Systems and Symmetry Breaking Koji Umemoto (YITP) Based on Arpan - - PowerPoint PPT Presentation

Koji Umemoto (YITP)

Entanglement of Purification in Many Body Systems and Symmetry Breaking

QIST 2019 June 6th, 2019

Based on Arpan Bhattacharyya (YITP), Alexander Jahn (Freie U.) Tadashi Takayanagi (YITP) and KU [1902.02369]

𝐹𝑋 = 𝐹𝑄 conjecture

2

Takayanagi-KU ’17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ’17

𝐹𝑋 𝜍𝐵𝐶 ≔ min

Σ𝐵𝐶

Area(Σ𝐵𝐶) 4𝐻𝑂

Σ𝐵𝐶

min

𝐵 𝐶

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Entanglement wedge cross section

𝐹𝑋 = 𝐹𝑄 conjecture

3

Takayanagi-KU ’17, Nguyen-Devakul-Halbasch-Zaletel-Swingle ’17

𝐹𝑋 𝜍𝐵𝐶

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

𝐹𝑄 𝜍𝐵𝐶 ≔ min

𝜔 𝐵𝐵′𝐶𝐶′𝑇𝐵𝐵′

Reflected entropy (2𝐹𝑋)

？

=

𝑇𝑆 𝜍𝐵𝐶 ≔ 𝑇 𝐵𝐵∗

√𝜍𝐵𝐶

Entanglement of purification

Logarithmic negativity (

3 2 𝐹𝑋)

ℰ𝑂 𝜍𝐵𝐶 ≔ log 𝜍𝐵𝐶

𝑈𝐵 1

Odd entanglement entropy (𝐹𝑋 + 𝑇𝐵𝐶)

𝑇𝑝 𝜍𝐵𝐶 ≔ lim

𝑜𝑝:𝑝𝑒𝑒→1

Tr(𝜍𝐵𝐶

𝑈𝐵 𝑜𝑝 − 1]

1 − 𝑜𝑝

Dutta and Faulkner ’19 Tamaoka ’18 Kudler-Flam and Ryu ’18

Σ𝐵𝐶

min

𝐵 𝐶

Entanglement of Purification (EoP)

4

Definition: 𝐹𝑄 𝜍𝐵𝐶 ≔ min

𝜔 𝐵𝐵′𝐶𝐶′

𝑇(𝜍𝐵𝐵′) (𝜍𝐵𝐵′ ≔ Tr𝐶𝐶′[ 𝜔 𝜔 𝐵𝐵′𝐶𝐶′])

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Entanglement of Purification (EoP)

5

Definition:

• In practice, hard to compute

𝐹𝑄 𝜍𝐵𝐶 ≔ min

𝜔 𝐵𝐵′𝐶𝐶′

𝑇(𝜍𝐵𝐵′) (𝜍𝐵𝐵′ ≔ Tr𝐶𝐶′[ 𝜔 𝜔 𝐵𝐵′𝐶𝐶′])

• Thus we still don’t know much about EoP in physical

many body systems e.g. CFTs

Related papers: Terhal-Horodecki-Leung-DiVincenzo ’02, Chen-Winter ‘12 Nguyen-Devakul-Halbasch-Zaletel-Swingle ’17, …

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

6

To compute EoP in many body systems (on a lattice) by numerically performing the minimization

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Our goal

𝐹𝑄 𝜍𝐵𝐶 ≔ min

𝜔 𝐵𝐵′𝐶𝐶′

𝑇(𝜍𝐵𝐵′)

7 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Our Targets

• 2d (massless) free scalar field theory on a lattice
• 2d transverse-field (critical) Ising model

Method: Minimal Gaussian purification ansatz Method: Full minimization without ansatz Purifications Ψ 𝐵𝐵′𝐶𝐶′ is assumed to be Gaussian with 𝐵′𝐶′ ≔ |𝐵𝐶| A theorem [Ibinson-Linden-Winter ’06] guarantees that it is sufficient to search dimℋ𝐵′ ≤ rank𝜍𝐵𝐶, dimℋ𝐶′ ≤ rank𝜍𝐵𝐶 for minimizing 𝑇𝐵𝐵′

