• It is essential to consider reduced transition matrices rather than areduced density matrix• Our method works for non-holographic TNs: This suggests geometryof TN is intimately related to distillation for generic TNs
Holography beyond AdS/CFT
Emergent geometry from distillation
Future directions• Operational interpretation of geometric distillation: local conf. trf., cornertransfer matrix, modular ow? [Milsted-Vidal; Nishino-Okunishi; Okunishi-Seki]• Analytic “proof” beyond MPS using analytic MERA rep. [Evenbly-White]• CFT realization? Quant. adiabatic comp. (~annealing) with T T̄
22/36
[McGough-Mezei-Verlinde]fl➡︎

Email: moritaka-AT-post.kek.jp

Appendix

Example: Matrix Product States| Ψ⟩ =∑{s}

Tr (As1⋯As6) | s1⋯s6⟩

γA

=

A

Ā

Proposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TN
3. To retain the amount of entanglement, take its square rootfi(Note: We will use the word RT surface and min. bond cut surface interchangeably)

24/36

Example: Matrix Product States

γA
foliationat τ = 0(τ: ow time)

Proposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TN

flfi

3. To retain the amount of entanglement, take its square root

25/36

Example: Matrix Product StatesAĀ
foliationat τ = 0= | Ψ⟩ (original state)
Proposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TNfi

3. To retain the amount of entanglement, take its square root

26/36

Example: Matrix Product States

γA

foliation atnite τ

Proposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TNfi

fi

3. To retain the amount of entanglement, take its square root

27/36

Example: Matrix Product States
foliation atA

nite τ

Ā

Proposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TNfi

fi

3. To retain the amount of entanglement, take its square root

28/36

Example: Matrix Product States

γA

foliation atnite τ

Proposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TNfi

fi

3. To retain the amount of entanglement, take its square root

29/36

Example: Matrix Product States
foliation at max τγAProposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TNfi

3. To retain the amount of entanglement, take its square root

30/36

Example: Matrix Product States
foliation at max τA ĀσProposal: Holographic entanglement distillation [Manabe-Matsueda-TM wip]
1. Push the boundaries for each A and Ā towards the RT (min bond cut)surface (cf. surface/state correspondence [Miyaji-Takayanagi])
2. De ne a new state on a pushed boundary (“foliation”) by removing theunreached part of TN and taking an inner product with the original TN
3. To retain the amount of entanglement,firemove singular value matrix σA at γA

31/36

Example: Matrix Product States
foliation at max τ

A Ā

=
χ

∑
i

=

σi | i⟩A | i⟩Ā =

=

σ

σ

=

=

( mixed canonical form)

When ES is like a plateau,
σ can be considered as
σnnearly uniform by taking
χ = n0, which well approximate
32/36
n
n0
the original stateExample: Matrix Product States
foliation at max τ

A Ā
σ

∑
i

σi | i⟩A | i⟩Ā ===( mixed canonical form)
If the ES is like the left gure,σ cannot be uniform althoughσnχ = n0 gives a good approx.

σ
n0

fi

χ

=

=

=

n

(We try to approx. the state bygeometry not with each tensors
as much as possible)

33/36

Example: Matrix Product States
foliation at max τ

A Ā

1
dA

=

=

=

=

Flat ES (leading order in holography)~ iid [Bao et al. ʼ19]

1
dA

χ

=

( mixed canonical form)

1| i⟩A | i⟩Ā
=
dA ∑
i

Maximally entangled state is distilled!
(χ∼e

SA

for exact-MPS state, e.g. VBS)

34/36

Example: Matrix Product States
Another candidate for HED:

foliation at max τ1σ− 12 +sdA

σ− 12 −s

=

=

=

σ

=

=

χ

1

1
| i⟩A | i⟩Ā
=
dA ∑
i

dA

✓This is nothing but a distillation by bulk modular ow✓The amount of entanglement is not preserved; it maximizes EE✓This distillation is consistent with iid limit (inverse needs ∞-ly many copies) 35/36
fl[Quintino, et al. ʼ19][Yoshida, et al. ʼ21]

Example 2: Random MERA

• The distribution of the real part of coe cients (wave function)in | Ψ⟩ (original state) and | Ψ(τfin)⟩ (distilled one) for ℋA and ℋĀ :(larger: red, lower: blue, almost zero: white)

χ=2| Ψ(τfin)⟩

| Ψ⟩
ffi

(Images are coarse-grained by 48 48 for visualization)

36/36

Example 2: Random MERA

• The distribution of the real part of coe cients (wave function)in | Ψ⟩ (original state) and | Ψ(τfin)⟩ (distilled one) for ℋA and ℋĀ :(larger: red, lower: blue, almost zero: white)χ=4| Ψ⟩ (too scattered)| Ψ(τfin)⟩

ffi(Images are coarse-grained by 48 48 for visualization)

