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 Hawking 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) � c 3 1 T Hawking = 8 π M G N k B Tearing apart of quantum vacuum fluctuations |0> t But: trans-Plankian problem, information puzzle etc. r Non-linear interactions (“scrambling”) → multi-partite entanglement Fakultät für Physik 4. 7. 2017 www.uni-due.de

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 � = |↑↑� + |↓↓� � ��� �� ̺ ˆ ̺ ∗ ˆ � √ , C 2 [ˆ ̺ ] = f Eigenvalues ˆ R ˆ R ˆ ̺ 2 convex roof construction: minimization over all decompostitions into pure states! tri-partite entanglement → GHZ states three-tangle τ 3 of three spins in pure state | GHZ � 3 = |↑↑↑� + |↓↓↓� √ 2 √ W-states | W � 3 = ( |↑↓↓� + |↓↑↓� + |↓↓↑� ) / 3 ??? quadri-partite entanglement → GHZ 4 states four-tangle(s) τ 4 of four spins in pure state entanglement entropy ↔ entanglement between two sub-systems of pure state | ψ AB � S = − Tr { ˆ ̺ A ln ˆ ̺ A } with ̺ A = Tr B {| ψ AB � � ψ AB |} ˆ Fakultät für Physik 4. 7. 2017 www.uni-due.de

Quantum Ising Model in Transverse Field Hamiltonian N N C max ˆ � σ z σ z � σ x H = − J ˆ i ˆ i + 1 − ˆ i 0.25 i = 1 i = 1 0.2 J = 0: paramagnetic state |→→→ . . . � C 2 (1) → no entanglement 0.15 J → ∞ : ferromagnetic state 0.1 |↑↑↑ . . . � + |↓↓↓ . . . � √ 0.05 2 0 → N -partite entanglement (GHZ type) J max 0 0.5 1 1.5 2 J J = 1: critical point (phase transition) → entanglement entropy S ∼ ln N (violation of “area” law. . . ) cannot be explained by bi-partite entanglement C 2 alone → multi-partite entanglement! Fakultät für Physik 4. 7. 2017 www.uni-due.de

Avalanche of Entanglement Reduced density matrices ˆ ̺ i = Tr lattice \{ i } {| Ψ � � Ψ |} , ̺ ij = Tr lattice \{ ij } {| Ψ � � Ψ |} , etc. ˆ Diagonalization (exact) � � � � ̺ ij ... = � ˆ α p α � Ψ α Ψ α � � ij ... ij ... � 2 First two eigenvalues are dominant 0.25 � � � � ̺ ij ... ≈ � 2 ˆ � Ψ α Ψ α α = 1 p α � � ij ... ij ... � 0.2 → rank-two matrices Tangle → calculation of tangles 0.15 first bi-partite C 2 then tri-partite √ τ 3 3 0.1 later quadri-partite τ 4 0.05 4 . . . finally N -partite 0 0 1 2 0.5 1.5 → avalanche of entanglement J Fakultät für Physik 4. 7. 2017 www.uni-due.de

Entanglement → Correlations 1.0 Correlations, e.g., j � corr = � ˆ σ z σ z σ z σ z σ z σ z � ˆ i ˆ i ˆ j � − � ˆ i �� ˆ j � 0.8 b Correlated density matrices 0.6 ̺ corr √ τ 3 ˆ = ˆ ̺ ij − ˆ ̺ i ˆ ̺ j etc. ij 0.4 a Upper bound (spectral norm) j � corr = Tr { ˆ σ z σ z σ z σ z ̺ corr ̺ corr � ˆ i ˆ i ˆ j ˆ } ≤ || ˆ || 1 ij ij 0.2 Concurrence for two spins 0.0 ̺ corr C 2 [ˆ ̺ ij ] ≤ || ˆ || 1 ij 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 → entanglement ↔ correlations ρ corr | | ˆ | | 1 3 Question: three (or more) spins? � ̺ corr → three-tangle τ 3 [ˆ ̺ ijk ] as approximate lower bound for maximum correlation || ˆ || 1 ijk Questions: outlier states? four spins? . . . Fakultät für Physik 4. 7. 2017 www.uni-due.de

Hierarchy of Correlations 2 Small J : hierarchy of correlations 4 two-point ≫ three-point ≫ four-point ̺ corr ̺ corr ̺ corr || ˆ || 1 ≫ || ˆ || 1 ≫ || ˆ || 1 1.5 ij ijk ijkl Violation of hierarchy at J ≈ 0 . 7 → approximation schemes 2 , 3 , 4 || 1 1 � σ x i σ x j σ x k σ x l � = ρ corr 2 � σ x i �� σ x j �� σ x k �� σ x || ˆ l � + 0.5 � σ x i σ x j � corr � σ x k �� σ x l � + · · · + � σ x i σ x j σ x k � corr � σ x l � + · · · + 3 � σ x i σ x j σ x k σ x l � corr 0 0 0.5 1 1.5 J Fakultät für Physik 4. 7. 2017 www.uni-due.de

Bose-Hubbard Model Hamiltonian N N + 1 � � ˆ � b † ˆ i ˆ b i + 1 + ˆ b † i + 1 ˆ � b † ˆ i ˆ b † i ˆ b i ˆ H = − J b i b i 2 i = 1 i = 1 2.0 Small J : paramagnetic → Mott insulator Large J : ferromagnetic → superfluid 1.5 4 2 , 3 , 4 || 1 Small J : hierarchy of correlations ρ corr 1.0 two-point ≫ three-point ≫ four-point || ˆ 3 ̺ corr ̺ corr ̺ corr 2 || ˆ || 1 ≫ || ˆ || 1 ≫ || ˆ || 1 ij ijk ijkl 0.5 Violation of hierarchy at J ≈ 0 . 16 well before the critical point 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (here J crit ≈ 0 . 3) J Fakultät für Physik 4. 7. 2017 www.uni-due.de

Summary T Hawking K.V. Krutitsky, A. Osterloh, R.S., Nature Scientific Reports 7 , 3634 (2017) Motivation: bi-partite entanglement ↔ pairs (partners) |0> multi-partite entanglement ↔ ??? t r Ising model: avalanche of entanglement → correlations Hubbard model 2 1.0 2.0 4 2 0.25 0.8 1.5 1.5 4 b 0.2 0.6 2 , 3 , 4 || 1 √ τ 3 Tangle 0.15 2 , 3 , 4 || 1 ρ corr 1.0 1 0.4 a ρ corr || ˆ 2 3 3 0.1 || ˆ 2 0.5 0.2 0.5 0.05 4 0.0 3 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.5 1 1.5 2 0 0.5 1 1.5 ρ corr | | ˆ | 1 | J J 3 J SFB TR12 Fakultät für Physik 4. 7. 2017 www.uni-due.de

Accuracy of Approximation C: exact value approximation 1) 0.02 (1) C exc -C C: approx.1) 0.25 C: approx. 2) (2) -C exc 0.015 C � � � � � � � � 0.01 � Ψ 1 Ψ 1 � Ψ 2 Ψ 2 ˆ ̺ ij ≈ p 1 � + ( 1 − p 1 ) � � � � ij ij ij ij � 0.2 0.005 0 0 0.5 1 1.5 2 approximation 2) 0.15 J � � � � � � � � � Ψ 1 Ψ 1 � Ψ 2 Ψ 2 p 1 � + p 2 0.1 � � � � ij ij ij ij � ̺ ij ≈ ˆ p 1 + p 2 0.05 note that p 3 < 0 . 5 % 0 0 0.5 1 1.5 2 J Fakultät für Physik 4. 7. 2017 www.uni-due.de