Ground state reduced matrix 𝜍𝐵𝐶 is Gaussian

𝐵 ∪ 𝐶 𝐵′ ∪ 𝐶′ 𝐵 𝐶 rank𝜍𝐵𝐶 = 4 dimℋ𝐵′ = dimℋ𝐶′ = 4

E.g. 2 qubits

8 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Our Targets

 EoP can increase with the physical distance.  Even if 𝜍𝐵𝐶 = 𝜍𝐶𝐵, the optimal purification can break its symmetry i.e. 𝜔∗

𝐵𝐵′𝐶𝐶′ ≠ 𝜔∗ 𝐶𝐶′𝐵𝐵′.

EoP behaves similarly in both models. Common results

9 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Setup

[1+1d, vacuum, periodic boundary condition]

𝑂: 16 𝑂: total sites number

10 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Setup

𝐵 𝐶

[1+1d, vacuum, periodic boundary condition]

𝑂: 16 𝑥: 2 𝑥: size of subsystem 𝑂: total sites number

11 Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Setup

𝐵 𝐶

𝑒

[1+1d, vacuum, periodic boundary condition]

𝑂: total sites number 𝑥: size of subsystem 𝑒: distance between 𝐵 and 𝐶 𝑂: 16 𝑥: 2 𝑒: 3

Results

12

E.g. 𝑂 = 60 free scalar EoP

(Mutual information 𝐽 𝐵: 𝐶 = 𝑇𝐵 + 𝑇𝐶 − 𝑇𝐵𝐶)

Half of mutual information

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Plateau-like behavior

Results

13

EoP Log negativity

(Log negativity 𝐹𝑂 = log 𝜍𝐵𝐶

𝑈𝐵 1)

E.g. 𝑂 = 60 free scalar

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Plateau-like behavior

Results

14

(𝑥 = 4)

E.g. 𝑂 = 60 free scalar

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Results

15

(𝑥 = 1)

E.g. 𝑂 = 10 Ising

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

(periodic b.c.)

Non-monotonicity

16

EoP increases with the distance 𝑒 (For small 𝑂, at 𝑒 = 1) Ising model 𝑥 = 2 Free scalar 𝑥 = 2

(dimℋ𝐵′𝐶′ ≔ dimℋ

𝐵𝐶) Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Non-monotonicity

17

EoP increases with the distance 𝑒 (For small 𝑂, at 𝑒 = 1) Ising model 𝑥 = 2 Mutual information

(dimℋ𝐵′𝐶′ ≔ dimℋ

𝐵𝐶) Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Non-monotonicity

18

It’s so weird… Perhaps the minimization does not work well? Theorem If 𝜍𝐵𝐶 has support only on the (anti-) symmetric subspace of ℋ

𝐵𝐶 = ℋ 𝐵 ⊗ ℋ𝐶,

then 𝐹𝑄 𝐵: 𝐶 = 𝑇𝐵 = 𝑇𝐶.

• We can show this behavior analytically in some cases

Christandl-Winter ’05

We can show 𝐹𝑄(𝑒 = 1) = 𝑇𝐵 = 𝑇𝐶 by a thm. and 𝐹𝑄(𝑒 = 0) < 𝑇𝐵 by numerics

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

E.g) in 𝑂 = 4 Ising model

Z2 symmetry breaking

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The optimal purifications do not necessarily have the exchange symmetry (𝐵𝐵′ ↔ 𝐶𝐶′)

𝜍𝐵𝐶 = 𝜍𝐶𝐵 𝜔opt.

𝐵𝐵′𝐶𝐶′ ≠ 𝜔opt. 𝐶𝐶′𝐵𝐵′ But Optimal purification

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

In some cases,

Z2 symmetry breaking

20

E.g. Ising model, 𝑂 = 10, 𝑥 = 1 The optimal purifications do not necessarily have the exchange symmetry (𝐵𝐵′ ↔ 𝐶𝐶′) 𝑇max ≔ max 𝑇𝐵′, 𝑇𝐶′ , 𝑇min ≔ min 𝑇𝐵′, 𝑇𝐶′

𝑇(𝜍𝐵′

• pt.) ≠ 𝑇(𝜍𝐶′
• pt.) at 𝑒 = 1

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation

21

 Interplay between quantum entanglement and classical correlations Classical correlations: typically in separable states