37/36

Example 2: Random MERA

• The distribution of the real part of coe cients (wave function)in | Ψ⟩ (original state) and | Ψ(τfin)⟩ (distilled one) for ℋA and ℋĀ :(larger: red, lower: blue, almost zero: white)

χ=6| Ψ(τfin)⟩
ffi(Images are coarse-grained by 48 48 for visualization)

38/36

Example 2: Random MERA

• The distribution of the real part of coe cients (wave function)in | Ψ⟩ (original state) and | Ψ(τfin)⟩ (distilled one) for ℋA and ℋĀ :(larger: red, lower: blue, almost zero: white)

χ=8| Ψ(τfin)⟩
ffi(Images are coarse-grained by 48 48 for visualization)

39/36

Example 2: Random MERA

• The distribution of the real part of coe cients (wave function)in | Ψ⟩ (original state) and | Ψ(τfin)⟩ (distilled one) for ℋA and ℋĀ :(larger: red, lower: blue, almost zero: white)χ = 20| Ψ(τfin)⟩
ffi(Images are coarse-grained by 48 48 for visualization)

40/36

Beyond MPS: MERA with pure BH analogy•

Whatʼs important here: RT surface (min bond cut
surface) in TN is NOT unique

Why does this happen and can we see this in the AdS/CFT?
Identifying this phenomenon must be extremely importantas Σ̂ includes all the information about singular values

U

Σ̂

V

✓ The enclosed region by each boundary and each RT surface is isometric; beyond that(even in EW) is NOT isometric (in the direction from one boundary to the other)✓ Σ̂ region necessarily contains the top tensor✓ By taking the in nite volume and continuum limit, two RT surfaces should get closer
and each slope at boundary should becomes orthogonal; But the each RT surface isbounded by the location of the top tensor✓ By smoothly changing the subregion, other possible candidates fail to give minimum

fi

41/36

Beyond MPS: MERA with pure BH analogy• This picture quite resembles to the pure
BH. An RT surface can continuously movefrom one side to the other without any
obstructions (because the state is pure)• TN suggests Σ̂ region can exists at theAΣ̂Dashed: causal cone for A

origin of the bulk even for pure AdS as apoint-like region.

42/36

Beyond MPS: MERA with pure BH analogy
O(ϵ)

A

Σ̂

Σ̂

Σ̂

O(ϵ)
Dashed: causal cone for A

Pythonʼs lunch (region sandwiched byextremal surfaces) or entanglement
shadow appear due to discretization
This is very close to EWCS=EoP setup.
The important di erences are:・This exists even in the pure state・If you take su ciently small
subregion, the RT surface is get
disconnected for minimization

fi

ff

ffi

condition whereas Σ̂ remains nite

cont. limit

Σ̂

r=0
43/36

Entanglement distillation (concentration)= [cos θ | 00⟩ + sin θ | 11⟩]
Suppose we want to extract maximally entangled pairs from the following state: | Ψ⟩⊗n
AB

⊗n
AB

Expanding terms: | 00⋯00⟩, ⋯, | 11⋯11⟩ n+1 di erent coe cients. Regard them as n+1 di erent orthogonal states.
nWithin each state, there arebasis.
(k)(k)
n

The probability to extract k-th density matrix (via projective measurement | ϕk⟩ =∑i=1| i⟩ | i′⟩): pk =ncos2(n−k) θ sin2k θ(k)Then the averaged # of EPRs (ebits) are
n
n
n
n
2
2
p log
=
exp log+ (n − k)log cos θ + k log sin θ log
∑ k
(k) ∑
[ (k)
]
(k)
k=0
k=0
n

(Stirlingʼs formula log n! ∼ n log n − n ⇒ log=nnn∼ n log n − k log k − (n − k)log(n − k) = k log + (n − k)log
)
(k)
k
n−k

n 2
n
n
exp k log
sin θ + (n − k)log
cos2 θ log∑[(k)(n − k
)]
(k)
n

k=0

The saddle point approx. for the blue part: k = n sin2 θ∼ − n cos2 θ log cos2 θ − n sin2 θ log sin2 θ = nSA

ff

ffi

ff

￼

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Entanglement distillation in holography• In contrast, holography (or tensor network), we only have a single
state. Then, why can one argue distillation?
First, the distillation or modular ow is not state-independent. (TTbar
might o er a state-independent (but Hamiltonian-dependent)
construction.)
Second, it has been argued that a holographic theory has a atentanglement spectrum, i.e. ∂nSn = 0, (to leading order). This isequivalent to preparing iid. [Bao, et al.]
(In contrast, when we consider back reaction at ES is false because
gravitational Renyi entropy is a ected by backreacting cosmic brane.)

