Monogamy of entanglement and mean-field ansatz for spin lattices - PowerPoint PPT Presentation

Monogamy of entanglement and mean-field ansatz for spin lattices Ralf Sch¨ utzhold Fakult¨ at f¨ ur Physik Universit¨ at Duisburg-Essen Monogamy of entanglement and mean-field ansatz for spin lattices – p.1/11

Spin Lattice Regular lattice of 1 / 2 -spins (qubits) µ ν σ x σ y σ z Pauli matrices ˆ σ µ = (ˆ µ , ˆ µ , ˆ µ ) Coordination number Z (neighbours) H = 1 ˆ � � σ µ · J · ˆ ˆ B · ˆ σ ν + σ µ Z <µ,ν> µ In general very complicated → mean-field ansatz � | Ψ mf � = | ψ µ � , | Ψ mf � = |↑� |↑� |↑� |↑� . . . e . g ., µ Variational mean-field energy per lattice site � ˆ H � mf = 1 2 � ˆ σ µ � · J · � ˆ σ ν � + B · � ˆ σ µ � N Neglect of entanglement!? Monogamy of entanglement and mean-field ansatz for spin lattices – p.2/11

Entanglement for Pure States Consider two spins (qubits) µ and ν : not entangled iff |↑� µ + |↓� µ |↑� ν + |↓� ν √ √ | Ψ µν � = | ψ µ � | ψ ν � , e . g ., 2 2 Maximum entanglement (Bell state) |↑� µ |↑� µ + |↓� ν |↓� ν √ | Ψ µν � = = | Bell � µν 2 General state with concurrence C with 0 ≤ C ≤ 1 √ √ C ˆ U µ ˆ | Ψ µν � = 1 − C | ψ µ � | ψ ν � + U ν | Bell � µν W.K. Wooters, Phys. Rev. Lett. 80 , 2245 (1998). Note: qutrits or three qubits are more complicated | Ψ GHZ � = |↑� |↑� |↑� + |↓� |↓� |↓� √ 2 Monogamy of entanglement and mean-field ansatz for spin lattices – p.3/11

Mixed State of Two Spins µ ν General decomposition (not unique) � � Ψ I Ψ I � � � � ρ <µν> = ˆ p I µν µν � I Problem: consider all possible decompositions � � Ψ I Ψ I �� � � � �� ent(ˆ ρ <µν> ) = min p I , | Ψ I p I ent µν � µν µν I Problem solved for concurrence C (ˆ ρ <µν> ) (2 qubits) W.K. Wooters, Phys. Rev. Lett. 80 , 2245 (1998); A. Uhlmann, Phys. Rev. A 62 , 022307 (2000). Symmetric decomposition for general mixed states 4 � � : C (ˆ � Ψ I Ψ I � Ψ I Ψ I � � �� � � � � � �� ρ <µν> = ˆ p I ρ <µν> ) = C ∀ I µν µν µν µν I =1 Again: for 2 qubits only... Monogamy of entanglement and mean-field ansatz for spin lattices – p.4/11

Monogamy of Entanglement Upper bound for concurrence of qubit-pairs T Hawking � C 2 (ˆ ρ µ ) ≥ τ 1 (ˆ ρ µ ) = 4 det(ˆ ρ <µν> ) ν ρ µ ) ≤ 1 with one-tangle τ 1 (ˆ |0> V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61 , 052306 (2000); t T.J. Osborne, F. Verstraete, Phys. Rev. Lett. 96 , 220503 (2006). r Lattice isotropy � � τ 1 1 C (ˆ ρ <µν> ) ≤ Z ≤ µ ν Z Entanglement decreases for large Z Expectation: mean-field ansatz becomes better Monogamy of entanglement and mean-field ansatz for spin lattices – p.5/11

Ground State Energy µ ν H = 1 ˆ � � σ µ · J · ˆ ˆ B · ˆ σ ν + σ µ Z <µ,ν> µ 4 � with � � � Ψ I Ψ I � � � Insert ˆ ρ <µν> = p I µν µν I =1 √ √ C ˆ µ ˆ � Ψ I � ψ I � ψ I U I U I � � � � � � 1 − C ν | Bell � µν = + µν µ ν → estimate for ground-state energy 4 � ˆ √ H � p I � σ I σ I σ I σ I � � �� � ˆ µ � · J · � ˆ ν � + B · � ˆ µ � + � ˆ ν � + O ( = C ) N 2 I =1 � ˆ σ I ψ I � � � ψ I � � with � ˆ µ � = → mean-field ansatz σ µ µ µ Ergo: Z ≫ 1 → C ≪ 1 → mean-field behaviour Monogamy of entanglement and mean-field ansatz for spin lattices – p.6/11