𝜍𝐵𝐶 = 𝑞𝑗

𝑗

𝜍𝐵

𝑗 ⊗ 𝜍𝐶 𝑗

Suppose that total correlation, half of MI, is just a sum of them

𝐽 𝐵: 𝐶 2 ~𝐹 𝐵: 𝐶 + 𝐷(𝐵: 𝐶)

Squashed entanglement 𝐹𝑡𝑟 will be a good candidate (∵ 𝐹𝑡𝑟 ≤ 𝐽/2) Remainings (𝐷 ≔

𝐽 2 − 𝐹𝑡𝑟)

Try to understand the qualitative aspects of results

(Of course not precise)

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation

22

1) EoP coincides with 𝑇𝐵 for pure states

Claim:

𝐹𝑄 𝐵: 𝐶 ≳ 1 𝐹 𝐵: 𝐶 + 2 𝐷(𝐵: 𝐶)

When 𝐷 𝐵: 𝐶 = 0, 𝐹𝑄 𝐵: 𝐶 = 𝑇𝐵 = 𝐹 𝐵: 𝐶

∴ 𝑏 = 1

2) EoP is at least as large as 𝐽(𝐵: 𝐶) for separable states When 𝐹 𝐵: 𝐶 = 0, 𝐹𝑄 𝐵: 𝐶 ≥ 𝐽 𝐵: 𝐶 = 2𝐷(𝐵: 𝐶)

∴ 𝑐 ≥ 2

Terhal-Horodecki-Leung-DiVincenzo ’02

• Q. What is the coefficients for EoP?

𝐹𝑄 𝐵: 𝐶 ~ 𝑏 𝐹 𝐵: 𝐶 + 𝑐 𝐷(𝐵: 𝐶)

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

A qualitative interpretation

23

Classical correlation is strong Entanglement is strong

𝐹𝑄 𝐵: 𝐶 ≳ 1 𝐹 𝐵: 𝐶 + 2 𝐷(𝐵: 𝐶)

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

𝐽 𝐵: 𝐶 2 ~ 1 𝐹 𝐵: 𝐶 + 1 𝐷(𝐵: 𝐶)

A qualitative interpretation

24

A toy model explaining why 𝑎2 symmetry is broken only at 𝑒 = 1:

Focus on the nearest neighbor entanglement  EoP converts classical correlation into entanglement in the purified system

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Summary

25

• We computed entanglement of purification in 2d free scalar field

and 2d Ising model by numerically performing the minimization.

• We found that EoP can increase with the physical distance.

It is quite different from other measures such as mutual information.

• The optimal purifications are not necessarily symmetric under

exchange 𝐵𝐵′ ↔ 𝐶𝐶′ even if the original state satisfies 𝜍𝐵𝐶 = 𝜍𝐶𝐵

• Both can be interpreted as an interplay between entanglement and

classical correlation

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Appendices

Entanglement of Purification (EoP)

27

𝐹𝑄 𝜍𝐵𝐶 ≔ min

𝜔 𝐵𝐵′𝐶𝐶′

𝑇(𝜍𝐵𝐵′) (𝜍𝐵𝐵′ ≔ Tr𝐶𝐶′[ 𝜔 𝜔 𝐵𝐵′𝐶𝐶′])

Terhal-Horodecki-Leung-DiVincenzo ’02

• 𝐹𝑄 ≥ 0 and 𝐹𝑄 = 0 if and only if 𝜍𝐵𝐶 = 𝜍𝐵 ⊗ 𝜍𝐶

Definition

• It monotonically decreases under local operations

∴A measure of total correlation (not just entanglement)

• Cf. mutual information 𝐽 𝐵: 𝐶 ≔ 𝑇𝐵 + 𝑇𝐶 − 𝑇𝐵𝐶

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

EoP for free scalar field

28

• Vacuum is Gaussian state

Ψtotal 𝜚 ∝ exp − 1 2 𝜚𝑈𝑋𝜚

where 𝑋𝑜𝑜′ = 1 𝑂

4sin2(

𝜌𝑙 𝑂 ) + 𝑛2𝑏2 𝑂 𝑙=1

𝑓

2𝜌𝑗𝑙(𝑜−𝑜′) 𝑂

(small) mass ~10−4 Lattice cutoff = 1 d.o.f. on the sites

=: exp − 1 2 𝜚𝐵𝐶, 𝜚other

𝑈

𝑄 𝑅 𝑅𝑈 𝑆 𝜚𝐵𝐶, 𝜚other

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Minimal Gaussian Purification ansatz