fl

fl

ff

fl

fl

ff

ff

Finally, holographic toy models like HaPPY code, random TN have at
ES [Dong-Harlow-Marolf]. Thus it is convincing that they o er acomplete distillation. In general, we expect quasi randomness or quasi
perfectness (~approx. QEC); such cases might result decrease in the
success prob. for distillation.
45/36

Beyond MPS: MERA with pure BH analogy• So far this works for MPS,σAor more generally the state of the type | ψ⟩ = (UA ⊗ VĀ)A

• Q. Can we extend our analysis for MERA?
A. At least we can make a guess!

Uσσ | αα⟩A Ā
∑ α
α

Ā

V

• The domain of dependence (≠ entanglement wedge) for eachregion corresponds to U and V for MERA

σ
U

V

⟺

σ
U

V
46/36

Beyond MPS: MERA with pure BH analogy• The region σ can appear because we take D(A), D(Ā), not EW• RT surface is not unique in TN;σEven within the pure state 2 RT surfaces
seemingly does not match

UV• But by smoothly changing the subregion, other possible
candidates fail to give minimum

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Beyond MPS: MERA with pure BH analogy• This picture quite resembles to the pure BH. A
An RT surface can continuously move fromone side to the other without any
obstructions (because the state is pure)• But now, the pure BH region is very imp; it
accounts for σ. Beyond that region, theσDashed: causal cone for A

channel from the boundary is no moreisometry (unless tensors have special
properties like perfect, dual-unitary, etc.)• TN suggests σ region can exists at the originof the bulk even for pure AdS (but maybe
just because of sub-AdS breakdown)

48/36

Beyond MPS: MERA with pure BH analogy• This is special to TN. Since the angle between
the minimal bond cut surface and the boundary
near the endpoint of the subregion is always
bounded by one unit of the isometry. It meansAσthe every RT surface sharing a common
endpoint looks locally the same near the point.

Dashed: causal cone for A

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Beyond MPS: MERA with pure BH analogy
A• Global coordinates?
Coordinate trf (for a parameter ∀α ∈ ℝ: r = 0 ⇔ z 2 − t 2 = α 2, x = 0):
z2 + x2 − t2 + α2
1 + r cos τ =;2αzt
1 + r 2 sin τ = ;zz2 + x2 − t2 − α2−r cos θ =;2αzxr sin θ = ;z

2σGeodesic in Poincare coordinates: z 2 + (x − l0)2 = (l + l0)2 where A : [−l, l + 2l0]
ϵ = 0 makes the all the slopes of those geodesics same but at the same time theslope becomes orthogonal to the boundary. Furthermore, for a nite ϵ, the slopes are

Dashed: causal cone for A

di erent from each other. This is di erent from TN.
But isnʼt this natural from RG? Di erent lattice regularization leads di erent theory
up to correction vanishing as ϵ → 0.

Continuum limit→Then, the sigma region becomes thinner and thinner.

O(ϵ)

O(ϵ)

σ

σ

σ = causal cone − RT region

ff

fi

ff

ff

ff

50/36

Path integral approach to distillation in MERA

• As we have discussed, modular ow works for acertain circumstance. In such a case, the (openboundary) MERA can be thought as a state preparedby Euclidean path integral on a semi-disc.

• Then, the modular ow=HED• However, it is a bit di erent fromordinary BCFT as the boundary

| 0⟩
| 0⟩

is (usually) not conformally invariant.

| 0⟩
| 0⟩
| 0⟩

| 0⟩

| 0⟩

The boundary state is given by many products ofancillae (thus spatially separable).

fl

ff

fl

Path integral optimization-> Liouville action

51/36

Random MERA: Renyi-2 entropy• In general, Renyi-2 entropy ofa TN state (blue) is less than

Renyi-2 entropy
S2 = − log TrρA2

the expected result from RT
formula (green)• In the large bond dim limit,
they are expected to coincide• The Renyi-2 entropy ofholographically distilled state
on γA (orange) is close to theoriginal one (blue) probably

Bond dim.

due to doubled singular value
52/36

Random MERA: Renyi-2 entropy• In general, Renyi-2 entropy of S /Sa TN state (blue) is less than

2

hol
2

the expected result from RT
formula (green)• In the large bond dim limit,
they are expected to coincide• The Renyi-2 entropy ofholographically distilled state
on γA (orange) is close to theoriginal one (blue) probably

Bond dim.

due to doubled singular value
53/36