Intermediate Summary Concurrence C measures deviation from mean-field � ˆ − � ˆ √ H � mf H � exact ≤ ( || J || + 2 || B || ) C + O ( C ) N N → C = 0 only if mean-field yields exact ground state √ Large Z ≫ 1 → small C ≤ 1 / Z ≪ 1 → mean-field becomes better for large Z ≫ 1 Note: different from quantum de Finetti theorem (full permutational invariance vs lattice symmetry) E.g., Lipkin-Meshkov-Glick model H = 1 ˆ � � σ µ · J · ˆ B · ˆ ˆ σ ν + σ µ N µ,ν µ → large spin Σ = � µ ˆ σ µ /N Monogamy of entanglement and mean-field ansatz for spin lattices – p.7/11

Example: Ising Model µ ν H = − J ˆ � � σ x σ x σ z ˆ µ ˆ ν − B ˆ µ Z <µ,ν> µ Mean-field ansatz: paramagnetic for B > | J | | Ψ mf � = |↑↑↑ . . . � Estimate for exact on-site density matrix √ C ) = |↑� �↑| + O (1 /Z 1 / 4 ) ρ µ = |↑� �↑| + O ( ˆ � → iterate monogamy argument C ≤ τ 1 /Z C ≤ O ( Z − 2 / 3 ) , ρ µ ) ≤ O ( Z − 1 / 3 ) τ 1 = 4 det(ˆ Hierarchy of correlations suggests C = O (1 /Z ) , τ 1 = O (1 /Z ) P. Navez, F. Queisser, R.S., J. Phys. A 47 ,225004 (2014). Monogamy of entanglement and mean-field ansatz for spin lattices – p.8/11

Improved Mean-Field Ansatz Idea: add a little bit of entanglement for 2 spins | Ψ µν � = N (1 + ˆ σ µ · ξ · ˆ σ ν ) |↑� µ |↑� ν Generalisation to spin lattices � � � � σ x σ x � � | Ψ � imf = N |↑� µ , exp ξ ˆ µ ˆ ν <µ,ν> µ Variational ansatz � cos(2 ℑ ξ ) � Z � ˆ H � imf = − J 2 tanh(2 ℜ ξ ) − B cosh(2 ℜ ξ ) N Energy minimum for J 4 BZ + O (1 /Z 2 ) ❀ C = O (1 /Z ) ξ min = Monogamy of entanglement and mean-field ansatz for spin lattices – p.9/11

XY-Model H = − J � 1 + γ ν + 1 − γ � ˆ � � σ x σ x σ y σ y σ z ˆ µ ˆ ˆ µ ˆ − B ˆ ν µ Z 2 2 <µ,ν> µ Scaling with anisotropy parameter γ J 4 BZ + O (1 /Z 2 ) ξ min = γ Scaling variable ζ = Z | ξ | C = 2 ζ − ζ 2 Θ(1 − ζ ) + O (1 /Z 2 ) Z → ξ min and C vanish in isotropic limit γ = 0 | Ψ imf � = | Ψ mf � = |↑↑↑ . . . � ↔ paramagnetic state is exact for B > | J | Monogamy of entanglement and mean-field ansatz for spin lattices – p.10/11

Conclusions & Outlook µ ν Concurrence C measures deviation from mean-field � ˆ − � ˆ √ H � mf H � exact ≤ ( || J || + 2 || B || ) C + O ( C ) N N • C = 0 ↔ mean-field yields exact ground state √ • monogamy: Z ≫ 1 → C ≤ 1 / Z ≪ 1 • unique mean-field ground state: C = O ( Z − 2 / 3 ) • improved mean-field ansatz: C = O (1 /Z ) (note: not rigorously proven) Outlook: bi-partite → tri-partite entanglement... A. Osterloh, R.S., arXiv:1406.0311 Monogamy of entanglement and mean-field ansatz for spin lattices – p.11/11