29

Ψ

𝐵𝐵′𝐶𝐶′ Gaussian 𝜚 ∝ exp − 1

2 𝜚𝐵𝐶, 𝜚𝐵′𝐶′

𝑈

𝐾 𝐿 𝐿𝑈 𝑀 𝜚𝐵𝐶, 𝜚𝐵′𝐶′

𝐵𝐶 = |𝐵′𝐶′|

• Tr𝐵′𝐶′|Ψ𝐻⟩⟨Ψ𝐻|𝐵𝐵′𝐶𝐶′ = 𝜍𝐵𝐶 ⇒ only 𝐿 is free parameters
• Perform the minimization of 𝑇𝐵𝐵′(𝐿) over the minimal Gaussian

purification ansatz by changing the components of 𝐿

(We can further reduce the numbers of parameters by using a symmetry of EE) 𝜍𝐵𝐶 𝜚𝐵𝐶, 𝜚′𝐵𝐶 ∝ exp − 1 2 𝜚𝐵𝐶, 𝜚𝐵𝐶

𝑈

𝑄 − 1 2 𝑅𝑆−1𝑅𝑈 − 1 2 𝑅𝑆−1𝑅𝑈 − 1 2 𝑅𝑆−1𝑅𝑈 𝑄 − 1 2 𝑅𝑆−1𝑅𝑈 𝜚𝐵𝐶, 𝜚′𝐵𝐶

Minimal Gaussian Purification ansatz

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

EoP for Ising model

30

• The critical 2d Ising model
• Total vacuum state Ω total → 𝜍𝐵𝐶 → 𝜔0 𝐵𝐵′𝐶𝐶′ (any purification)

𝐼total = − 𝜏𝑗

𝑨 ⊗ 𝜏 𝑘 𝑨 − 𝜏𝑗 𝑦 𝑗 <𝑗,𝑘>

• All possible purifications = All isometry maps (embedding + unitary)

𝜍𝐵𝐶 → 𝐽𝐵𝐶 ⊗ 𝑊

𝐵′𝐶′ 𝑗𝑡𝑝 𝜔0 𝐵𝐵′𝐶𝐶′

• Minimize 𝑇𝐵𝐵′(𝑊

𝐵′𝐶′ 𝑗𝑡𝑝 ) without any ansatz

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

EoP for Ising model

31

We do not need to consider arbitrary large Hilbert space ℋ𝐵′𝐶′

Ibinson-Linden-Winter ’06

𝐵 ∪ 𝐶 𝐵′ ∪ 𝐶′ 𝐵 𝐶

Theorem In a finite dimensional case, the minimum of 𝑇𝐵𝐵′ can be achieved by

dimℋ𝐵′ ≤ rank𝜍𝐵𝐶 and dimℋ𝐶′ ≤ rank𝜍𝐵𝐶 rank𝜍𝐵𝐶 = 4 dimℋ𝐵′ = dimℋ𝐶′ = 4

E.g. 2 qubits

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Z2 symmetry breaking

32

The Z2 symmetry breaking and quantum phase transition 𝐼total = − 𝜏𝑗

𝑨 ⊗ 𝜏 𝑘 𝑨 − ℎ 𝜏𝑗 𝑦 𝑗 <𝑗,𝑘>

(𝑂 = ∞, thermal ground state Ω = lim

𝛾→∞𝑓−𝛾𝐼/𝑎(𝛾) )

(𝑥 = 1, 𝑒 = 1)

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto

Implications to holography

33

EWCS MI/2

• Reflection symmetry could also break in excited states or 𝑃(1) correction

𝐵 𝐶 𝐵 𝐶

𝑒

• We know that EWCS behaves differently from MI around the transition point

Entanglement of Purification in Many Body Systems and Symmetry Breaking - Koji Umemoto