Avalanche of Entanglement Ralf Schtzhold, Fakultt fr Physik 4. 7. - - PowerPoint PPT Presentation

Avalanche of Entanglement

Ralf Schützhold, Fakultät für Physik

• 4. 7. 2017

Motivation

t r |0> THawking

Linear fields: Gaussian (squeezed) states → pairs of particles → bi-partite entanglement M. Hotta, R.S., W.G. Unruh, Phys. Rev. D 91, 124060 (2015) E.g., Hawking radiation S. W. Hawking, Nature 248, 30 (1974); Comm. Math. Phys. 43, 199 (1975) THawking = 1 8πM c3 GNkB Tearing apart of quantum vacuum fluctuations But: trans-Plankian problem, information puzzle etc. Non-linear interactions (“scrambling”) → multi-partite entanglement

Fakultät für Physik

• 4. 7. 2017

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Entanglement for Spins (QuBits)

Invariant under local unitary operations & non-increasing for local decoherence/dissipation etc. bi-partite entanglement → Bell states concurrence for arbitrary mixed states ˆ ̺ of two spins |Bell = |↑↑ + |↓↓ √ 2 , C2[ˆ ̺] = f

• Eigenvalues
• ˆ

̺ ˆ R ˆ ̺∗ ˆ R

• ˆ

̺

• convex roof construction: minimization over all decompostitions into pure states!

tri-partite entanglement → GHZ states three-tangle τ3 of three spins in pure state |GHZ3 = |↑↑↑ + |↓↓↓ √ 2 W-states |W3 = (|↑↓↓ + |↓↑↓ + |↓↓↑)/ √ 3 ??? quadri-partite entanglement → GHZ4 states four-tangle(s) τ4 of four spins in pure state entanglement entropy ↔ entanglement between two sub-systems of pure state |ψAB S = −Tr {ˆ ̺A ln ˆ ̺A} with ˆ ̺A = TrB {|ψAB ψAB|}

Fakultät für Physik

• 4. 7. 2017

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Quantum Ising Model in Transverse Field

Hamiltonian ˆ H = −J

N

• i=1

ˆ σz

i ˆ

σz

i+1 − N

• i=1

ˆ σx

i

J = 0: paramagnetic state |→→→ . . . → no entanglement J → ∞: ferromagnetic state |↑↑↑ . . . + |↓↓↓ . . . √ 2 → N-partite entanglement (GHZ type) J = 1: critical point (phase transition) → entanglement entropy S ∼ ln N (violation of “area” law. . . ) cannot be explained by bi-partite entanglement C2 alone → multi-partite entanglement! 0.5 1 1.5 2 J 0.05 0.1 0.15 0.2 0.25 C2(1) Jmax Cmax

Fakultät für Physik

• 4. 7. 2017

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Avalanche of Entanglement

0.5 1 1.5 2 J 0.05 0.1 0.15 0.2 0.25 Tangle 2 3 4

Reduced density matrices ˆ ̺i = Trlattice\{i}{|Ψ Ψ|}, ˆ ̺ij = Trlattice\{ij}{|Ψ Ψ|}, etc. Diagonalization (exact) ˆ ̺ij... =

α pα

• Ψα

ij...

Ψα

ij...

• First two eigenvalues are dominant

ˆ ̺ij... ≈ 2

α=1 pα

• Ψα

ij...

Ψα

ij...

• → rank-two matrices

→ calculation of tangles first bi-partite C2 then tri-partite √τ3 later quadri-partite τ4 . . . finally N-partite → avalanche of entanglement

Fakultät für Physik

• 4. 7. 2017

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Entanglement → Correlations

a b

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 √τ3 | |ˆ ρcorr

3

| |1

Correlations, e.g., ˆ σz

i ˆ

σz

j corr = ˆ

σz

i ˆ

σz

j − ˆ

σz

i ˆ

σz

j

Correlated density matrices ˆ ̺corr

ij

= ˆ ̺ij − ˆ ̺i ˆ ̺j etc. Upper bound (spectral norm) ˆ σz

i ˆ

σz

j corr = Tr{ˆ

σz

i ˆ

σz

j ˆ

̺corr

ij

} ≤ ||ˆ ̺corr

ij

||1 Concurrence for two spins C2[ˆ ̺ij] ≤ ||ˆ ̺corr

ij

||1 → entanglement ↔ correlations Question: three (or more) spins? → three-tangle

• τ3[ˆ

̺ijk] as approximate lower bound for maximum correlation ||ˆ ̺corr

ijk

||1 Questions: outlier states? four spins? . . .

Fakultät für Physik

• 4. 7. 2017

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Hierarchy of Correlations 0.5 1 1.5 J 0.5 1 1.5 2

2 3 4

||ˆ ρcorr

2,3,4||1 Small J: hierarchy of correlations two-point ≫ three-point ≫ four-point

||ˆ ̺corr

ij

||1 ≫ ||ˆ ̺corr

ijk

||1 ≫ ||ˆ ̺corr

ijkl

||1 Violation of hierarchy at J ≈ 0.7 → approximation schemes σx

i σx j σx k σx l =

σx

i σx j σx k σx l +

σx

i σx j corrσx k σx l + · · · +

σx

i σx j σx k corrσx l + · · · +

σx

i σx j σx k σx l corr Fakultät für Physik

• 4. 7. 2017

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Bose-Hubbard Model

Hamiltonian ˆ H = −J

N

• i=1
• ˆ

b†

i ˆ

bi+1 + ˆ b†

i+1ˆ

bi

• + 1

2

N

• i=1

ˆ b†

i ˆ

b†

i ˆ

bi ˆ bi 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

J ||ˆ ρcorr

2,3,4||1

Small J: paramagnetic → Mott insulator Large J: ferromagnetic → superfluid Small J: hierarchy of correlations two-point ≫ three-point ≫ four-point ||ˆ ̺corr

ij

||1 ≫ ||ˆ ̺corr

ijk

||1 ≫ ||ˆ ̺corr

ijkl

||1 Violation of hierarchy at J ≈ 0.16 well before the critical point (here Jcrit ≈ 0.3)

Fakultät für Physik

• 4. 7. 2017

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Summary

K.V. Krutitsky, A. Osterloh, R.S., Nature Scientific Reports 7, 3634 (2017) Motivation: bi-partite entanglement ↔ pairs (partners) multi-partite entanglement ↔ ???

t r |0> THawking

Ising model: avalanche of entanglement → correlations Hubbard model

0.5 1 1.5 2 J 0.05 0.1 0.15 0.2 0.25 Tangle 2 3 4

a b

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 √τ3 | |ˆ ρcorr

3

| |1

0.5 1 1.5 J 0.5 1 1.5 2 2 3 4 ||ˆ ρcorr

2,3,4||1

2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 J ||ˆ ρcorr

2,3,4||1

SFB TR12

Fakultät für Physik

• 4. 7. 2017

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Accuracy of Approximation

0.5 1 1.5 2 J 0.05 0.1 0.15 0.2 0.25

C: exact value C: approx.1) C: approx. 2) 0.5 1 1.5 2 J 0.005 0.01 0.015 0.02 Cexc-C

(1)

C

(2)-Cexc

approximation 1) ˆ ̺ij ≈ p1

• Ψ1

ij

Ψ1

ij

• + (1 − p1)
• Ψ2

ij

Ψ2

ij

• approximation 2)

ˆ ̺ij ≈ p1

• Ψ1

ij

Ψ1

ij

• + p2
• Ψ2

ij

Ψ2

ij

• p1 + p2

note that p3 < 0.5 %

Fakultät für Physik

• 4. 7. 2017

www.uni-due.de